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*** START OF THE PROJECT GUTENBERG EBOOK 75807 ***





[Illustration: _Wide World Photos._

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


                            [Illustration]


                               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.

[Illustration: THE MEASURE OF A MAN

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 [Illustration]

 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 [Illustration]

 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 [Illustration]

 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.

[Illustration: HOW TO DRAW A FOUR-DIMENSIONAL FIGURE

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.]

[Illustration: THE TESSERACT

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.

[Illustration: In space of three dimensions we cannot make a right hand
glove and a left hand glove look the same no matter how we turn them
around. But if we turn one glove inside out it will match the other
except that the lining now appears on the outside.]

[Illustration: Our two hands cannot be turned inside out so as to look
the same in three dimensions, though they might in four 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.

[Illustration: By movement in one dimension we cannot make the lines
_AB_ and _B′A′_ coincide for if we drag _B′A′_ straight
on to _AB_ the ends will not match. But if we swing _B′A′_
around through the second dimension we bring it on _AB_ so the
letters correspond.]

[Illustration: In space of two dimensions, such as a table top, we
cannot bring these two triangles into the same position. If we drag one
straight over on to the other (movement in one dimension) they will
not fit together. If we swing one triangle around (movement in two
dimensions) they still do not fit. But if we take one triangle off the
table and turn it over (movement in the third dimension) we can then
lay it by the side of the other and they will match perfectly.]


                    A WORM’S-EYE VIEW OF THE WORLD

[Illustration]

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.

Now the sun being a big heavy body must extend its gravitational
strain for a considerable distance around and a ray of light passing
through this crumpled up space would not be able to pursue a straight
course. And, according to the eclipse observations, it does not.
Light like everything else follows “the easiest way” and this is not
always the straight and narrow path. A river takes the easiest, not the
shortest, way to the sea and this leads it through many meanderings.
Most of us, I suppose, have a mental image of Newton’s gravitation as
a sort of rope by which the sun pulls the earth into its orbit when
it is disposed to fly off on a tangent. But from Einstein’s viewpoint
we should rather think of the earth as picking its way as best it
can through a space-and-time combination that has been strained and
distorted by the power of the sun. I visualize Einstein’s solar system
as a spider web with the sun in the middle like the spider and the
planets like flies trying to get around through the tangled strands.
But it is more complicated than that for each planet has its own lesser
web of radiating influence to drag about with it wherever it goes.

Newton’s idea is simpler, but unfortunately light at least seems to
follow Einstein’s law, not Newton’s. That is why Einstein is such
a troublesome fellow. If he would confine himself to metaphysical
speculation nobody need bother about these strange notions of his. But
when he points how they can be proved and then British astronomers and
American physicists find things according to his deductions he cannot
be ignored. The man does not seem to have that decent respect for the
opinions of mankind that leads most of us to limit our logic to the
sphere of common sense. When he gets an idea in his head he follows it
wherever it leads him even though he bumps up against Euclid and Newton
and the rest of us. For instance, if you admit the second of his two
fundamental postulates, that the speed of light is constant, regardless
of the velocity of its source, you are irresistibly led--unless you
let go of his hand somewhere on the way--to the conclusion that time
is a local affair; that there is no way of telling by light signals
whether two clocks at a distance are keeping the same time, or whether
two events at different places occur simultaneously. You could not tell
this even if you could shoot a watch from one place to the other with
the speed of light, for no matter how many seconds--or years--the watch
might be on its way it would register the same time. If instead of a
watch a man could travel at that speed he would not grow old on the
way. According to Einstein no man, watch or any other material thing
can travel with the speed of light, for it would require an infinite
force to give the smallest particle such a velocity. But let us suppose
that a hollow projectile holding a man, such as Jules Verne and Wells
used on their voyages to the moon, should be sent off into space with
a velocity one twenty-thousandth less than light. If at the end of a
year the projectile should be caught like a comet by the gravitation of
some star and be swung around and sent back to the earth, the man on
stepping out of his shell would be two years older but he would find
the world two hundred years older. This would be, as Professor Langevin
suggests in _Scientia_, 1911, an interesting way to study history,
but it would be risky, not to say impossible. Still French scientists,
like Napoleon, have no place in their dictionaries for so stupid a word
as “impossible” and M. Esnault-Pelterie has figured out that a thousand
pounds of radium would be sufficient to carry a man to Venus in 35
hours if a hollow projectile could be fitted up like a rocket with the
radium in the rear sending out a rapid fire of electrons.


                         TURNING TIME BACKWARD

To loosen up our conventional ideas of the fixity of time and space we
may accept the aid of the scientific romancers. Camille Flammarion, the
famous French astronomer, wrote a fantastic little book called “Lumen”
which tells of a man who died in 1864. His soul flew straight to its
heaven which was one of the planets of Capella, the largest star in
the constellation Auriga. Here he found the benevolent inhabitants of
that sphere, who were endowed with superhuman powers of sight, watching
with great distress the bloody scenes of the French revolution of 1793,
and wondering how it would come out. To the visitor from the earth
this was an old story, to the people of Alpha Aurigae it was a present
spectacle, for the distance of the star was such that it took light 72
years to travel from the earth, so they were 72 years belated in their
observation of current events on our planet.

The spirit of the defunct Parisian, having the power of flying through
empty space at any speed he chose, found that he had thereby also
acquired control of time and could hasten, retard, stop or reverse the
course of events at will by simply varying his speed. If he remained
stationary, scenes on the earth would unfold at their normal rate and
in regular order. If he traveled away from the earth with the speed
of light everything seemed to stand still. If he traveled faster than
light he overtook the rays that had left the earth farther and farther
back in the past so he saw through them events in the reverse order.
For instance when he looked down on Waterloo he saw the battlefield
strewn with corpses and Napoleon walking toward Waterloo backward
pushing his horse by the bridle. This is how the battle looked to the
interspatial observer:

 When my sight was sufficiently habituated to the scene, I perceived
 some soldiers coming to life out of the eternal night, and by a single
 effort standing up. The dead horses revived like the dead cavaliers,
 and the latter remounted them. As soon as two or three thousand men
 had returned to life, I saw them form unconsciously in line of battle.
 The two armies took their places fronting one another, and began to
 fight desperately with a fury that one might have taken for despair.
 As the combat deepened on both sides, the soldiers came to life more
 rapidly....

 At each gap made by the cannon in the serried ranks a group
 of resuscitated dead filled up the gaps immediately. When the
 belligerents had spent the whole day in tearing one another to pieces
 with grape-shot, with cannons and bullets, with bayonets, sabers and
 swords--when the great battle was over, there was not a single person
 killed, no one was even wounded; even uniforms that before it were
 torn and in disorder were in good condition, the men were safe and
 sound, and the ranks in correct form. The two armies slowly withdrew
 from one another, as if the heat of the battle and all its fury had
 no other object than the restoration to life, amid the smoke of the
 combat, of the two hundred thousand corpses which had lain on the
 field a few hours before. What an exemplary and desirable battle it
 was!

Another literary curiosity on the same theme is “Ignis” by Comte Didier
de Chousy. This tells of certain engineers who attempted to utilize
the internal heat of the earth by running the waters of a lake into a
deep boring. The result was an explosion that blew off a piece of the
planet. But the passengers on this artificial asteroid on looking down
through their well at the earth they had left could see the lake and
city undisturbed and watch themselves at work as they were before the
place blew up. The explanation was that this fragment of the earth was
projected into space more rapidly than the speed of light and so was
catching up with the rays that had gone out before the explosion; these
rays, of course, carried the picture of earlier scenes. But Einstein
would say that this story--as we might ourselves have suspected--must
be fiction for according to his theory the speed of light is the
absolute limit of motion, the infinity of velocity, which no material
body may excel or attain. He does not, however, say anything about the
possible speed of a disembodied spirit such as Flammarion employed in
his imaginary exploration of space.


                     THE METAPHYSICS OF THE MOVIES

But from such fantasies we can see that the order in which we view
events depends upon how fast and in what direction we are moving and
that past and future may be reversed to our vision. This is easily
made apparent by means of motion pictures. If the film is reeled off
in the wrong direction the action is reversed. So we see divers rising
gracefully out of the water and landing on the spring board. Newly
hatched chickens, dismayed at the sight of this unfriendly world,
calmly tuck themselves back into their broken shells which close in
upon them. When we have come to the close of a perfect Thanksgiving
Day the obliging operator may give us an encore of the dinner reversed
by running his machine backward. Then we see pieces of turkey politely
picked out of the mouths of the diners with their forks and replaced
upon the plates. When these are passed back to the carver he puts the
slices neatly in their places and the fowl is then sent back to the
oven to be unroasted. The cook then sticks on the feathers. The hired
man carries the turkey out to the chopping block where with one swift
stroke he restores the head and the fowl runs off backwards. This
is just as correct as the ordinary order. The sequence of events is
the same. Cause and effect are linked together as firmly as before,
only they have exchanged places. A scientist knowing nothing of our
world except from watching such reversed motion pictures might deduce
from them the same consistent and logical system of natural laws that
we now have although some of them, for instance, the second law of
thermo-dynamics, would be reversed in form.

The motion-picture man has also the power to alter the speed of the
passage of time as he will by turning the crank faster or slower.
Sometimes he is quite too careless in the way he employs this
prerogative. If he is behind time on his schedule he will rush through
a lazy siesta scene in a Mexican plaza with all the fury of a Mack
Sennett farce. But this telescoping of time can be used to advantage
as when he shows us the growth of a plant, the unfolding of its flower
and the ripening of its fruit, all in fifteen minutes. On the other
hand motion may be slowed up by taking twice as many pictures a minute
as usual and projecting them at the ordinary rate. For instance, if
it is a dog jumping up to grab a piece of meat from his master’s
hand, we see the dog rise slowly from the ground and, while poised in
mid-air, eye the meat carefully to select the best point of attack,
then deliberately take it between his jaws and gradually descend.
Now notice that this is just as true a picture of the dog’s jump as
any other. The movie man has simply expanded time measurements as he
expands space measurements when he shows us a close-up. A close-up with
a face covering a sixteen-foot screen is just as true as a smaller
picture. It is what we should always see if the lens of our eyes were
a bit more convex. We look through the small end of an opera-glass and
objects seem magnified. We look through the large end and objects seem
minified. This is not an illusion. The opera-glass _does_ actually
enlarge or reduce _what we see_.

So, too, time intervals can be lengthened or shortened. Take a dose
of hashish--no, don’t--I should say, if you did take a dose you would
find that your perception of duration was prolonged. If while under the
influence of the drug you drop a book it will seem an hour getting to
the ground. De Quincey describes such experiences in his “Confessions
of an Opium Eater.” But without entering into such abnormal states we
all know by everyday experience how time flies or lags according to
the number of our sensations. Bergson’s philosophy is built upon the
distinction between the idea of duration as experienced by all of us
and the idea of time as established by the physicists for comparative
measurements.

    We live in deeds not years; in thoughts not breaths;
    In feelings not in figures on a dial.
                                           --_Festus._

For all we know an ephemeral insect that dies in a day may live a
longer life than a Galapagos turtle that exists for two centuries.

What Mark Twain said about classical music applies also to science; “It
is not so bad as it sounds.” The thing that the chemist calls “sodium
chloride” other folks call “salt”--and so does he when he is off duty.
Don’t let the scientist bluff you by his polysyllabic propensity. Just
try to see what he means by such language. Now what these new-fashioned
non-Euclidean geometricians call “the four-dimensional space-time
continuum” is essentially the same system of reference as you have used
ever since you could toddle. Minkowski did not invent it. Everybody
thinks that way unless he is an idiot. Each one of us has had to build
up his own philosophy of the universe long before we went to school,
mostly before we could talk. We had to study geometry while we were
in our cradles--worse than that we had to work out a practical system
of geometry for ourselves without the help of Euclid or anyone else.
We had to excogitate a system of relationship between the sights and
sounds and touches that came to us before we could get along in the
world. Probably we all solve this riddle of the universe in about the
same way although since there is no way of directly comparing notes we
cannot be sure about that.


                 THE EGOCENTRIC THEORY OF THE UNIVERSE

But the framework that we construct to hold everything outside of
ourselves is essentially of the following form:

You are the center of your universe. Everything and every event that
you are considering is related to you here and now. Starting from
this, your point of place and time, you imagine eight straight lines
stretching out toward infinity in eight directions as divergent
as possible. These lines--call them destinations or directions or
dimensions or coördinates as you please--consist of four opposing
pairs, right and left, up and down, forward and back, future and past.
Somewhere along or between these four dimensional lines that cross
in your brain you can find a place for anything that you need; your
pencil, the discovery of America, the sun and next Friday. You can
connect up all these things by lines which may represent changes, that
is the tracks of movements in space and time. To connect the pencil
in your hand with the discovery of America you would have to count
back 428 years on the time line and measure off on the east-west and
north-south lines whatever distance you may be from San Salvador--not
to consider the motion of the earth.

Anything that exists, that is to say, persists, is moving along the
time dimension at what appears to be a uniform rate. Of course you
can, if you like, conceive of time itself as a stream flowing through
things. Since all motion is relative, that way of looking at it is
just as “true” as the other. But it is simpler and more sensible to
think of things moving through a stationary time just as we think of
them moving through a stationary space. A material point that is at
rest, such as the dot of an _i_ on this page, (we continue to
disregard the motion of the earth) is not moving about in space but
is moving forward in time. Its track then is a straight line along
the time dimension. That is, a material point is a line in the fourth
dimension. If you move the page to the right the forward movement
of the dot of the _i_ in the time dimension is combined with
the sideways motion in a single slanting line. If you move the page
simultaneously upward, rightward and backward the track of the point is
a line combining the movement in all four dimensions. Such a track of a
point moving through space and time is called its “world-line.” It is a
continuity of one dimension. Any event is the point of intersection of
one or more such world lines and we can never observe anything except
such intersections. That is to say, everything happens somewhere and
sometime.

A picture flashed on a cinema screen has three dimensions. It is, say,
10 feet long and 6 feet high and lasts ¹⁄₁₆ of a second, but it has no
thickness. A man necessarily has four dimensions. He may measure from
24 to 72 inches in one dimension, from 8 to 18 inches in the second,
from 4 to 9 inches in the third and 70 years in the fourth.

After all, the idea of the relativity of time ought to be easier to
accept than that of space for it is in accord with experience instead
of contrary to it. We drop off to sleep and wake the next instant if we
credit our personal perceptions. Why should we believe the sun and the
clock in preference to ourselves?

Bergson bases his whole philosophy upon the distinction between
_duration_ as it is felt by the individual while he is living
through it and _time_ as it is employed by the physicist in his
calculations. The latter conception, physical time, is, as Bergson
says, a mere invention of man and virtually a fourth dimension of
space, so he concludes:

 To sum up; every demand for explanation in regard to freedom comes
 back, without our suspecting it, to the following question: “Can time
 be adequately represented by space?” To which we answer: Yes, if you
 are dealing with time flown; No, if you speak of time flowing.[1]

Past and future are alike to the physicist, differing only in
direction, like east and west. But to the living person they are
altogether different things. For man rolls up his past, as a tourist
his rug, and carries it with him wherever he goes. That is why Wells’s
“Time Machine” and the reversed reels of the movies are so funny. There
is nothing absurd about running a wheel backward but there is about
running a man backward.[2] The physicist feels no reluctance about
turning the stream of time backward for all physical phenomena are
reversible under the proper conditions. If we interpret the universe
as merely matter in motion and imagine at a certain instant that every
individual particle reverses its motion and goes in just the opposite
direction at the same speed, then the whole history of the world would
be reënacted in the opposite order and the earth would return to its
primeval nebulæ.

In Wells’s story, “The New Accelerator,” a professor invents an
elixir that speeds up the rate of living a thousandfold. A person
taking a dose of it sees people as wax figures apparently motionless
in the midst of violent action. Falling objects seem to stand still
in the air. The music of a band is reduced to “a low-pitched, wheezy
rattle” or “the slow muffled ticking of some monstrous clock.” But in
compensation for this the accelerated drug-fiend could watch at leisure
the slow flapping of a bee’s wings.

But even Wells with his seven-league-boots imagination finds it
difficult to keep ahead of the march of science. What he then saw only
with his mind’s eye we can actually observe. By moving the accelerating
lever on your phonograph toward the S end of the scale you can slow up
the tune and lower its pitch until it becomes inaudible as music. The
new Pathé ultra-rapid camera can take pictures at the rate of 160 to
the second. When these are projected on the screen at the usual rate
of 16 to the second all movement takes place ten times slower than in
actual life. This gives opportunity for the study in detail of the
action of a ballplayer pitching a curve or of the wing motion of a
humming bird or of the splash of a marble falling into water or of the
flight of a bullet. We can magnify motion or minify it as much as we
will. The cinematograph owes its origin the desire of Senator Leland
Stanford to study the movement of a horse’s legs so as to find out why
one racer went faster than another.

Such playful flights of the scientific imagination as Wells and
Flammarion indulge in and such freaks of projection as the camera
man amuses us with are of use to those of us who find difficulty
in translating a mathematical formula into terms of everyday life.
There is no better place to study metaphysics than in the world of
the flickering screen, for there man has complete control of time and
space. He can enlarge and reduce any object. He can hasten, retard
or reverse any action. He can throw upon the screen at the same time
events happening months and miles apart. Therefore to those of us
who have had the advantage of an education in the movies, Einstein’s
ideas of the relativity of time and space do not seem startling or
inconceivable.

Kant not only conceived the possibility of more than three dimensions
but believed in the probability of it. His argument is based on greater
insight into the intentions of the Almighty than we of this day would
claim:

 “If it is _possible_ that there be developments of other
 dimensions in space, it is also very _probable_ that God has
 somewhere produced them. For His works have all the grandeur and
 variety that can possibly be conceived.”

In this temporal, spatial and material world of ours reality requires
that the four dimensions should hang together. But at an infinite
distance from all matter this fourfold combination would be dissolved
into a three-dimensional space and a one-dimensional time. In that
extra-mundane realm time ceases to flow, gravitation no longer drags
downward, matter is non-existent, light is immovable and change is
impossible.[3] Thus the new mathematics leads to a state curiously
like the conventional conception of heaven.

We talk as our forefathers did about “the ends of the earth” but
we know that one might start from his home and walk forever in any
direction without coming to an end of it. But though the earth’s
surface is infinite in the sense of endless, yet one never can get more
than 8,000 miles away from home where’er he may roam. If a man stood
on the top of the highest mountain on earth and aimed a level gun in
any direction, the bullet, if it could be given sufficient velocity to
counteract the influence of gravity, would go around the world and hit
him in the back of the head. Or if light were sufficiently deflected
by gravitation to follow a level line around the earth--another absurd
assumption--the man looking through a level telescope in any direction
could see how his hair was combed in the back. Such happenings, though
impossible, are not inconceivable but are logical consequences of our
knowledge that the world is round and that what we call straight or
level lines as measured on plain or sea are really great circles around
a center four thousand miles below.

Now is it not also conceivable that the lines we call straight
in astronomical space may also have an imperceptible curvature in
some unknown fourth dimension? If this curve is closed like the
circumferences of the earth a ray of light pursuing a straight course
in a certain direction might eventually return upon its track, even
though not refracted or reflected by the matter it passes through or
by. A telescope of unlimited power pointed into space at a tangent
might then show the observer his own back, if light were transmitted
instantaneously, but, since it is not and since the curvature of space,
if there be any, is exceedingly minute, what the observer would see,
assuming that the earth had come back to its former position, might be
the scenes of some geological age millions of years ago.


                        NON-EUCLIDEAN GEOMETRY

The idea that space itself may be curved and that the axioms and
assumptions on which our geometry since the time of Euclid have been
based, may not be absolutely and exactly and eternally and universally
true has been diligently studied during the last fifty years. The
Russian Lobatchewsky, the Hungarian Bolyai and the German Riemann
have developed systems of geometry by starting from premises the
opposite of those of Euclid and these systems are just as logical and
consistent with themselves as the ordinary or Euclidean geometry.
These non-Euclidean geometries were at first commonly regarded as mere
freaks of the mathematical imagination, but they have already proved
valuable in leading to a reconsideration of the fundamental principles
of our thinking and, if Einstein is right, they may be necessary
to explain physical phenomena. It is hard for the mathematician to
discover anything useless. A distinguished American mathematician in
announcing a new theorem exclaimed: “And thank Heaven, no possible use
can ever be found for it.” But, whatever it was, he made a rash boast
for nowadays the mechanic treads on the heels of the mathematician and
uses imaginary quantities, actual only in the fourth dimension, like
√-1, in figuring out the winding of his dynamo.

Readers whose mathematical faculty is weak or undeveloped and who like
something concrete with “human interest” in it will find what they
want in “Flatland by A Square,” a book published in Boston in 1891.
The author, who turned out to be the Reverend Edwin Abbott, tells of a
land in only two dimensions. The ruling class consisted of polygons,
the bourgeoisie of squares and equilateral triangles, the lower class
of isosceles triangles of narrow base, while the criminals had more
irregular forms and the women were mere needles. Since all were
confined to a surface, four lines set in a square made a tight prison.
The inhabitants of Flatland, even the aristocratic and intellectual
individuals who had so many sides as to be almost circular, could not
conceive of a third dimension from which a person like ourselves could
look down and see at a glance the insides of their houses, their safes
and their bodies just as a being in the fourth dimension could see the
insides of ours. The narrator, that is, A Square of Flatland, visits
as a missionary the land of two dimensions where all the people lie
in a line and refuse to believe in anything outside it and finally he
encounters and endeavors to convert a solitary point of no dimensions
but finds him, as we should expect, an incorrigible solipsist.

We should all of us have been familiar with the fourth dimension for
years if Slade had not turned out a trickster. Slade was an American
medium--the original of Browning’s “Mr. Sludge”--who fooled Professor
Zöllner by giving him what purported to be experimental evidence of
the fourth dimension. Zöllner was a distinguished German physicist,
Professor of Astronomy in the University of Leipzig, old, near-sighted,
pre-disposed to spiritualism, and unskilled in legerdemain. Any proofs
that Zöllner asked for, Slade was usually able at the next séance to
produce. All the things that one might do in four dimensions but could
not do in three were forthcoming by the obliging spirits whom Slade had
at call.

[Illustration: In space of one dimension (a straight line) there could
be neither bend, loop nor knot in a string.]

[Illustration: In space of two dimensions (a flat surface) a double
bend could be made in the string but no loop or knot could be made.]

[Illustration: But if we raise one string (into the third dimension)
and lay it over the other like this:]

[Illustration: We get a loop but cannot form a knot without using the
ends.]

[Illustration: A knot like this cannot be made in a string so long as
the ends are held by the hands. But if we could use a fourth dimension
we could tie such a knot as easily as we made a bend by the use of the
second dimension and a loop by the use of the third. If such a knot
could be tied in a string so held it would be experimental evidence of
the existence of four-dimensional space.]

Zöllner tied the ends of a string together and sealed them on the table
top, letting the loop hang down under the table out of sight. He then
asked to have a single knot tied in the string and the spirits tied
four. Zöllner also reports that the coins he put into a sealed box were
taken out and writing produced inside sealed slates.

On the basis of these experiments Zöllner wrote a volume on
“Transcendental Physics” to prove the existence of another world in
the fourth dimension. But when Slade tried his tricks in London he
was caught at them by Professor E. Ray Lankester. He was convicted of
deception with intent to defraud in the Bow Street Police Court and
sentenced to three months’ imprisonment with hard labor. Nowadays the
apparatus for Slade’s famous slate-writing trick can be purchased at
any conjurer’s shop.

It is vain to expect anything scientific to come out of the séance
room where the alleged phenomena are not reproducible under specified
conditions but appear only occasionally and under circumstances
prescribed by the medium who always may be and often is proved to be a
sleight-of-hand--or sleight-of-foot--performer. The fourth dimension
which Einstein and other scientists are now considering is not
conceived of as the abode of departed spirits, a spare room for ghostly
visitants, but merely as a new factor in a mathematical formula. It
offers us no hope of ever being able to take coin out of a closed safe
or put coin into an unopened coconut but it does promise to explain
certain optical phenomena which, though rare and minute, are yet open
to the observation of anybody, be he skeptical or credulous.


                         SOME SIMPLE EXAMPLES

Lisbon lies nearly straight east of New York but when a ship captain
wants to go to Lisbon he does not sail straight east but sets his
course a little northward in the beginning and a little southward
toward the end and so gets there quicker than if he had followed a line
of latitude. Draw his course on a flat map and you would think he was
taking a roundabout route, but trace it on a globe and you will see
that he is following a great circle, the geodetic line, which is the
shortest distance between any two points on the earth’s surface.

An airman looking down on a rocky, hilly, woody country sees it as a
flat plain and if he watched a hunter returning home with his bag of
game would wonder that he did not go straight instead of wandering
around in such an irregular way. Yet the hunter, being tired, is
taking what is for him the shortest way home as he dodges rocks and
circumambulates the hills. The easiest way is the shortest way.

A river in its desire to reach the sea always takes the shortest
possible way. Its meanderings are not meaningless but determined by
a law as rigid as a law of geometry, that is, the law of gravitation
which prevents the river from taking a short cut over the hill.

If you look at a landscape over a heated plain or bonfire or through
uneven glass you will see that the image is distorted and confused
because the rays of light are refracted and entangled as they pass
through this unequal medium. Yet each ray is going just as straight as
it can toward your eye.

Now to such familiar cases where a ray of light is bent out of its
straight course by the uneven density of the air or glass through
which it passes Einstein has added another and unsuspected effect,
namely, that light is likewise deflected in passing through a strong
gravitational field such as the vicinity of a large body like the sun.

It has long been known that the displacement of the earth in space and
time (that is to say, its motion) causes an apparent displacement of
the stars in space.

The astronomer does not point his telescope straight at a star. If he
did, he would not see it, for, owing to the forward motion of the
earth, the telescope moves out of range of the rays that otherwise
would have reached it.

[Illustration: Everyone knows that a ray of light is bent out of its
straight course as it passes from the air into a denser medium like
water or glass, and that this deflection apparently shifts the position
of the object from which the light comes. Einstein’s theory and the
British eclipse observations prove, what was not known before, that a
ray of light as it passes through the gravitational field of a large
body like the sun, is also perceptibly bent out of its straight course
and likewise makes an apparent shift in the position of its source,
the star.--From Black & Davis’ “Practical Physics.” Published by The
Macmillan Company.]

If you have ever tried to shoot a bird on the wing, or, better, a
prairie-dog from a train you will get the idea. Or, if you have not had
this experience, you have doubtless watched the raindrops running down
a car window and have noticed that when the rain is falling straight
down the drops strike the pane on a slant when the car is moving
forward. The faster the car moves the greater the deviation from the
perpendicular. If the train runs backward the rain-streaks slant in
the opposite direction. If then you should be asked to point out the
direction of the cloud from which the rain is coming you would--unless
you knew and made allowance for the movement of the train--point in
a line with the streaks on the pane, sometimes backward, sometimes
forward, but not straight upward where the raincloud really is.

Now the astronomer is on a moving train, the earth, which is rushing
around a ring about 186,000,000 miles across. Consequently every star
appears to wabble around in a little ellipse and the astronomer has
to aim his telescope, now on one side, then on the other, of the real
position of the star in order to bring it on the cross-hairs of his
object glass. This apparent displacement of the stars is known as “the
aberration of light” was explained by Fresnel in 1818--to everybody’s
satisfaction until recently--on the assumption that all space is filled
with an immovable medium, the ether, which transmits the rays of light
in straight lines in the form of wave motion, and that the earth moves
through the ether without displacing it, somewhat as an airplane moves
through still air. But the aviator knows how fast he is moving by the
current of air streaming back in his face. Why then, since the ether is
in perfect repose, could we not determine the absolute motion of the
earth through space by measuring the drift of the ether as it streams
through the pores of the earth? Light appears to afford us a means of
measuring such a drift of the ether through matter, if there be such.
Since light is conveyed by the ether we should naturally expect it to
take less time to travel a certain distance if the receiving instrument
is carried toward the source of the light by the earth motion than
if it is being carried away from it. This question was put to the
crucial test by two American physicists, Michelson and Morley, who
devised an instrument so delicate that it could detect differences of
one-25,000,000th of an inch in the path of a light ray. But although
this delicacy was ten times greater than was necessary to detect the
ether drift, if there were any, no evidence of such drift could be
discovered.


                       THE ECLIPSE OBSERVATIONS

In the history of science the year 1919 is likely to be known, not as
the year of the overthrow of the German Empire, but as the year of the
overthrow of Newton’s law of gravitation. The British astronomers who
went to Africa to observe the eclipse of the sun May 29, 1919, came
back with the proof that a ray of light passing close by the sun is
bent out of its straight course. The photographs taken during the
six minutes when the sun was shadowed show the surrounding stars in
different positions from where they are seen when the sun’s disk is not
in their midst. This is the second time that Einstein has scored over
Newton. The first was in regard to the orbit of Mercury. If the sun and
Mercury were alone in the universe the planet, according to Newton’s
law, would revolve forever around the sun in the same elliptical track.
But the presence of the other planets makes Mercury deviate from this
regular route so the ellipse it describes is never quite the same but
slowly shifts around so that in the course of centuries its longer
diameter would be pointing in a different direction. Calculating by
Newton’s law, the influence exerted by the other planets astronomers
found that it would shift the orbit of Mercury 532 seconds of arc in a
century. But when they took observations on Mercury they found that its
orbit was shifting at the rate of 574 seconds. The discrepancy between
observation and theory, 42 seconds, is thirty times greater than could
be accounted for by errors of instruments or observation. But according
to Einstein’s theory, if the sun and Mercury were alone in space with
no other planets interfering, the orbit of Mercury would not remain
the same, but would advance at the rate of 43 seconds a century. This,
as the reader will observe, is in substantial agreement with the
discrepancy which has for two centuries puzzled astronomers, since it
was inexplicable on the Newtonian theory.

The electro-magnetic theory of light, thought out by Clerk Maxwell
forty-five years ago, has proved to be an excellent guide to
research and led to many practical applications, such as wireless
telegraphy. According to this theory the miles-long Marconi waves,
the infinitesimal waves that we feel as heat or see as light and the
still more minute waves of the X-rays are movements of the same sort,
though differing in length, and all travel at the same speed in space
of 186,000 miles a second. It was one of the implications of Maxwell’s
theory, though it was not perceived until later, that light and all
such waves must exercise a certain pressure upon a body against which
they strike, just as a jet of water from a fireman’s hose pushes
against the side of a house. The pressure of light is so exceedingly
slight that it had never been noticed, but it has been actually
detected and measured by Professors E. F. Nichols of Yale and G. F.
Hull of Dartmouth. The sunshine falls upon the earth with a force of
160 tons. Both theory and experiment have shown that a beam of light
has inertia or mass, that is to say, a beam of light pushes like a
water jet, and it has now been proved, by the eclipse expedition, that
the pull of gravity deflects a beam of light as it does a water jet.
That is to say, a beam of light has weight, is attracted by gravity.
This deflection of a beam of light by gravity is extremely small, but
photographs taken during the recent total eclipse of the sun show that
star beams that passed near the sun are bent out of a straight path.

[Illustration: The eclipse expedition found that the stars seen about
the sun appear slightly shifted from the positions they occupy on a
map of the same region of the sky when the sun is not in their midst.
This shows that a ray from a star is refracted or bent as it passes
close to the sun and confirms Einstein’s theory that light is affected
by gravitation. The observed angle of deflection agrees closely with
that predicted by Einstein but is twice as great as that required by
Newton’s theory of gravitation. In this diagram of course the angle
of the deflected ray and the size of the sun and earth relative to
distance are greatly exaggerated.]

A better illustration of the eclipse observation than I could word is
given by Sir Oliver Lodge in his interesting article on “The New Theory
of Gravitation” in The _Nineteenth Century_ of December, 1915,
from which I therefore quote:

 Take a fine silk thread of indefinite length, and stretch it straight
 over the surface of a smooth table or floor. Imagine a star at one end
 of the thread, and an eye at the other; and let the thread typify one
 of the rays of light emitted in all directions by the star, viz. the
 ray emitted in the direction of the observing eye.

 Now take a halfpenny [or an American quarter,] place it on the
 table close to the thread, so that the eye end of the thread is ten
 feet away; and then push the halfpenny gently forward, till it has
 displaced the thread the barely perceptible amount of one thousandth
 of an inch. The eye looking along the thread will now see that the
 ray is no longer absolutely straight; in other words, the star whose
 apparent position is determined by that ray will appear slightly
 shifted. The scale is fixed by the size of the halfpenny, whose
 diameter, one inch, is used to represent the Sun’s diameter of 800,000
 miles. The ten-foot distance between eye and Sun practically supposes
 that the eye is on the Earth, which would be a spot one hundredth of
 an inch in diameter, or about the size of this full stop.

 As for the distance of the star, at the other far end of the thread,
 that does not matter in the least: but, on this scale, it may be
 interesting to note that one of the nearest stars, about eight
 light-years away, would require the thread to be a thousand miles long.

 The ray is now bent or deflected as it passes the neighborhood of the
 Sun on its long journey, so that it is out of place one thousandth of
 an inch at a distance of ten feet; and the effect of this tilt of the
 ray, upon the observer, is to make him just able to see a star upon
 the Sun’s ‘limb’ when it is really behind it, or to make him see a
 star slightly further off the ‘limb’ or rim of the Sun than it really
 is. The shift of one thousandth of an inch at a distance of ten feet
 corresponds to an angle of one and three-quarter seconds of arc, which
 is just the optical shift that actually ought to occur, according to
 Einstein, when a ray from a star nearly grazes the Sun’s limb on its
 way to a telescope; and this is the optical shift which we now know
 does occur. That may be taken as the definite result of the recent
 eclipse observations. The effect, both in magnitude and direction, had
 been predicted four years before, on the strength of a mathematical
 investigation, by Professor Einstein.

The images of two stars, one on each side of the sun’s disk, will
apparently be crowded a little apart when the sun comes between them. A
star that would be just eclipsed by the edge of the sun’s disk if its
rays came straight may still be visible since the rays are curved. In
other words we can “see around a corner” as every good teacher is said
to do. If the sun were encircled by a ring of stars, or a nebula, like
a halo, the circle of light would be contracted as it passed the sun
and would come to a focus at a place seventeen times the distance of
Neptune, or 47,600,000,000 miles beyond the sun.

The observations made by the British expeditions during the eclipse
of May 29, 1919, were not altogether satisfactory. At Principe, on
account of a cloud that drifted by at an inopportune time, only a few
photographs could be obtained. At Sobral one of the object glasses
gave distorted plates, but the other gave a very good series of seven
star images. These when measured at the Greenwich Observatory gave
the following figures which are in accordance with those calculated by
Einstein’s formula:


           RADIAL DISPLACEMENT OF STARS IN SECONDS OF ANGLE

As observed by the British astronomers:
      .20   .32   .56   .54   .84   .97   1.02

As predicted by Einstein:
      .32   .33   .40   .53   .75   .85    .88

This is regarded by the astronomers of the British Eclipse Expedition
as sufficiently close to confirm Einstein’s law but those who hesitate
to accept so far-reaching and subversive a theory on the basis of these
few minute measurements may hold their judgment in suspense until
1922 when the next solar eclipse, visible in Australia, takes place.
Or possibly some means may be found to take star photographs close to
the sun while shining. Our California mountain observatories may be
of service in this since they are perched above much of the dust and
mist and denser air that cause a strong light to irradiate and fog the
photographic plate. Doubtless, too, the old photographs of earlier
eclipses will now be got out to see if they contain any stars suitable
for measuring.

Some of the opponents of Einstein suggest that the observed deflection
of the starlight may be due to a solar atmosphere that refracts the
rays like our earthly air. But it is hardly probable that an enveloping
atmosphere sufficiently dense and so far-extending as to produce such
an effect would have remained unobserved and it is highly improbable
that the density of such an atmosphere should have just the density and
decrease with the distance at just the rate to produce the deflection
predicted by Einstein’s calculation.[4]

The discovery is rather disconcerting to astronomers, for all their
calculations for the last three hundred years have been based upon the
assumption that light travels in straight lines at even speed through
empty space or, what is the same thing, through the ether. If now light
is pulled aside by gravitation as it goes by a solid body the rays
from a distant star having to pass through the tangled throng of the
Milky Way might travel a very devious route and the star would appear
to us to be located in a different place from where it really is. In
fact it is possible that a star which we see double may actually be
single but that rays starting out from it in different directions may
be so deflected by passing near other stars that when they reach us
they appear to come from different points of space and so appear to us
as twin stars. There may, too, be dead or dark stars on the way whose
existence we cannot discern and allow for.

Now those of us who are not astronomers are not much concerned over
a discrepancy of a few hundredths of a second in the measurement of
an angle by the telescope. We do not care much where Mercury will be
five centuries hence, for we do not know quite where it is now. If
astronomers made the laws of Nature instead of merely discovering
them we might be afraid that at their next congress they might repeal
Newton’s law of gravitation and send us all flying off into space. But
fortunately they have no such power and even though they should all
become adherents of Einstein’s most revolutionary theories, Newton’s
laws of mechanics and Euclid’s laws of geometry would remain as true
as they ever were, not perhaps absolutely and universally true, as we
have assumed, but sufficiently accurate for all practical purposes.
Deviations from them can only become detectable when we come to
consider movements as swift as light waves or electrons.

How a heavy object can alter space relations may be seen from this
simple illustration: Stretch a sheet of rubber over a hoop like a
drumhead. It is now level and flat and if parallel lines are drawn
across it in two directions so as to divide it up into squares like a
checkerboard all these lines are straight and equidistant and all the
squares are of equal size.

A row of worms, starting in an even rank and crawling along the
parallel lines across the drumhead, would keep even all the way. Now
lay a bullet on the center of the drumhead. The rubber sags down and
stretches, most in the middle, least at the edges. The “parallel”
lines are no longer equidistant. The squares are no longer equal. The
lines are no longer of the same length. If now we repeat our worm race
we shall find that those worms following lines close to the weight
have to go down hill and up again and so travel a greater distance to
traverse the same number of squares than those following lines nearer
the edge which lie comparatively flat and are nearly as short as
before. Consequently the worms will be slowed up in proportion to their
nearness to the center and the row of their heads will be swung around
at an angle to their former frontage.

We might “explain” this by assuming that the worms on seeing the bullet
to one side were drawn by their curiosity a little toward it, those
nearest of course being drawn the most. Or if we had got beyond this
crude animistic method of explanation we might assume that the bullet
was attached to the head of each worm by an invisible lariat which
being pulled by the bullet drew the worms more or less to one side, the
shorter the lariat the stronger the pull. Or if we had outgrown this
crude mechanical method of explanation we might assume the existence
of a “force” in the lead which in some mysterious manner attracts the
heads of the worms inversely as the square of their distance. But
instead of inventing a wormhead psychology or an invisible cord or an
incomprehensible force is it not simpler to consider the space between
and to suppose that the lines to be traversed are lengthened in the
neighborhood of the weight?

Now these four successive methods of explanation have been used
to account for gravitation. First it was assumed by the ancient
Babylonians and Hebrews that the sun and stars were living beings, gods
or angels, moving of their own volition around the earth, or at least
that each was guided in its orbit by its particular god or angel. The
later Greeks of Ptolemy’s time supposed the heavenly bodies to be set
in concentric crystal spheres and so revolved; I presume by somebody
turning a crank behind the scenes. Then came Newton and said: “Let’s
discard the Ptolemaic spheres and all mechanical connection and assume
a _force_ of gravitation attracting all bodies in proportion to
their masses and inversely proportional to the squares of the distances
separating them.” Now comes Einstein and says: “Let’s discard this
hypothetical force and simply assume that the field of time and space
traversed by a moving body is altered if there is another body in
the vicinity.” In Einstein’s view gravitation is not a force; it is
a distortion of space and time in the presence of matter. A comet
sweeping past the sun cannot pursue a straight course, as it could in
interstellar space, but follows a curved path about the sun which is
for the comet the shortest way it can go under the circumstances.

So, too, a row of light waves coming from a distant star keeps an
even front as they pass through empty space but as they come close
to the sun they find their paths impeded, or, we may say, stretched.
Those going nearest the sun are slowed up the most; those farthest off
the least. Consequently the wavefront is slued around a bit and the
direction of the ray is slightly altered.

If now light waves have difficulty getting past the sun we should
expect that they would experience like difficulty getting away from the
sun. They would be slowed up a bit by its gravitational pullback. The
frequency would be reduced; the interval of time between wave-crests
lengthened. This means, in the case of sound, lowering the pitch. Touch
your finger to the turntable of your phonograph and you flat the tone.
In the case of light, it means change of color toward the red. This
effect, according to Einstein, should be, but has not been, observed.

“If Einstein’s third prediction is verified,” says Sir Oliver Lodge,
“Einstein’s theory will dominate all higher physics and the next
generation of mathematical physicists will have a terrible time of
it. For university courses and for all practical purposes we shall
have the Galilean and Newtonian dynamics but they will reign as a
limited monarchy and sooner or later the Einstein physics cannot fail
to influence every intelligent man. If these complications are to
come into science we must leave them to the younger men. I hope that
gravitation, now that it has begun to interact with light, will begin
to give up its secrets, but in my time I must be content to get secrets
out dynamically and leave transcendental methods to others.”

One English scientist, Thomas Case, writes to _The Times_ to
protest that it would have been in much better taste for the Royal
Society to have adjourned its discussion “before bringing into question
the reputation of Newton, who was President of the Royal Society for
the last twenty-five years of his life and raised the society to the
acme of its fame.”


                           WHO IS EINSTEIN?

Albert Einstein was born in Germany in 1874. He early showed the bent
of his genius and at the age of twelve, when his fellow pupils were
plodding along with their daily tasks, he was plunging through works
of higher mathematics borrowed from his teacher. He was only eighteen
when he conceived the outlines of his theory and ten years later it
was ready to give to the world. He left Germany for Switzerland at
the age of sixteen and became naturalized as a Swiss citizen. His
first academic position was the Professorship of Mathematical Physics
at the Zürich Polytechnic. Then the founding of the Kaiser Wilhelm
Academy for Research at Berlin gave him opportunity to work out his
theories undisturbed by other duties. Shortly before the war he was
called to Berlin to succeed the famous Dutch physicist, Professor
van’t Hoff in the Academy. The object of this institution was the
same as Carnegie had when he founded his institution for scientific
research at Washington, which was to seek out the exceptional man
wherever he may be found and set him at his peculiar tasks. At Berlin
Einstein receives a salary of $4,500 and has nothing to do but sit and
think. This he continued to do all through the five years of war and
revolution as quietly and persistently as Kant at Königsberg during
the wars and revolutions of a century before. Or as Archimedes at the
siege of Syracuse who was absorbed in drawing geometrical figures in
the sand--his blackboard--when a Roman soldier ran him through with a
spear. On two occasions he took part in the world-struggle going on
about his study, both actions greatly to his credit. In the beginning
he refused to sign the manifesto of the German men of science denying
all the charges against Germany, and at the time of the armistice he
signed an appeal in favor of the revolution. He is an ardent Zionist
and has promised to aid the Hebrew university which is to be founded at
Jerusalem.

According to tradition, Isaac Newton was led to his theory of
gravitation by observing an apple falling from a tree in his garden.
The newspaper correspondents start a similar tradition by reporting
that Einstein got his theory of gravitation by observing a man falling
from the roof of a building in Berlin. Now a man has the advantage
of an apple in that he is able to tell his sensations. When Dr.
Einstein, who had seen the accident from his library window in the
top story of a neighboring apartment house, reached the spot he found
the man had hit upon a pile of soft rubbish and had escaped almost
without injury. Asked how it felt to fall he told Dr. Einstein that
he had no sensation of downward pull at all. This led Dr. Einstein to
consider whether the relativity theory, which he had applied only to
the case of uniform motion in a straight line, could not be extended
to difform or accelerated motion by gravitation. So the special
relativity theory which he had enunciated in 1905 developed ten years
later into a generalized relativity theory (_Verallgemeinerten
Relativitätstheorie_).


                          HOW TO LOSE WEIGHT

A man falling out of an airplane is obeying a natural impulse, namely,
the force of gravitation. So long as he does not resist he is free
as air, light as a feather, and altogether comfortable. He can look
down with complacency and contempt on the poor mortals below him who
are trying to stand up against this natural impulse and laboriously
dragging one foot after another as they crawl about the earth when they
might be flying through space without effort as he is. It is only when
he tries to stop his free fall by bumping against the ground that he
gets into trouble on account of gravitation. It was in this way that
the Calvinists, who were a sort of mathematical theologians, conceived
of the fall of man. The sinner is simply obeying the force of natural
depravity, namely, moral gravity, and so long as he is conscienceless
and does not consider his inevitable end he has no knowledge of the
moral law and is quite happy in his downfall.

A person falling freely loses all his weight. His hat does not press
down on his head. His feet do not press down on his shoes. If he lets
go of his walking-stick it does not “fall down” at his feet. It stands
upright and simply travels along with him. For, as Galileo showed when
he dropped his big and little cannon ball off the Leaning Tower of
Pisa, all bodies fall with the same speed.

If he were in a falling elevator with an opaque door he would not
know he were falling unless he surmised it from the _absence of
gravitation_ as evidenced by his own feeling of lost weight and the
queer behavior of the objects in the car. He might fall all his life
and never find it out. The law of gravitation is like criminal law; you
don’t feel it till you come into conflict with it.

Or if our illustration requires too tall a skyscraper, let us imagine
that a comet as it flies by knocks a chip off the earth with a group
of people on it. This terrestrial fragment, cast loose in space, gets
caught by the attractive force of some gigantic and distant star and
falls toward it with ever-increasing velocity for thousands of years.
The inhabitants of this errant orb would never know it from their own
feelings or any observations they could make _on their own little
world_. Does that seem incredible to you? Then tell me how do you
know but this our world is such a planet and together with the solar
system has been falling for thousands of years toward some center of
attraction? Astronomers, indeed, say that we are moving at tremendous
speed toward Canis Major, in other words that the world is going to the
dogs.

All this means that uniformly accelerated motion, such as gravitation
imparts to a freely falling body, is, like uniform translatory motion,
a question of relativity and cannot be discovered by an observer
carried along by such movement.

The idea that uniform translation, like the moving train we have
considered, is merely relative motion, is an old idea and not hard
to understand or accept. But when we try to extend the principle of
relativity to acceleration, that is, to a rate of motion that is
continuously increased or retarded, we get a new and revolutionary
conception of the universe and are drawn into some very startling
conclusions. Einstein took this step five years ago and that is what
has caused the present excitement. For Einstein when he once gets
hold of an idea follows it wherever it leads him with the undaunted
determination of a Nantucket sailor towed by a harpooned whale. It was
a whale of an idea that he harpooned in 1915 and it carried him into
strange waters. It led directly to a contradiction or correction of
one of the two fundamental postulates which he had laid down as the
foundation of his theory of the universe in 1905, namely, that the
velocity of light in space is a constant. But he promptly abandoned
this idea with cheerful nonchalance in favor of the new notion that the
velocity of light is affected by gravitation.


                       A SUBSTITUTE FOR GRAVITY

Let us then follow Einstein and apply his Principle of Equivalence to
accelerated motion and see what it leads to. Imagine yourself shut up
inside a closed chamber, like an elevator car, somewhere out in space
away from the gravitational forces of the earth or sun. Suppose this
chamber to be rising with a constantly increasing velocity. We can,
if we want to be definite about it, assume that the chamber is a big
shell pulled up by a cable coiling around a conical windlass that hauls
it up faster all the time. Or we can assume that it is propelled from
behind by the continuous backfire of explosives, like the rocket which
Professor Goddard proposes to send to the moon. All we need is some
force, not gravitation, capable of giving the chamber every second an
additional velocity of thirty-two feet a second. Now the point is that
if you were in such an upward-moving chamber you would not know but
what you were resting on the earth. Everything would behave exactly
the same. If you now weigh one hundred and fifty pounds on the scales,
that is, if your shoe soles press down with that force, the floor of
the rising chamber would press upward with that same force and you
would not know the difference. If you let loose a ball from your hand
the floor would rise up to meet it and it would appear to fall. If you
threw the ball upward with a velocity greater than the velocity of the
chamber at the moment, the ball would rise, but since the velocity of
the chamber was constantly increasing the floor would gain on the ball
and catch up with it. This would look to you just the same as when on
earth you threw a ball into the air and it fell back to the ground,
drawn, as you are accustomed to think, by “the force of gravitation.”
But here we have no “force,” but merely a mode of motion.

Under such circumstances it would seem that all Nature conspired to
keep you in the dark. You appeal to the ether, that supposedly stable
and stationary medium that fills all space, but that also fails you.
You try the Michelson-Morley experiment to see if you are moving
through the ether or at rest on the earth but your apparatus expands or
contracts just enough to deceive you.

You now try observing horizontal rays of light but they seem to bend;
that is, a beam of sunshine entering a pinhole on one side of your
_camera obscura_ will not strike the wall at a spot exactly
opposite but a little below it, if you have instruments sufficiently
delicate to show this. You try vertical rays of light in this fashion:
You examine with the spectroscope rays of light coming from two sources
below (behind) your instrument, one at a distance and the other nearer.
Now since you are moving away with increasing speed, the light from
the farther source will have to take longer strides to catch up. Or
in other words, its frequency will be reduced and it will be shoved
toward the red end of the spectrum where the longer waves are. You will
have noticed that when a whistling train rushes past the train you are
on, the whistle as it comes toward you is raised in pitch (decreased
wave-length) and as it recedes from you is lowered in pitch (increased
wave-length).

Now, says Einstein to himself, if my Principle of Equivalence is
correct and there is no difference between (1) weight and (2) the
accelerated upward movement of an observer, then all the optical
effects that I have thought out in the second case must apply to the
first, that is, to gravitation. It must follow that a ray of light
passing through a gravitational field will be bent out of its course
as though it were attracted by the heavy body. This prediction has
been verified. It must further follow that light proceeding from a
heavy body like the sun or a star will be held back or slowed up by the
attraction of gravitation, and the spectral lines will be displaced
toward the left as compared with the same lines in the spectrum of
an earthly light. Now such displacement has been observed in stellar
spectra but it does not seem to be of the right value to satisfy
Einstein’s equation and it has not been observed in sunlight.

The remarkable thing about it is that Einstein, by following a line of
reasoning somewhat like that which I have crudely outlined, not merely
supplied an explanation for phenomena that had been observed but could
not be explained (such as the discrepancy in the orbit of Mercury) but
he provided in advance the explanation for phenomena that had never
been observed until he directed attention to it (such as the deflection
of starlight by the sun). Sir Oliver Lodge says of this:[5]

 Before Einstein’s prediction nothing of the kind had been seen,
 nothing of the kind had been looked for, nor, so far as it is known,
 had such an amount of deflection been suspected.

Whatever may ultimately be thought of the validity of Einstein’s views
as a whole it is evident that he has worked out a mathematical method
of unprecedented power and wide usefulness.

Professor Bumstead of Yale says:

 Einstein’s theory is important in that it exemplifies a method which
 is in many respects new in theoretical physics and which may prove to
 be a very powerful method for advancing scientific knowledge. There
 was no idea that the prediction of the bending of light would fix
 up Mercury’s perihelion and incidentally explain a two-century old
 astronomical difficulty. That came straight out of a blue sky.


                 MECHANICAL VERSUS MATHEMATICAL MINDS

We sometimes hear it said that “Einstein has overthrown Newton’s theory
of gravitation.” That is impossible because Newton did not have any
theory of gravitation. He merely laid down the law of gravitation. He
told how bodies behaved toward their neighbors; he did not tell why.
Newton was not content with the idea of action at a distance through
empty space and he tried to explain gravitation by the pressure of the
ether on material bodies but he was not satisfied with the results
and did not publish them. In the 234 years since many men have tried
their hands at devising some sort of machinery that will “explain”
gravitation. For human beings are like Toddie of “Helen’s Babies”
and want to have the watch opened so they can “see the wheels go
wound.” At least Anglo-Saxons have that desire. Poincaré, the French
physicist, said this is the distinction between the Anglo-Saxon and
Latin minds; the former are uneasy until they can imagine a mechanical
model to represent natural phenomena, the latter are satisfied with
a mathematical formula expressing the action. The ether, which was
invented to explain light, also required “explanation.” Lord Kelvin
imagined it to consist of spinning tops which have a sort of mobile
stability. Sir Oliver Lodge has filled it with a complicated structure
of interlocking geared wheels to account for electro-magnetic action.
These are typical Anglo-Saxon modes of thinking. On the other hand,
Einstein, who, in spite of his Hebrew blood and German training, has
preëminently what Poincaré claims as the Latin temperament, does not
have any use for the ether and does not care at all whether he can
“picture” the fourth dimensions on paper or not.

Now some of us are excessively Anglo-Saxon in our attitude toward
mathematics. It is with a fellow-feeling for such folks that I have
filled this little volume with such crude and absurd analogies as
trains and elevators and projectiles flying through space and Coney
Island mirrors. To the mathematically minded such illustrations are not
simplifications but complications, not representations but caricatures.
Mathematics is the proper language of physics as the five-barred staff
is the proper language of music. Ask a musician to explain a symphony
in plain everyday English and he cannot do it, though he carry the
Oxford Dictionary in his head. He can have the music played for us
or he can show us the printed score but he could never convey it in
ordinary language however long he might be willing to talk or we to
listen. But we must not do the musician or the mathematician the
injustice to suspect that his notions are hazy or absurd because he
cannot explain (_i.e._ translate) them to us.

Nor should we assume that the new ideas, because they are more
difficult for us to grasp, are necessarily more complicated or
extravagant than the old. A friend of mine who is familiar with both
tells me that Einstein’s papers are easier reading than Newton’s
“Principia.”

The aim of science is simplification through generalization and
this is the widest generalization yet attempted. It promises to
bring gravitation into relationship with other forces. One great
generalization, the law of the conservation of energy worked out by
Joule and others in the forties, brought heat and work and chemical
power all into one simple system. Clerk Maxwell in the seventies
brought together in one beautiful formulation all the diverse phenomena
of light, electricity and magnetism.

But gravitation has always stood out against any such league of natural
forces. It refused to come into the combine. It remained unique,
independent, irreducible, unalterable and inexplicable. Everything
else is correlated and interactive. Heat destroys magnetism, magnetism
produces electricity; electricity dissolves chemical combination;
chemical combination produces heat; heat causes motion; motion makes
magnetism; magnetism produces heat; and so on in endless round, each
affecting all the others. Different substances behave very differently;
one is more easily heated than another; some are readily magnetized or
electrified, others are not so susceptible; certain elements rush into
each others’ arms, others cannot be forced into combination.

But gravitation seemed indifferent to all these things; it showed no
prejudices or preferences. It attracted with equal force all sorts of
substances, no matter whether they were hot or cold, shiny or black,
moving or still, electrified or magnetized or neither. Other forces and
effects too required time for action at a distance. Sound travels at
the rate of 1,100 feet a second in ordinary air. Light travels at the
rate of 186,337 miles a second in a vacuum. But the force of gravity
seemed not to require any time but to be everywhere, acting all the
while, and nothing could shield it off or shut it out or in any way
interfere with it. The substance or mass of a body as measured by its
weight (the gravitational pull of the earth) was always identical with
its mass as measured by its inertia (its resistance to being set in
motion). All the energies are interchangeable. All other forces could
be reduced or increased, annulled or brought into effect at will.
Not so gravitation. Any bodies of a certain mass placed at a certain
distance apart are always drawn by the same attraction. That is,
gravitation is affected by nothing except geometrical relationships.

This naturally leads us to suspect that gravitation is nothing but
a geometrical relationship, that it is somehow a peculiarity of
space itself. If so, our demand of the physicist that he show us
gravitation,--drag out this mysterious force from its hiding-place
and let us see it--is altogether irrational. It is like a blind man
hunting in a dark cellar at midnight for a black cat that isn’t there.
The geometrician tells us that the internal angles of any triangle
are equal to two right angles. Shall we ask him, what is the force
that makes it so? Shall we refuse to ride on a trolley car until the
electrician can answer our persistent question; “but what _is_
electricity?” When we ask such a question we are really asking him to
tell us what electricity _is not_. To show us what electricity is
he can keep his mouth shut and simply point to the dynamo that produces
it, the wire that conveys it and the motor that consumes it. But what
we secretly mean is that he show us a mechanical model that imperfectly
imitates some of the actions of electricity or a mathematical formula
that will calculate its effects.

Now Einstein seems in the way of making gravitation the foundation of a
new system of geometry. Instead of “explaining” gravitation in terms of
something else he will explain other things in terms of gravitation,
or rather of his space-time manifold of which gravitation is one of the
properties.

Einstein’s _law_ of gravitation proves to be more accurate than
Newton’s law, but the correction is trifling except in rare cases. But
Einstein’s _theory_ of gravitation is fundamental and far-reaching
and if it is substantiated it will revolutionize physics and radically
affect our ordinary conceptions of the universe. The verification of a
prediction does not necessarily prove the truth of the hypothesis that
led to the prediction. Many a scientific discovery has come out of a
false assumption. Just as a miner may reopen an abandoned gold mine
or work over his dump heap to get more out of it, so scientists often
return to an old theory which they had abandoned for a more fruitful
hypothesis.


                          THE WEIGHT OF LIGHT

It is interesting to see that our modern physicists show a disposition
to adopt a corpuscular or emission theory of light not unlike the
conception which Newton steadfastly and vainly defended against the
undulatory theory. Professor Thomson, of Cambridge, reminds us that the
crucial experiment between the two theories was the test made by Bennet
in 1792 to determine if light exerted any pressure on a body when it
struck it as it would if light consisted of minute particles driven
straight forward with great velocity. Bennet found no such pressure
and the corpuscular theory was regarded as disproved. But it was later
found that the undulatory theory also involved such a pressure, and
recent experimenters have proved and measured it. As Professor Thomson
says:

 It is perhaps fortunate that Bennet had not at his command more
 delicate apparatus. Had he discovered the pressure of light, it would
 have shaken confidence in the undulatory theory and checked that
 magnificent work at the beginning of the last century which so greatly
 increased out knowledge of optics.

Of course any modern form of the emission theory of light must
account, as Newton’s did not, for such phenomena as interference and
polarization, which are so satisfactorily handled by the undulatory
theory. That is, it must combine the best features of both. Professor
Thomson shows that only an exceedingly small fraction of the ether is
concerned in the forward movement of light, in other words, “the wave
front must be more analogous to bright specks on a dark ground than to
a uniformly illuminated surface.” He does not, however, go so far as
Planck in regarding it as proved that radiant energy of all kinds has
a unit or atomic structure, the color of the light depending on the
size of these particles.

The discovery of the pressure of a beam of light has led to some
startling conclusions. For example, what shall be done with Newton’s
law that action and reaction are equal? When a gun is fired the kick of
the gun is balanced by the momentum of the projectile. When a reflector
throws a beam of light into space, the kick of it is there all right
but where is the projectile, if light is merely the undulation of an
imponderable fluid? We may suppose that the light strikes some dark
body out in space, transmits its impulse to that and Newton’s laws is
satisfied, but it may be a long time before such a body is encountered
and it may never be: at any rate a law that remains in a state of
innocuous desuetude for several thousand years is not good for much.
We must then assume that light has mass since it has inertia and
momentum. But if light has mass it must have weight; that is, it must
be attracted by gravitation. The eclipse observations confirmed this
deduction. Newton would have expected something of this, for he says in
his _Opticks_:[6]

 Query 1.--Do not Bodies act upon Light at a distance, and by their
 action bend its Rays, and is not this action (_caeteris paribus_)
 strongest at the least distance?

The observed deflection of light due to the sun’s gravitation is
greater than Newton would have anticipated but it would have been still
more disconcerting to the nineteenth-century physicists, for in giving
up Newton’s emission theory they had come to regard light as merely a
form of motion in a weightless medium, the ether. Disembodied energy,
like heat and light in ethereal space, was regarded as having no mass
or weight. Twentieth-century physicists are coming to the opposite
view, that the mass of a body is the measure of its internal energy.
If so, mass is not constant but changes with composition, temperature,
structure, electrification and motion.

As Einstein himself expresses it:

 It is evident that it is not possible to attribute an absolute
 sense to the notion of acceleration, no more than to the notion
 of velocity. It is only possible to speak of the acceleration of
 a material point in connection with a body taken as the body of
 reference. It follows from this that there is no sense in attributing
 to a body a “resistance to acceleration” in the absolute sense, like
 the resistance of inertia in the classical mechanics. Further, this
 resistance of inertia ought to be so much the greater when there is,
 in the neighborhood of the body, more inert masses not in accelerated
 movement. On the other hand, this resistance ought to disappear when
 these masses participate in the acceleration of the body.

 Now it is altogether remarkable that the equations of the
 gravitational field contain these different aspects of the resistance
 of inertia, which one might call the _relativity of inertia_.

The progress of science is continually toward a dematerialization
of matter. Physical analysis resolves the crude, heavy, solid stuff
that our senses show us into finer and finer particles farther and
farther apart until these practically disappear into mere points of
irradiating influence. First the mass is divided into the molecule and
this again into the atom, assumed, at the time it was invented, to be
the ultimate unit of matter. But recently the atom has been shown to
be a sort of solar system, but more complex, composed of hundreds of
electrons, corpuscles of electricity, whose radius is calculated to be
1/10,000,000,000,000 of a centimeter (a centimeter is so ---- long).
“But the size of the centers of disturbance, which in Einstein’s theory
are associated with matter, bears to the size of the electron about the
same proportion as the size of the smallest particle visible under the
most powerful microscope to that of the earth itself.”[7]

The old axiom was, “matter cannot act where it is not.” The new version
might rather read: “matter cannot act except where it is not.” That is
to say, attention is now directed to the space surrounding a material
body or electrical corpuscle.

Although we laymen are not concerned with the niceties of astronomical
measurements there is an aspect of this conflict of theories that
does interest us. The theory of Newton or, to go back further, of
Galileo, that the earth moves around the sun, altered profoundly the
philosophical and religious beliefs of the world, and the theory
of Einstein is much more far-reaching and revolutionary in its
metaphysical implications than the former. Professor Planck, who has
just received the Nobel Prize for his discoveries in physics, said of
Einstein’s first paper:

 It surpasses in boldness everything previously suggested in
 speculative natural philosophy and even in the philosophical theories
 of knowledge. Non-Euclidean geometry is child’s play in comparison....
 The revolution introduced into the physical conceptions of the world
 is only to be compared in extent and depth with that brought about by
 the Copernican system of the universe.


                   MUTABLE THEORIES AND STABLE FACTS

There is a feeling very prevalent among the general public interested
in such things that the foundations of modern science are being swept
away by the recent discoveries. The layman has been led to believe
that such laws as gravitation, the conservation of matter and the
immutability of the elements are the most certain and absolute truths
of science. But now he hears reputable men of science talk calmly about
the decay of matter and the transformation of one element into another,
and gravely consider a theory which makes invalid Newton’s three laws
of motion. It surprises, even shocks, him, as much as it would to
have a convention of bishops discuss the question of whether there
is a God, or the Supreme Court agree to set aside the Constitution
of the United States, or a congress of physicians resolve that all
medicine does more harm than good. He knows that the mere broaching of
such heretical views in these assemblies would be met with a storm of
indignation and that all the weapons of contempt, ridicule and even
personal spite would be directed against the rash innovator. Therefore
he is astonished and puzzled to see that in the scientific world these
revolutionary theories are received with interest and even pleasure,
and in the criticism to which they are subjected there is scarcely a
trace of animosity. And he does not see why men of science who have
accepted doctrines apparently contradictory to their former teachings
do not appear shamefaced and apologetic before the public, like augurs
whose tricks had been exposed.

The difficulty of the layman arises from his not understanding how a
scientist looks at his science; not realizing how firmly he holds to
its facts and how loosely he holds to its theories. The scientist never
bothers his head with the question whether a particular theory is true
or false. He considers it simply as more or less useful, more or less
adequate, succinct and comprehensive. A theory is merely a tool, and
he drops one theory and picks up another at will and without a thought
of inconsistency, just as a carpenter drops his saw and picks up his
chisel. He will say that the earth moves around the sun one moment, and
the next will revert to the theory of Chaldean astronomers, because it
is more convenient, and say “the sun rises.”

Really, the new discoveries are not so upsetting to science as they
appear to the general public. Unexpected and revolutionary as they are,
no page of millions that record the experiments and observations of
science is invalidated. No man’s work is proved wrong. Revolutions in
science do not destroy; they extend.

In the reaction of public opinion toward any novel and revolutionary
idea there are three stages observable.

1. That it is not true.

2. That it is not new even if it is true.

3. That it does not make any difference anyhow.

The first is merely the natural and instinctive reaction against any
disturbing intellectual innovation. It is a flat denial inspired by
that unconscious neophobia or xenophobia that possesses all of us
more or less. The second stage is the effort at compromise in which
usually both the advocates and opponents of the new idea coöperate by
endeavoring to prove that it is not so novel and unprecedented as was
at first assumed but fits in very fairly with our accepted notions, in
fact may be regarded as a supplement or even a natural development of
them. The third stage, like the second, is designed as an attempt to
disarm opposition by allaying alarm in the conservative mind.

The second line of argument has a good deal of validity, for even the
most startling and original idea will be found on closer examination
to have its roots deep in the ground of the past and to have been
approximately anticipated many times before. The third line of argument
also contains some truth for we find everyday life does go on in much
the same way, although it may seem that the foundations have been
knocked from under our mental, moral or social universe by some new
notion. Yet as the popular mind gradually accepts and adapts itself to
the novel conception we generally find that its influence is even more
far-reaching than was at first anticipated.

In the case of the Copernican theory it took about two centuries for
the controversy to pass through the three stages and the mind of the
public to become readjusted to the new conception of the earth’s
revolution. In the case of the Darwinian theory of evolution the
process was accomplished in about fifty years. The Einstein theory
is more subversive of ordinary ideas than either of the others so it
would naturally take longer to soak in. But the modern mind seems to
be subject to acceleration and we see in the two months since the
notion has been sprung upon the public that all three of the lines of
argument are appearing at once and so the controversial period may run
its course in five years though it will be longer before its indirect
influence upon our fundamental philosophy and habits of thought are
fully felt.


                     SCIENTIFIC VERSUS LEGAL LAWS

In all such discussions we must bear in mind that “law” in the
scientific sense of the word means, not a commandment or a rule, but
merely a way of working. It is a concise description of how things
behave. There are no laws _in_ Nature; there are only laws
_of_ Nature; that is to say, laws drawn out of Nature (or, if you
prefer Latin to Anglo-Saxon, laws deduced from Nature) by man for his
own convenience in thinking. Physical laws are therefore essentially
psychological; mere memory schemes, calculating machines. The law
of gravitation is no more gravity than the funny wriggles that my
stenographer is making in her notebook are the sounds I am uttering.
To change geometries does not require any such effort as to change
cars. It means merely changing our minds. But this is harder for some
of us than it ought to be. Here is where the theory of relativity
will be of use to us. Poincaré, the French mathematician, cousin of
the late President, said: “These two propositions, ‘the earth turns
round’ and ‘it is more convenient to suppose the earth turns round’
have the same meaning. There is nothing more in the one than in the
other.” If Galileo and his inquisitors had understood the Principle of
Relativity it might have saved them both trouble; the former temporary
imprisonment and the latter everlasting disgrace. A revolution in
science is simply a change in mental attitude. Maybe a political
revolution is no more.

It is disconcerting to the layman to be told, first, that matter
consists of solid round atoms in empty space; next, that it is made
of mere particles of electricity and negative at that; then that it
is constituted out of strains in the ether; again, that the atoms are
bubbles in the ether; and finally, that there is not any ether. But
these various hypotheses are like the crayon strokes that an artist
makes about a figure he is trying to draw. They are all attempts at
preliminary sketches for mental pictures of natural phenomena. We do
not call the geographers inconsistent and contradictory because one
colors Massachusetts red on the map and another colors it green. All
scientific hypotheses are put to the pragmatic test of which works the
best in unlocking the secrets of Nature. Is “wheat” or “sesame” the
magic word? Whether we call a dog “Fido” or “Towser” depends not on
which name is shorter or sounds better but on which the dog answers to.
If gravitation comes to heel better when we say “Einstein” than when
we say “Newton,” all right, we’ll change. I trust that these frivolous
illustrations will not lead my readers to accuse me of treating gravity
with levity.

The layman--and with him must be included all those who have merely
learned science but not used it--talks a great deal about “the laws
of Nature,” which he regards as abstract, immutable, universal and
eternal edicts, part of which are transcribed into the textbooks. To
the working scientist they are only more or less convenient formulas;
in the ultimate analysis only mnemonic symbols for stringing together
facts to make them easier to handle, like _vibgyor_, for the
spectrum colors. He knows that most of them are limited in their
scope and only approximate in their accuracy. His chief delight is in
discovering these limitations and irregularities. He regards these
“laws” with no awe or reverence. He has no attachment for any of
them--unless it happens to be one that he has formulated himself. If he
finds a new hypothesis that works better he throws the old one aside
as he does his old model dynamo, or keeps it around as handy still for
doing some of the common work of the laboratory. It is, to recur to
our example, just as “true,” using the word in its ordinary sense, to
say that the sun goes around the earth as to say that the earth goes
around the sun, for all motion is relative, and we can regard either
body as the stationary one or both as moving, as we choose. When we say
that the statement that the earth moves around the sun is the “true”
one, we merely mean that it is the more convenient form of expression,
for on this hypothesis the paths of the earth and the other planets
become circles (or more accurately speaking, irregular and eccentric
spirals) while on the other and older hypothesis their paths are very
complicated and difficult to handle mathematically. The theory that
the earth moves is not only simpler than that of a stationary earth,
but it is wider in its scope. It explains more, that is, it connects up
with other knowledge, such as the flattening at the poles. Copernicus,
then, did not discover a new fact about the solar system. He only
invented a lazier way of thinking about it.

The man of science invents an hypothesis whenever he needs one in his
business. It is to him merely a new tool, a _novum organum_. If
there is not an ether it would be necessary to create one. So he did
it. He had to have a noun for the verb “undulate.” When he had created
it he saw it was not good. The properties with which he endowed it were
self-contradictory, and it refused either to move with the earth or to
pass through it. But these theoretical inconsistencies do not bother
the physicist much. In spite of them the ether is a handy thing to have
about the laboratory. The scientist does not abandon a theory because
it has inconsistencies any more than he divorces his wife because she
has inconsistencies. Certainly the physicist did not consider himself
presumptuous in thus inventing ether for his own convenience. He knew
that the ordinary man had in the same way invented “matter” long ago
for his own convenience. It is a crude, inadequate and impossible idea,
this naïve conception of matter as something solid, heavy, hard,
inert, indestructible, impenetrable, colored and surfaced; but it is
good enough for part of the people all of the time and for all of the
people part of the time. The physicist himself uses it for everyday.
Only in his rigorous moments does he come down to bed-rock and say,
with Poincaré, “Mass is a co-efficient which it is convenient to
introduce into calculations.”

But when the physicist thus reduces matter to a small italic _m_
some people are sure to say that he is denying the existence of
matter. What would they say about Riemann who considers matter to be
holes in the ether? A definition is a different thing from a denial.
There are people among us who deny the existence of matter and they
call themselves “Scientists,” too, but they are not the ones who are
devoting their days and nights to the study of the workings of matter
in order to make it the servant of man.

A professor of chemistry would not think of asking his students if
the atomic theory is true any more than he would ask them if the
atomic theory is blue. He does not care whether they believe the
atomic theory or not. He only wants them to be able to use the atomic
theory for getting certain valuable results. Consequently, he watches
with interest and without apprehension the progress of discovery in
radio-activity which is undermining the old conception of the atom.
He would be glad to get rid of the atomic theory if he could find
something better because after all it is a clumsy thing and will not
hold half the facts he wants to put into it. He would have no more
hesitation about dropping it than he has in setting down one beaker
to pick up a larger one when what he has in the first is frothing
over. He does not want to spill anything, but he does not care what
vessel it is in. Revolutions in science never go backward and they
differ from political revolutions in that nothing worth saving is
lost in transition. The new theory must always include all that the
old one does and more. In their struggle for existence, formulas
fight like snakes; the one that can swallow the other beats. Now a
four-dimensional universe can take in a three-dimensional universe and
have space to spare for whatever the narrower conception could not
include so it seems likely to prevail.

We now know how to sympathize with those poor frightened people who
lived in the times of Copernicus and Galileo when they were told that
the solid earth on which they stood was not supported by anything,
but whirling about and rushing around through empty space and that
half the time they hung with their heads down over immeasurable space
with nothing to hold on to. But they got used to it in time and lived
happily ever after. So may we.

For the benefit of those who want to get their information at first
hand I append an article by Dr. Einstein himself which appeared in the
London _Times_ of December 13, 1919, and in _Science_ of
January 6, 1920:

                     TIME, SPACE, AND GRAVITATION

                       _By Dr. Albert Einstein_


 I respond with pleasure to your Correspondent’s request that I should
 write something for the _Times_ on the Theory of Relativity.

 After the lamentable breach in the former international relations
 existing among men of science, it is with joy and gratefulness that I
 accept this opportunity of communication with English astronomers and
 physicists. It was in accordance with the high and proud tradition of
 English science that English scientific men should have given their
 time and labor, and that English institutions should have provided the
 material means, to test a theory that had been completed and published
 in the country of their enemies in the midst of war. Although
 investigation of the influence of the solar gravitational field on
 rays of light is a purely objective matter, I am none the less very
 glad to express my personal thanks to my English colleagues in this
 branch of science; for without their aid I should not have obtained
 proof of the most vital deduction from my theory.

 There are several kinds of theory in Physics. Most of them are
 constructive. These attempt to build a picture of complex phenomena
 out of some relatively simple proposition. The kinetic theory of
 gases, for instance, attempts to refer to molecular movement the
 mechanical, thermal, and diffusional properties of gases. When we say
 that we understand a group of natural phenomena, we mean that we have
 found a constructive theory which embraces them.

 But in addition to this most weighty group of theories, there is
 another group consisting of what I call theories of principle. These
 employ the analytic, not the synthetic method. Their starting-point
 and foundation are not hypothetical constituents, but empirically
 observed general properties of phenomena, principles from which
 mathematical formulæ are deduced of such a kind that they apply to
 every case which presents itself. Thermodynamics, for instance,
 starting from the fact that perpetual motion never occurs in ordinary
 experience, attempts to deduce from this, by analytic processes, a
 theory which will apply in every case. The merit of constructive
 theories is their comprehensiveness, adaptability, and clarity, that
 of the theories of principle, their logical perfection, and the
 security of their foundation.

 The theory of relativity is a theory of principle. To understand it,
 the principles on which it rests must be grasped. But before stating
 these it is necessary to point out that the theory of relativity is
 like a house with two separate stories, the special relativity theory
 and the general theory of relativity.

 Since the time of the ancient Greeks it has been well known that in
 describing the motion of a body we must refer to another body. The
 motion of a railway train is described with reference to the ground,
 of a planet with reference to the total assemblage of visible fixed
 stars. In physics the bodies to which motions are spatially referred
 are termed systems of coördinates. The laws of mechanics of Galileo
 and Newton can be formulated only by using a system of coördinates.

 The state of motion of a system of coördinates cannot be chosen
 arbitrarily if the laws of mechanics are to hold good (it must be
 free from twisting and from acceleration). The system of coördinates
 employed in mechanics is called an inertia-system. The state of
 motion of an inertia-system, so far as mechanics are concerned, is
 not restricted by nature to one condition. The condition in the
 following proposition suffices: a system of coördinates moving in
 the same direction and at the same rate as a system of inertia
 is itself a system of inertia. The special relativity theory is
 therefore the application of the following proposition to any natural
 process:--“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.”

 The second principle on which the special relativity theory rests is
 that of the constancy of the velocity of light in a vacuum. Light
 in a vacuum has a definite and constant velocity, independent of
 the velocity of its source. Physicists owe their confidence in this
 proposition to the Maxwell-Lorentz theory of electro-dynamics.

 The two principles which I have mentioned have received strong
 experimental confirmation, but do not seem to be logically compatible.
 The special relativity theory achieved their logical reconciliation by
 making a change in kinematics, that is to say, in the doctrine of the
 physical laws of space and time. It became evident that a statement
 of the coincidence of two events could have a meaning only in
 connection with a system of coördinates, that the mass of bodies and
 the rate of movement of clocks must depend on their state of motion
 with regard to the coördinates.

 But the older physics, including the laws of motion of Galileo
 and Newton, clashed with the relativistic kinematics that I have
 indicated. The latter gave origin to certain generalized mathematical
 conditions with which the laws of nature would have to conform if
 the two fundamental principles were compatible. Physics had to be
 modified. The most notable change was a new law of motion for (very
 rapidly) moving mass-points, and this soon came to be verified in the
 case of electrically-laden particles. The most important result of
 the special relativity system concerned the inert mass of a material
 system. It became evident that the inertia of such a system must
 depend on its energy-content, so that we were driven to the conception
 that inert mass was nothing else than latent energy. The doctrine of
 the conservation of mass lost its independence and became merged in
 the doctrine of conservation of energy.

 The special relativity theory, which was simply a systematic extension
 of the electro-dynamics of Maxwell and Lorentz, had consequences which
 reached beyond itself. Must the independence of physical laws with
 regard to a system of coördinates be limited to systems of coördinates
 in uniform movement of translation with regard to one another? What
 has nature to do with the coördinate systems that we propose and with
 their motions? Although it may be necessary for our descriptions
 of nature to employ systems of coördinates that we have selected
 arbitrarily, the choice should not be limited in any way so far as
 their state of motion is concerned. (General theory of relativity.)
 The application of this general theory of relativity was found to
 be in conflict with a well-known experiment, according to which it
 appeared that the weight and the inertia of a body depended on the
 same constants (identity of inert and heavy masses). Consider the
 case of a system of coördinates which is conceived as being in stable
 rotation relative to a system of inertia in the Newtonian sense. The
 forces which, relatively to this system, are centrifugal must, in
 the Newtonian sense, be attributed to inertia. But these centrifugal
 forces are, like gravitation, proportional to the mass of the bodies.
 Is it not, then, possible to regard the system of coördinates as at
 rest, and the centrifugal forces as gravitational? The interpretation
 seemed obvious, but classical mechanics forbade it.

 This slight sketch indicates how a generalized theory of relativity
 must include the laws of gravitation, and actual pursuit of the
 conception has justified the hope. But the way was harder than was
 expected, because it contradicted Euclidean geometry. In other
 words, the laws according to which material bodies are arranged in
 space do not exactly agree with the laws of space prescribed by the
 Euclidean geometry of solids. This is what is meant by the phrase “a
 warp in space.” The fundamental concepts “straight,” “plane,” etc.,
 accordingly lose their exact meaning in physics.

 In the generalized theory of relativity, the doctrine of space and
 time, kinematics, is no longer one of the absolute foundations of
 general physics. The geometrical states of bodies and the rates of
 clocks depend in the first place on their gravitational fields, which
 again are produced by the material systems concerned.

 Thus the new theory of gravitation diverges widely from that of Newton
 with respect to its basal principle. But in practical application the
 two agree so closely that it has been difficult to find cases in which
 the actual differences could be subjected to observation. As yet only
 the following have been suggested:--

 1. The distortion of the oval orbits of planets round the sun
 (confirmed in the case of the planet Mercury).

 2. The deviation of light-rays in a gravitational field (confirmed by
 the English Solar Eclipse expedition).

 3. The shifting of spectral lines toward the red end of the spectrum
 in the case of light coming to us from stars of appreciable mass (not
 yet confirmed).

 The great attraction of the theory is its logical consistency. If
 any deduction from it should prove untenable, it must be given up. A
 modification of it seems impossible without destruction of the whole.

 No one must think that Newton’s great creation can be overthrown in
 any real sense by this or by any other theory. His clear and wide
 ideas will forever retain their significance as the foundation on
 which our modern conceptions of physics have been built.

 A final comment. The description of me and my circumstances in _The
 Times_ shows an amusing feat of imagination on the part of the
 writer. By an application of the theory of relativity to the taste of
 readers, today in Germany I am called a German man of science, and in
 England I am represented as a Swiss Jew. If I come to be regarded as
 a _bête noire_, the descriptions will be reversed, and I shall
 become a Swiss Jew for the Germans and a German man of science for the
 English!


                              FOOTNOTES:


[Footnote 1: Bergson: “Time and Free Will,” p. 221.]

[Footnote 2: Bergson in his “Laughter” traces all humor back to this
fundamental absurdity of making a man act mechanically.]

[Footnote 3: “We can thus say that all these paradoxical phenomena (or
rather negations of phenomena) which have been enumerated above can
only happen after the end or before the beginning of eternity” (De
Sitter).]

[Footnote 4: If you insist upon seeing just what is the difference
between Einstein’s and Newton’s laws of gravitation here it is as given
in _The Scientific Monthly_ of January, 1920:

Any particle or light pulse moves so that the integral of _ds_
between the two points of its path (in four dimensions) is stationary
where

                        (according to Einstein)
_ds^{2}_ = -(1 - 2_m_/_r_)^{-1}_dr^{2}_-_r^{2}_ _d θ^{2}_ + (1 - 2_m_/_r_)_dt_

                       or (according to Newton)
_ds^{2}_ = _dr^{2}_ - _r^{2}_ _d θ^{2}_ + (1 - 2_m_/_r_)_dt_

These expressions are in polar coördinates for a particle of
gravitational mass _m_.

The new factor introduced by Einstein is, as shown above,

1/{1-(2_m_/_r_)}
]

[Footnote 5: _Nineteenth Century_, December, 1919.]

[Footnote 6: Quoted by Eddington in _Contemporary Review_,
December, 1919.]

[Footnote 7: Sir Joseph Thomson in _Nature_, December 4, 1919.]




And finally


IF YOU WANT TO READ MORE ABOUT THE EINSTEIN THEORIES


  _For the non-mathematical reader_:

  ABBOTT, EDWIN.


  Flatland, by A Square. Boston, 1891.

  An amusing way of leading up to the fourth dimension.


  CAMPBELL, NORMAN.


  The Commonsense of Relativity. _Philosophical Magazine_,
  April, 1911.


  CARR, WILDON.


  The Metaphysical Implications of the Theory of Relativity.
  _Philosophical Review_, Jan., 1915.


  CARUS, PAUL.


  The Principle of Relativity. Chicago: Open Court Publishing
  Co., 1913.


  COMSTOCK, D. F.


  The Principle of Relativity. _Science_, May 20, 1910,
  vol. 31, p. 767.


  CUNNINGHAM, E.


  Einstein’s Relativity Theory of Gravitation. _Nature_,
  Dec. 4, 11, and 18, 1919.

  An interesting non-mathematical discussion of the latest
  phases of the theory.


  EDDINGTON, A. S.


  Einstein’s Theory of Space and Time. _Contemporary_
  _Review_, Dec, 1919.

  Good popular article.


  EDDINGTON, A. S.


  Gravitation. _Scientific American Supplement_, July 6 and
  13, 1918.

  An excellent popular explanation by the leading British
  disciple of Einstein.


  EINSTEIN, A.


  Time, Space, and Gravitation. _Science_, Garrison, 1920,
  Jan. 3, n.s., vol. 51, p. 8-10.

  My Theory. _Living Age._ Boston, 1920, vol. 304, p. 41-3,
  Jan. 3.


  FLAMMARION, CAMILLE.


  Lumen. New York: Dodd, Mead and Co., 1897.

  Contains nothing about Einstein but presents the relativity
  of time in fantastic form.
  KEYSER, C. J.


  Concerning the Figure and the Dimensions of the Universe
  of Space. _Science_, June 13, 1913.


  LODGE, SIR OLIVER.


  The New Theory of Gravity. _Nineteenth Century_, Dec.,
  1919.

  The Ether versus Relativity. _Fortnightly Review_, Jan.,
  1920.

  Admirable article by a courteous opponent.


  POINCARÉ, HENRI.


  Science and Method; also contained in The Foundations
  of Science. New York: The Science Press, 1913.


  RUSSELL, BERTRAND.


  The Relativity Theory of Gravitation. _English Review_,
  Dec., 1919.

  A clear explanation by one of the foremost of British
  philosophers.


  THOMSON, J.


  Deflection of Light by Gravitation and the Einstein
  Theory of Relativity. _Scientific Monthly_, Garrison, N. Y.,
  1920, vol. 10, p. 79-85, Jan.


  VARIOUS WRITERS.


  The Fourth Dimension Simply Explained. New York:
  Munn and Co., 1910.

  The essays submitted for a prize offered by the _Scientific_
  _American_. Twenty-two mathematicians try their best to
  justify the title and if they do not succeed it is not their
  fault.


  WETZEL, REINHARD A.


  The New Relativity in Physics. _Science_, New York,
  1913. New ser., vol. 38, pp. 466-474.

  Explains the relativity of time with diagrams and references
  to the literature.




  Other articles of general interest may be found in:


  _Science_: July 16, 1909; May 20, 1910; June 20, 1913;
  April 24, 1914; Dec. 5, 1919.

  _Scientific American Supplement_: April 7, 1917; Dec. 17,
  1910.

  London _Nation_: Nov. 14 and Dec. 27, 1919.

  London _Times_: Nov. 8, 18, and 25, Dec. 4 and 19,
  1919.

  London _Nature_: Almost every number in Nov., Dec., 1919,
  Jan., Feb., 1920.

  New York _Times_: Nov. 7 and 16, Dec. 21, 1919.

  New York _Sun_: Nov. 10, 1919.

  _The New Republic_: Jan. 21, 1920.


  _For the mathematical reader_:




  EINSTEIN, ALBERT.


  Bases physiques d’une théorie de la gravitation. Société
  Astronomique de France. _Bulletin_, Paris, 1917, Tome 31,
  pp. 407-411.

  Die formale Grundlage der allgemeinen Relativitätstheorie.
  Königlich Preussischen Akademie der Wissenschaften.
  _Sitzungsberichte_, Berlin, 1904 (Juli-Dez.), pp.
  1030-1085.

  Die Grundlagen der allgemeinen. Relativitätstheorie. _Annalen_
  _der Physik_, Leipzig, 1916, Band 49, Folge 4, p. 769.

  Ist die Trägheit eines Korpers von seinem Energieinhalt
  abhängig? _Annalen der Physik_, Leipzig, 1905, Band 18,
  Folge 4, pp. 639-644.

  Lichtgeschwindigkeit und Statik des Gravitätionsfeldes.
  _Annalen der Physik_, Leipzig, 1912, Band 38, Folge 4, pp.
  355-369.

  Prinzipielles zur allgemeinen Relativitätstheorie. _Annalen_
  _der Physik_, Leipzig, 1918, Band 55, Folge 4, pp. 241-244.

  Uber das Relativitätsprinzip und die aus demselben
  gezogenen Folgerungen. _Jahrbuch der Radioaktivität und_
  _Elektronik_, Leipzig, 1908, Band 4, pp. 411-462.

  Uber den Einfluss der Schwerkraft auf die Ausbreitung
  des Lichtes. _Annalen der Physik_, Lypzig, 1911. Band 35,
  pp. 898-908.

  Uber die Möglichkeit einer neuen Prüfung des Relativitätsprinzips.
  _Annalen der Physik_, Leipzig, 1907, Band 23,
  pp. 197-208.

  Uber die vom Relativitätsprinzip geforderte Trägheit der
  Energie. _Annalen der Physik_, Leipzig, 1907, Band 23,
  Folge 4, pp. 371-384.

  Uber einen die Erzeugung und Verwandlung des Lichtes
  betreffendes heuristischen Gesichtspunkt. _Annalen der_
  _Physik_, Leipzig, 1905, Band 17, Folge 4, pp. 132-148.

  Zum gegenwärtigen Stande des Gravitätionsproblems.
  _Physikalische Zeitschrift_, Leipzig, 1913, Band 14, pp. 1249-1266.

  Zum Relativitäts Problem. _Scientia_, Bologna, 1914, vol.
  15, pp. 337-48.

  Zur Elektrodynamik bewegter Korper. _Annalen der_
  _Physik_, Leipzig, 1905, Band 17, Folge 4, pp. 891-921.

  Zur Theorie der Lichterzeugung und Lichtabsorption.
  _Annalen der Physik_, Leipzig, 1906, Band 20, Folge 4, pp.
  199-206.

  Spielen Gravitationsfelder im Aufbau der materiellen
  Elementarteilchen eine wesentliche Rolle? _Sitz. Preuss._
  _Akad. Wiss._, April 10, 1919.

  EINSTEIN, ALBERT, and MARCEL GROSSMAN.


  Entwurf einer verallgemeinerten Relativitätstheorie und
  einer Theorie der Gravitation. _Zeitschrift für Mathematik_
  _und Physik_, Leipzig, 1914, Band 62, pp. 225-261.


  EINSTEIN, LORENTZ and MINKOWSKI.


  Relativitätsprinzip, Das. Eine Sammlung von Abhandlungen.
  Mit Anmerkungen von A. Sommerfeld und Vorwort
  von O. Blumenthal. Leipzig: B. G. Teubner, 1913,
  89 p. (_Fortschritte der mathematischen Wissenschaften in_
  _Monographien_, Heft 2.)

  A reprint of these three famous fundamental papers.




  ABRAHAM, M.


  Die neue Mechanik. _Scientia_, Bologna, 1914, vol. 15,
  pp. 8-27.

  Nochmals Relativität und Gravitation. Bemerkungen zu A.
  Einsteins Erwiderung. _Annalen der Physik_, Leipzig, 1912,
  Folge 4, vol. 39, pp. 444-448.


  BACKLUND, A. V.


  Zusammenstellung einer Theorie der klassischen Dynamik
  und der neuen Gravitätionstheorie von Einstein. _Arkiv för_
 _mathematik, astronomie, och fysik_, Stockholm, 1919, Band
  14, no. 11, 64 seite.


  BATEMAN, H.


  General Relativity Theory. _Phil. Mag._, Feb., 1909, vol.
  37.

  Applies the theory to life and mind.


  BRILLOUIN, MARCEL.


  Propos Sceptiques au Sujet du Principe de Relativité.
  _Scientia_, Bologna, 1913, vol. 13, pp. 10-26.


  BROSE, HENRY L.


  Einstein’s Theory of Relativity (non-math. form). Lecture
  published in pamphlet by B. H. Blackwell. Noted in
  _English Mechanic_, Dec. 19, 1919.


  CARMICHAEL, ROBERT DANIEL.


  The Theory of Relativity. _Mathematical Monographs_,
  no. 12. New York: John Wiley, 1913.


  COBB, CHARLES W.


  Relativity. _Journal of Philosophy, Psychology, and Scientific_
  _Method_, Jan. 18, 1917.


  CONWAY, A. W.


  Relativity. _Edin. Math. Tracts_, no. 3, London, 1913.


  CROMMELIN, A. C. D.


  Results of the Total Solar Eclipse of May 29 and the
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                                THANKS


Most of the mathematical references cited above have been borrowed
bodily from the book list prepared by Miss Mary E. Todd of the Science
Room of the New York Public Library and published in the _Library
Journal_. As soon as the Einstein craze struck New York these books
were placed on a long table and it has been difficult to find a seat
at this table, day or evening, ever since. At Cambridge University
when Professor Eddington lectured on the Einstein theory the students
waiting for the opening of the hall doors formed a cue extending
half-way across Trinity Great Court. It is unusual in any university to
have “standing room only” at a lecture on mathematical physics.


About half of the present volume appeared in _The Independent_
of November 29, December 7, 13, and 20, 1919, and I am indebted to
Hamilton Holt, the editor, and to Karl V. S. Howland, the publisher
of that magazine, for the privilege of reprinting them in book form.
I am further grateful to several professors of physics, mathematics,
astronomy and philosophy who have been kind enough to criticize and
correct this material, but since it would not be fair to hold them
responsible for my personal views and unconventional language I shall
have to express my thanks to them in private.




                                 INDEX


 Bergson, Henri, 47, 51, 52


 Carlyle, iv

 Curvature of space, 26, 35, 38, 96


 Eclipse observations, 2, 65, 72

 Einstein, Albert
   Postulates of, 13, 83, 111
   Theory of time, 109
   Theory of gravitation, 39, 58, 76, 80, 86, 93, 96, 109
   Life of, 78
   Works of, 116, 117

 Equivalence, principle of, 14, 83, 86

 Ether, 8, 11, 88, 106


 Flammarion, Camille, 40

 Flatland, 35, 57

 Fourth dimension, 24, 34, 49


 Gravitation, 68, 74, 77, 80, 86, 90


 Laws, natural, 102, 109

 Light, deflection of ray, 11, 62, 79, 85

 Light, pressure of, 67, 93

 Lodge, Sir Oliver, 2, 68, 75, 87, 88

 Lorentz, 22, 111, 112, 118


 Macdonald, George, 34

 Maxwell, Clerk, 67, 111, 112

 Mercury, orbit of, 14, 66

 Michelson and Morley experiment, 11, 21, 22, 85

 Minkowski, iv, 18

 Mirror, images, 18

 Motion picture illustrations, 31, 44, 46, 50, 53


 Newton, 15, 23, 39, 66, 76, 79, 87, 90, 95

 Non-Euclidean geometry, 30, 48, 57, 112


 Planck, 98, 121

 Poincaré, H., 88, 107


 Relativity of measurements, 15, 20, 26

 Relativity of motion, 4, 8, 15, 23

 Relativity of time, 15, 24, 47, 51


 Slade, 58, 60

 Spectrum, shifting of lines of, 14, 77, 85


 Theories, reception of new, 98, 101

 Theory of relativity, 2, 15

 Thomson, Sir Joseph, 3, 93, 94, 95

 Time as fourth dimension, 30, 32, 49

 Time, reversal of, 41, 44


 Wells, H. G., 32, 34, 40, 52


 Zöllner, 58, 60




                         =TRANSCRIBER’S NOTES=


Simple typographical errors have been silently corrected; unbalanced
quotation marks were remedied when the change was obvious, and
otherwise left unbalanced.

Punctuation and spelling were made consistent when a predominant
preference was found in the original book; otherwise they were not
changed.

Inconsistent hyphens left as printed.



*** END OF THE PROJECT GUTENBERG EBOOK 75807 ***