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.