of refinement that the path from mirror M via mirrors 1, 2, 3, etc.,
back through M and into the telescope, were exactly the same length
as that from flame to telescope by way of the mirrors I, II, III,
etc.--exactly the same to a margin of error materially less than a
single wave-length of light--why, then, the two sets of interference
fringes would come out exactly superposed provided the motion of
the earth through the "ether" turn out to have no influence upon the
velocity of light; or, if such influence exist, these fringes would
be displaced from one another to an extent measuring the influence in
question. But our ability to set up this complicated pattern of mirrors
at predetermined distances falls far short of the wave-length as a
measure of error. So in practice all that we can say is that having
once set the instrument up, and passed a beam of light through it,
there will be produced two sets of parallel interference fringes. These
sets will fail of superposition--each fringe of one set will be removed
from the corresponding fringe of the other set--by some definite
distance. Then, any subsequent variation in the speed of light along
the two arms will at once be detected by a shifting of the interference
bands through a distance which we shall be able to measure.
THE VERDICT
Under the theories and assumptions governing at the time of the
original performance of this experiment, it will be readily seen that
if this machine be set up in an "ether stream" with one arm parallel
to the direction of the stream and the other at right angles thereto,
there will be a difference in the speed of the light along the two
arms. Then if the apparatus be shifted to a position oblique to the
ether stream, the excess velocity of the light in the one arm would
be diminished, and gradually come to zero at the 45-degree angle,
after which the light traveling along the other arm would assume the
greater speed. In making observations, therefore, the entire apparatus
was slowly rotated, the observers walking with it, so that changes
of the sort anticipated would be observed.
The investigators were, however, ignorant of the position in which the
apparatus ought to be set to insure that one of the arms lie across
the ether drift; and they were ignorant of the time of year at which
the earth's maximum velocity through the ether was to be looked for. In
particular, it is plain that if the solar system as a whole is moving
through the ether at a rate less than the earth's orbital velocity,
there is a point in our orbit where our velocity through the ether
and that around the sun just cancel out and leave us temporarily in
a state of "absolute rest." So it was anticipated that the experiment
might have to be repeated in many orientations of the machine and at
many seasons of the year in order to give a series of readings from
which the true motion of the earth through the ether might be deduced.
For those who have a little algebra the demonstration which Dr. Russell
gives on a subsequent page will be interesting as showing the
situation in perfectly general terms. It will be realized that the
more complicated arrangement of mirrors in the experiment as just
described is simply an eightfold repetition of the simple experiment
as outlined by Dr. Russell, and that it was done so for the mere
sake of multiplying by eight the distances travelled and hence the
difference in time and in phase.
And now for the grand climax. The experiment was repeated many times,
with the original and with other apparatus, indoors and outdoors, at
all seasons of the year, with variation of every condition that could
imaginably affect the result. The apparatus was ordinarily such that a
shift in the fringes of anywhere from one-tenth to one one-hundredth
of that which would have followed from any reasonable value for
the earth's motion through the ether would have been systematically
apparent. The result was uniformly negative. At all times and in all
directions the velocity of light past the earth-bound observer was
the same. The earth has no motion with reference to the ether!
[The amazing character of this result is not by any possibility to
be exaggerated.]* [According to one experiment the ether was carried
along by a rapidly moving body and according to another equally
well-planned and well-executed experiment a rapidly moving body did
not disturb the ether at all. This was the blind alley into which
science had been led.]232
THE "CONTRACTION" HYPOTHESIS
[Numerous efforts were made to explain the contradiction.]* [It
is indeed a very puzzling one, and it gave physicists no end of
trouble. However Lorentz and Fitzgerald finally put forward an
ingenious explanation, to the effect that the actual motion of
the earth through the ether is balanced, as far as the ability of
our measuring instruments is concerned, by a contraction of these
same instruments in the direction of their motion. This contraction
obviously cannot be observed directly because all bodies, including
the measuring instruments themselves (which after all are only
arbitrary guides), will suffer the contraction equally. According to
this theory, called the Lorentz-Fitzgerald contraction theory,]272
[all bodies in motion suffer such contraction of their length in the
direction of their motion;]283 [the contraction being made evident
by our inability to observe the absolute motion of the earth, which
it is assumed must exist.]272 [This would suffice to show why the
Michelson-Morley experiment gave a negative result, and would preserve
the concept of absolute motion with reference to the ether.]283
[This proposal of Lorentz and Fitzgerald loses its startling aspect
when we consider that all matter appears to be an electrical structure,
and that the dimensions of the electric and magnetic fields which
accompany the electrons of which it is constituted change with the
velocity of motion.]267 [The forces of cohesion which determine the
form of a rigid body are held to be electromagnetic in nature; the
contraction may be regarded as due to a change in the electromagnetic
forces between the molecules.]10 [As one writer has put it, the
orientation, in the electromagnetic medium, of a body depending for
its very existence upon electromagnetic forces is not necessarily a
matter of indifference.]*
[Granting the plausibility of all this, on the basis of an
electromagnetic theory of matter, it leaves us in an unsatisfactory
position. We are left with a fixed ether with reference to which
absolute motion has a meaning, but that motion remains undetected and
apparently undetectable. Further, if we on shore measure the length
of a moving ship, using a yard-stick which is stationary on shore,
we shall obtain one result. If we take our stick aboard it contracts,
and so we obtain a greater length for the ship. Not knowing our
"real" motion through the ether, we cannot say which is the "true"
length. Is it not, then, more satisfactory to discard all notion
of true length as an inherent quality of bodies, and, by regarding
length as the measure of a relation between a particular object and
a particular observer, to make one length as true as the other?]182
[The opponents of such a viewpoint contend that Michelson's result
was due to a fluke; some mysterious counterbalancing influence was
for some reason at work, concealing the result which should normally
have been expected. Einstein refuses to accept this explanation;]192
[he refuses to believe that all nature is in a contemptible conspiracy
to delude us.]*
[The Fitzgerald suggestion is further unsatisfactory because it
assumes all substances, of whatever density, to undergo the same
contraction; and above all for the reason that it sheds no light upon
other phenomena.]194 [It is indeed a very special explanation; that
is, it applies only to the particular experiment in question. And
indeed it is only one of many possible explanations. Einstein
conceived the notion that it might be infinitely more valuable to
take the most general explanation possible, and then try to find
from this its logical consequences. This "most general explanation"
is, of course, simply that it is impossible in any way whatever to
measure the absolute motion of a body in space.]272 [Accordingly
Einstein enunciated, first the Special Theory of Relativity, and
later the General Theory of Relativity. The special theory was so
called because it was, limited to uniform rectilinear and non-rotary
motions. The general theory, on the other hand, dealt not only with
uniform rectilinear motions, but with any arbitrary motion whatever.
TAKING THE BULL BY THE HORNS
The hypothesis of relativity asserts that there can be no such concept
as absolute position, absolute motion, absolute time; that space and
time are inter-dependent, not independent; that everything is relative
to something else. It thus accords with the philosophical notion of
the relativity of all knowledge.]283 [Knowledge is based, ultimately,
upon measurement; and clearly all measurement is relative, consisting
merely in the application of a standard to the magnitude measured. All
metric numbers are relative; dividing the unit multiplies the metric
number. Moreover, if measure and measured change proportionately,
the measuring number is unchanged. Should space with all its contents
swell in fixed ratio throughout, no measurement could detect this;
nor even should it pulse uniformly throughout. Furthermore, were
space and space-contents in any way systematically transformed (as by
reflection in curved mirrors) point for point, continuously, without
rending, no measurement could reveal this distortion; experience
would proceed undisturbed.]263
[Mark Twain said that the street in Damascus "which is called
straight," is so called because while it is not as straight as a
rainbow it is straighter than a corkscrew. This expresses the basic
idea of relativity--the idea of comparison. All our knowledge is
relative, not absolute. Things are big or little, long or short,
light or heavy, fast or slow, only by comparison. An atom may be
as large, compared to an electron, as is a cathedral compared to
a fly. The relativity theory of Einstein emphasizes two cases of
relative knowledge; our knowledge of time and space, and our knowledge
of motion.]216 [And in each case, instead of allowing the notions of
relativity to guide us only so far as it pleases us to follow them,
there abandoning them for ideas more in accord with what we find
it easy to take for granted, Einstein builds his structure on the
thesis that relativity must be admitted, must be followed out to the
bitter end, in spite of anything that it may do to our preconceived
notions. If relativity is to be admitted at all, it must be admitted
in toto; no matter what else it contradicts, we have no appeal from
its conclusions so long as it refrains from contradicting itself.]*
[The hypothesis of relativity was developed by Einstein through
a priori methods, not the more usual a posteriori ones. That is,
certain principles were enunciated as probably true, the consequences
of these were developed, and these deductions tested by comparison of
the predicted and the observed phenomena. It was in no sense attained
by the more usual procedure of observing groups of phenomena and
formulating a law or formula which would embrace them and correctly
describe the routine or sequence of phenomena.
The first principle thus enunciated is that it is impossible to
measure or detect absolute translatory motion through space, under
any circumstances or by any means. The second is that the velocity of
light in free space appears the same to all observers regardless of
the relative motion of the source of light and the observer. This
velocity is not affected by motion of the source toward or away
from the observer,]283 [if we may for the moment use this expression
with its implication of absolute motion.]* [But universal relativity
insists that motion of the source toward the observer is identical
with motion of the observer toward the source.]283
[It will be seen that we are at once on the horns of a dilemma. Either
we must give up relativity before we get fairly started on it, or we
must overturn the foundations of common sense by admitting that time
and space are so constituted that when we go to meet an advancing
light-impulse, or when we retreat from it, it still reaches us with
the same velocity as though we stood still waiting for it. We shall
find when we are through with our investigation that common sense is
at fault; that our fixed impression of the absurdity of the state of
affairs just outlined springs from a confusion between relativism
and absolutism which has heretofore dominated our thought and gone
unquestioned. The impression of absurdity will vanish when we have
resolved this confusion.]*
QUESTIONS OF COMMON SENSE
[But it is obvious from what has just been said that if we are to
adopt Einstein's theory, we must make very radical changes in some of
our fundamental notions, changes that seem in violent conflict with
common sense. It is unfortunate that many popularizers of relativity
have been more concerned to astonish their readers with incredible
paradoxes than to give an account such as would appeal to sound
judgment. Many of these paradoxes do not belong essentially to the
theory at all. There is nothing in the latter that an enlarged and
enlightened common sense would not readily endorse. But common sense
must be educated up to the necessary level.]141
[There was a time when it was believed, as a result of centuries of
experience, that the world was flat. This belief checked up with the
known facts, and it could be used as the basis for a system of science
which would account for things that had happened and that were to
happen. It was entirely sufficient for the time in which it prevailed.
Then one day a man arose to point out that all the known facts were
equally accounted for on the theory that the earth was a sphere. It
was in order for his contemporaries to admit this, to say that so far
as the facts in hand were concerned they could not tell whether the
earth was flat or round--that new facts would have to be sought that
would contradict one or the other hypothesis. Instead of this the
world laughed and insisted that the earth could not be round because
it was flat; that it could not be round because then the people would
fall off the other side.
But the field of experimentation widened, and men were able to observe
facts that had been hidden from them. Presently a man sailed west and
arrived east; and it became clear that in spite of previously accepted
"facts" to the contrary, the earth was really round. The previously
accepted "facts" were then revised to fit the newly discovered truth;
and finally a new system of science came into being, which accounted
for all the old facts and all the new ones.
At intervals this sort of thing has been repeated. A Galileo shows
that preconceived ideas with regard to the heavens are wrong, and
must be revised to accord with his newly promulgated principles. A
Newton does the same for physics--and people unlearn the "fact"
that motion has to be supported by continued application of force,
substituting the new idea that it actually requires force to stop a
moving body. A Harvey shows that the things which have been "known"
for generations about the human body are not so. A Lyell and a Darwin
force men to throw overboard the things they have always believed about
the way in which the earth and its creatures came into being. Every
science we possess has passed through one or more of these periods
of readjustment to new facts.
SHIFTING THE MENTAL GEARS
Now we are apt to lose sight of the true significance of this. It
is not alone our opinions that are altered; it is our fundamental
concepts. We get concepts wholly from our perceptions, making them
to fit those perceptions. Whenever a new vista is opened to our
perceptions, we find facts that we never could have suspected from the
restricted viewpoint. We must then actually alter our concepts to make
the new facts fit in with the greatest degree of harmony. And we must
not hesitate to undertake this alteration, through any feeling that
fundamental concepts are more sacred and less freely to be tampered
with than derived facts.]* [We do, to be sure, want fundamental
concepts that are easy for a human mind to conceive; but we also
want our laws of nature to be simple. If the laws begin to become,
intricate, why not reshape, somewhat, the fundamental concepts, in
order to simplify the scientific laws? Ultimately it is the simplicity
of the scientific system as a whole that is our principal aim.]178
[As a fair example, see what the acceptance of the earth's sphericity
did to the idea represented by the word "down." With a flat earth,
"down" is a single direction, the same throughout the universe; with
a round earth, "down" becomes merely the direction leading toward
the center of the particular heavenly body on which we happen to be
located. It is so with every concept we have. No matter how intrinsic
a part of nature and of our being a certain notion may seem, we can
never know that new facts will not develop which will show it to be a
mistaken one. Today we are merely confronted by a gigantic example of
this sort of thing. Einstein tells us that when velocities are attained
which have just now come within the range of our close investigation,
extraordinary things happen--things quite irreconcilable with our
present concepts of time and space and mass and dimension. We are
tempted to laugh at him, to tell him that the phenomena he suggests
are absurd because they contradict these concepts. Nothing could be
more rash than this.
When we consider the results which follow from physical velocities
comparable with that of light, we must confess that here are
conditions which have never before been carefully investigated. We
must be quite as well prepared to have these conditions reveal some
epoch-making fact as was Galileo when he turned the first telescope
upon the skies. And if this fact requires that we discard present
ideas of time and space and mass and dimension, we must be prepared
to do so quite as thoroughly as our medieval fathers had to discard
their notions of celestial "perfection" which demanded that there
be but seven major heavenly bodies and that everything center about
the earth as a common universal hub. We must be prepared to revise
our concepts of these or any other fundamentals quite as severely as
did the first philosopher who realized that "down" in London was not
parallel to "down" in Bagdad or on Mars.]*
[In all ordinary terrestrial matters we take the earth as a fixed
body, light as instantaneous. This is perfectly proper, for such
matters. But we carry our earth-acquired habits with us into the
celestial regions. Though we have no longer the earth to stand on,
yet we assume, as on the earth, that all measurements and movements
must be referred to some fixed body, and are only then valid. We cling
to our earth-bound notion that there is an absolute up-and-down,
back-and-forth, right-and-left, in space. We may admit that we can
never find it, but we still think it is there, and seek to approach
it as nearly as possible. And similarly from our earth experiences,
which are sufficiently in a single place to make possible this
simplifying assumption, we get the idea that there is one universal
time, applicable at once to the entire universe.]141 [The difficulty
in accepting Einstein is entirely the difficulty in getting away from
these earth-bound habits of thought.]*
IV
THE SPECIAL THEORY OF RELATIVITY
What Einstein's Study of Uniform Motion Tells Us About Time and Space
and the Nature of the External Reality
BY VARIOUS CONTRIBUTORS AND THE EDITOR
Whatever the explanation adopted for the negative result of the
Michelson-Morley experiment, one thing stands out clearly: the attempt
to isolate absolute motion has again failed.]* [Einstein generalizes
this with all the other and older negative results of similar sort into
a negative deduction to the effect that no experiment is possible upon
two systems which will determine that one of them is in motion and
the other at rest.]121 [He elevates the repeated failure to detect
absolute motion through space into the principle that experiment
will never reveal anything in the nature of absolute velocities. He
postulates that all laws of nature can and should be enunciated in
such forms that they are as true in these forms for one observer
as for another, even though these observers with their frames of
reference be in motion relative to one another.]264
[There are various ways of stating the principle of the relativity
of uniform motion which has been thus arrived at, and which forms
the basis of the Special Theory of Einstein. If we care to emphasize
the rôle of mathematics and the reference frame we may say that]*
[any coordinate system having a uniform rectilinear motion with
respect to the bodies under observation may be interchangeably used
with any other such system in describing their motions;]232 [or that
the unaccelerated motion of a system of reference cannot be detected
by observations made on this system alone.]194 [Or we can let this
aspect of the matter go, and state the relativity postulate in a form
more intelligible to the non-mathematician by simply insisting that
it is impossible by any means whatever to distinguish any other than
the relative motion between two systems that are moving uniformly. As
Dr. Russell puts it on a later page, we can assume boldly that the
universe is so constituted that uniform straight-ahead motion of an
observer and all his apparatus will not produce any difference whatever
in the result of any physical process or experiment of any kind.
As we have seen, this is entirely reasonable, on philosophical grounds,
until we come to consider the assumptions of the past century with
regard to light and its propagation. On the basis of these assumptions
we had expected the Michelson-Morley experiment to produce a result
negativing the notion of universal relativity. It refused to do this,
and we agree with Einstein that the best explanation is to return
to the notion of relativity, rather than to invent a forced and
special hypothesis to account for the experiment's failure. But we
must now investigate the assumptions underlying the theory of light,
and remove the one that requires the ether to serve as a universal
standard of absolute motion.
LIGHT AND THE ETHER
It is among the possibilities that the wave theory of light itself
will in the end be more or less seriously modified. It is even more
definitely among the possibilities that the ether will be discarded.]*
[Certainly when Lord Kelvin estimates that its mass per cubic
centimeter is .000,000,000,000,000,001 gram, while Sir Oliver Lodge
insists that the correct figure is 1,000,000,000,000,000 grams, it is
quite evident that we know so little about it that it is better to
get along without it if we can.]216 [But to avoid confusion we must
emphasize that Einstein makes no mention whatsoever of the ether;
his theory is absolutely independent of any theory of the ether.]139
[Save as he forbids us to employ the ether as a standard of absolute
motion, Einstein does not in the least care what qualities we assign
to it, or whether we retain it at all. His demands are going to be made
upon light itself, not upon the alleged medium of light transmission.
When two observers in relative motion to one another measure their
velocities with respect to a third material object, they expect to get
different results. Their velocities with regard to this object properly
differ, for it is no more to be taken as a universal super-observer
than either of them. But if they get different results when they come
to measure the velocity with which light passes their respective
systems, relativity is challenged. Light is with some propriety
to be regarded as a universal observer; and if it will measure our
velocities against each other we cannot deny it rank as an absolute
standard. If we are not prepared to abandon universal relativity,
and adopt one of the "fluke" explanations for the Michelson-Morley
result, we must boldly postulate that in free space light presents the
same velocity $C$ to all observers--whatever the source of the light,
whatever the relative motion between source and observer, whatever the
relative motion between the several observers. The departure here from
the old assumption lies in the circumstance that the old physics with
its ether assigned to light a velocity universally constant in this
ether; we have stopped talking about the medium and have made the
constant $C$ refer to the observer's measured value of the velocity
of light with regard to himself.
We are fortified in this assumption by the Michelson-Morley
result and by all other observations bearing directly upon the
matter. Nevertheless, as Mr. Francis says in his essay, we feel
instinctively that space and time are not so constituted as to
make it possible, if I pass you at 100 miles per hour, for the
same light-impulse to pass us both at the same speed $C$.]* [The
implicit assumptions underlying this feeling, be they true or false,
are now so interwoven with the commonly received notions of space and
time that any theory which questions them has all the appearance of
a fantastic and unthinkable thing.]115 [We cannot, however, go back
on our relativity; so when]* [Einstein shows us that an entirely new
set of time and space concepts is necessary to reconcile universe
relativity with this fundamental fact of the absolute constancy of
the observed velocity of light in vacuo,]18 [all that is left for us
to do is to inquire what revisions are necessary, and submit to them.]*
[The conceptual difficulties of the theory arise principally from
attributing to space and time the properties of things. No portion of
space can be compared with another, save by convention; it is things
which we compare. No interval of time can be compared with another,
save by convention. The first has gone when the second becomes
"now".]149 [It is events that we compare, through the intervention
of things. Our measurements are never of space or of time, but only
of the things and the events that occupy space and time. And since
the measurements which we deal with as though they were of space and
of time lie at the foundation of all physical science, while at the
same time themselves constituting, as we have seen, the only reality
of which we are entitled to speak, it is in order to examine with
the utmost care the assumptions underlying them. That there are such
assumptions is clear--the very possibility of making measurements is
itself an assumption, and every technique for carrying them out rests
on an assumption. Let us inquire which of these it is that relativity
asks us to revise.]*
THE MEASUREMENT OF TIME AND SPACE
[Time is generally conceived as perfectly uniform. How do we judge
about it? What tells us that the second just elapsed is equal to the
one following? By the very nature of time the superposition of its
successive intervals is impossible. How then can we talk about the
relative duration of these intervals? It is clear that any relationship
between them can only be conventional.]178 [As a matter of fact, we
habitually measure time in terms of moving bodies. The simplest method
is to agree that some entity moves with uniform velocity. It will be
considered as travelling equal distances in equal intervals of time,
the distances to be measured as may be specified by our assumptions
governing this department of investigation.]179 [The motions of the
earth through which we ultimately define the length of day and year,
the division of the former into 86,400 "equal" intervals as defined by
the motions of pendulum or balance wheel through equal distances, are
examples of this convention of time measurement. Even when we correct
the motions of the earth, on the basis of what our clocks tell us of
these motions, we are following this lead; the earth and the clocks
fall out, it is plain that one of them does not satisfy our assumption
of equal lengths in equal times, and we decide to believe the clock.]*
[The foregoing concerning time may be accepted as inherent in
time itself. But concerning lengths it may be thought that we
are able to verify absolutely their equality and especially their
invariability. Let us have the audacity to verify this statement. We
have two lengths, in the shape of two rods, which coincide perfectly
when brought together. What may we conclude from this coincidence? Only
that the two rods so considered have equal lengths at the same
place in space and at the same moment. It may very well be that each
rod has a different length at different locations in space and at
different times; that their equality is purely a local matter. Such
changes could never be detected if they affected all objects in the
universe. We cannot even ascertain that both rods remain straight
when we transport them to another location, for both can very well
take the same curvature and we shall have no means of detecting it.
Euclidean geometry assumes that geometrical objects have sizes and
shapes independent of position and of orientation in space, and
equally invariable in time. But the properties thus presupposed are
only conventional and in no way subject to direct verification. We
cannot even ascertain space to be independent of time, because when
comparing geometrical objects we have to conceive them as brought to
the same place in space and in time.]178 [Even the statement that
when they are made to coincide their lengths are equal is, after
all, itself an assumption inherent in our ideas of what constitutes
length. And certainly the notion that we can shift them from place
to place and from moment to moment, for purposes of comparison, is
an assumption; even Euclid, loose as he was from modern standards
in this business of "axioms," knew this and included a superposition
axiom among his assumptions.
As a matter of fact, this procedure for determining equality of lengths
is not always available. It assumes, it will be noted, that we have
free access to the object which is to be measured--which is to say,
it assumes that this object is at rest with respect to us. If it
is not so at rest, we must employ at least a modification of this
method; a modification that will in some manner involve the sending
of signals. Even when we employ the Euclidean method of superposition
directly, we must be assured that the respective ends of the lengths
under comparison coincide at the same time. The observer cannot be
present at both ends simultaneously; at best he can only be present
at one end and receive a signal from the other end.
THE PROBLEM OF COMMUNICATION
Accordingly, in making the necessary assumptions to cover the matter
of measuring lengths, we must make one with regard to the character
of the signals which are to be employed for this purpose. If we
could assume a system of signalling that would consume no time in
transmission all would be simple enough. But we have no experience with
such a system. Even if we believe that it ought to be possible thus
to transmit signals at infinite velocity, we may not, in the absence
of our present ability to do this, assume that it is possible. So we
may only assume, with Einstein, that for our signals we shall employ
the speediest messenger with which we are at present acquainted. This
of course is light, the term including any of the electromagnetic
impulses that travel at the speed $C$.
Of course in the vast majority of cases the distance that any light
signal in which we are interested must go to reach us is so small that
the time taken by its transmission can by no means be measured. We
are then, to all intents and purposes, at both places--the point of
origin of the signal and the point of receipt--simultaneously. But
this is not the question at all. Waiving the fact that in astronomical
investigations this approximation no longer holds, the fact remains
that it is, in every case, merely an approximation. Approximations are
all right in observations, where we know that they are approximations
and act accordingly. But in the conceptual universe that parallels the
external reality, computation is as good an agent of observation as
visual or auditory or tactile sensation; if we can compute the error
involved in a wrong procedure the error is there, regardless of whether
we can see it or not. We must have methods which are conceptually
free from error; and if we attempt to ignore the velocity of our
light signals we do not meet this condition.
The measurement of lengths demands that we have a criterion of
simultaneity between two remote points--remote in inches or remote
in light-years, it does not matter which. There is no difficulty in
defining simultaneity of two events that fall in the same point--or
rather, in agreeing that we know what we mean by such simultaneity. But
with regard to two events that occur in remote places there may be a
question. A scientific definition differs from a mere description in
that it must afford us a means of testing whether a given item comes
under the definition or not. There is some difficulty in setting up
a definition of simultaneity between distant events that satisfies
this requirement. If we try simply to fall back upon our inherent
ideas of what we mean by "the same instant" we see that this is not
adequate. We must lay down a procedure for determining whether two
events at remote points occur at "the same instant," and check up
alleged simultaneity by means of this procedure.
Einstein says, and we must agree with him, that he can find but one
reasonable definition to cover this ground. An observer can tell
whether he is located half way between two points of his observation;
he can have mirrors set up at these points, send out light-signals,
and note the time at which he gets back the reflection. He knows
that the velocity of both signals, going and coming, is the same;
if he observes that they return to him together so that their time of
transit for the round trip is the same, he must accept the distances as
equal. He is then at the mid-point of the line joining the two points
under observation; and he may define simultaneity as follows, without
introducing anything new or indeterminate: Two events are simultaneous
if an observer midway between them sees them at the same instant, by
means, of course, of light originating at the points of occurrence.]*
[It is this definition of simultaneity, coupled with the assumption
that all observers, on whatever uniformly moving systems, would obtain
the same experimental value for the velocity of light, that leads to
the apparent paradoxes of the Special Theory of Relativity. If it be
asked why we adopt it, we must in turn ask the inquirer to propose
a better system for defining simultaneous events on different moving
bodies.]198
[There is nothing in this definition to indicate, directly, whether
simultaneity persists for all observers, or whether it is relative,
so that events simultaneous to one observer are not so to another. The
question must then be investigated; and the answer, of course, will
hinge upon the possibility of making proper allowances for the time of
transit of the light signals that may be involved. It seems as though
this ought to be possible; but a simple experiment will indicate that
it is not, unless the observers involved are at rest with respect to
one another.
AN EINSTEINIAN EXPERIMENT
Let us imagine an indefinitely long, straight railroad track, with
an observer located somewhere along it at the point $M$. According to
the convention suggested above, he has determined points $A$ and $B$
in opposite directions from him along the track, and equally distant
from him. We shall imagine, further, than a beneficent Providence
supplies two lightning flashes, one striking at $A$ and one at $B$,
in such a way that observer $M$ finds them to be simultaneous.
While all this is going on, a train is passing--a very long train,
amply long enough to overlap the section $AMB$ of the track. Among
the passengers there is one, whom we may call $M'$, who is directly
opposite $M$ at the instant when, according to $M$, the lightning
strikes. Observe he is not opposite $M$ when $M$ sees the flashes,
but a brief time earlier--at the instant when, according to $M$'s
computation, the simultaneous flashes occurred. At this instant there
are definitely determined the points $A'$ and $B'$, on the train;
and since we may quite well think of the two systems--train-system
and track-system--as in coincidence at this instant, $M'$ is midway
between $A'$ and $B'$, and likewise is midway between $A$ and $B$.
Now if we think of the train as moving over the track in the direction
of the arrow, we see very easily that $M'$ is running away from the
light from $A$ and toward that from $B$, and that, despite--or if
you prefer because of--the uniform velocity of these light signals,
the one from $B$ reaches him, over a slightly shorter course, sooner
than the one from $A$, over the slightly longer course. When the light
signals reach $M$, $M'$ is no longer abreast of him but has moved along
a wee bit, so that at this instant when $M$ has the two signals, one
of these has passed $M'$ and the other has yet to reach him. The upshot
is that the events which were simultaneous to $M$ are not so to $M'$.
It will probably be felt that this result is due to our having,
somewhat unjustifiably and inconsistently, localized on the train
the relative motion between train and track. But if we think of
the track as sliding back under the train in the direction opposite
to the arrow, and carrying with it the points $A$ and $B$; and if
we remember that this in no way affects $M$'s observed velocity of
light or the distances $AM$ and $BM$ as he observes them: we can
still accept his claim that the flashes were simultaneous. Then we
have again the same situation: when the flashes from $A$ and from
$B$ reach $M$ at the same moment, in his new position a trifle to
the left of his initial position of the diagram, the flash from $A$
has not yet reached $M'$ in his original position while that from
$B$ has passed him. Regardless of what assumption we make concerning
the motion between train-system and track-system, or more elegantly
regardless of what coordinate system we use to define that motion,
the event at $B$ precedes that at $A$ in the observation of $M'$. If
we introduce a second train moving on the other track in the opposite
direction, the observer on it will of course find that the flash at $A$
precedes that at $B$--a disagreement not merely as to simultaneity but
actually as to the order of two events! If we conceive the lightning
as striking at the points $A'$ and $B'$ on the train, these points
travel with $M'$ instead of with $M$; they are fixed to his coordinate
system instead of to the other. If you carry out the argument now,
you will find that when the flashes are simultaneous to $M'$, the
one at $A$ precedes that at $B$ in $M$'s observation.
A large number of experiments more or less similar in outline to
this one can be set up to demonstrate the consequences, with regard
to measured values of time and space, of relative motion between two
observers. I do not believe that a multiplicity of such demonstrations
contributes to the intelligibility of the subject, and it is for
this reason that I have cut loose from immediate dependence upon the
essayists in this part of the discussion, concentrating upon the single
experiment to which Einstein himself gives the place of importance.
WHO IS RIGHT?
We may permit Mr. Francis to remind us here that neither $M$ nor
$M'$ may correct his observation to make it accord with the other
fellow's. The one who does this is admitting that the other is at
absolute rest and that he is himself in absolute motion; and this
cannot be. They are simply in disagreement as to the simultaneity of
two events, just as two observers might be in disagreement about the
distance or the direction of a single event. This can mean nothing
else than that, under the assumptions we have made, simultaneity is
not an absolute characteristic as we had supposed it to be, but, like
distance and direction, is in fact merely a relation between observer
and objective, and therefore depends upon the particular observer
who happens to be operating and upon the reference frame he is using.
But this is serious. My time measurements depend ultimately upon
my space measurements; the latter, and hence both, depend closely
upon my ideas of simultaneity. Yours depend upon your reading of
simultaneity in precisely the same way.]* [Suppose the observer on
the track, in the above experiment, wants to measure the length of
something on the car, or the observer on the car something on the
track. The observer, or his assistant, must be at both ends of the
length to be measured at the same time, or get simultaneous reports
in some way from these ends; else they will obtain false results. It
is plain, then, that with different criteria of what the "same time"
is, the observers in the two systems may get different values for
the measured lengths in question.]220
[Who is right? According to the principle of relativity a decision on
this question is absolutely impossible. Both parties are right from
their own points of view; and we must admit that two events in two
different places may be simultaneous for certain observers, and yet
not simultaneous for other observers who move with respect to the first
ones. There is no contradiction in this statement, although it is not
in accordance with common opinion, which believes simultaneousness to
be something absolute. But this common opinion lacks foundation. It
cannot be proved by direct perception, for simultaneity of events
can be perceived directly,]24 [and in a manner involving none of our
arbitrary assumptions,]* [only if they happen at the same place;
if the events are distant from each other, their simultaneity or
succession can be stated only through some method of communicating by
signals. There is no logical reason why such a method should not lead
to different results for observers who move with regard to one another.
From what we have said, it follows immediately that in the new theory
not only the concept of simultaneousness but also that of duration is
revealed as dependent on the motion of the observer.]24 [Demonstration
of this should be superfluous; it ought to be plain without argument
that if two observers cannot agree whether two instants are the same
instant or not, they cannot agree on the interval of time between
instants. In the very example which we have already examined, one
observer says that a certain time-interval is zero, and another gives
it a value different from zero. The same thing happens whenever the
observers are in relative motion.]* [Two physicists who measure the
duration of a physical process will not obtain the same result if
they are in relative motion with regard to one another.
They will also find different results for the length of a body. An
observer who wants to measure the length of a body which is moving
past him must in one way or another hold a measuring rod parallel to
its motion and mark those points on his rod with which the ends of
the body come into simultaneous coincidence. The distance between the
two marks will then indicate the length of the body. But if the two
markings are simultaneous for one observer, they will not be so for
another one who moves with a different velocity, or who is at rest,
with regard to the body under observation. He will have to ascribe
a different length to it. And there will be no sense in asking which
of them is right: length is a purely relative concept, just as well
as duration.]24
THE RELATIVITY OF TIME AND SPACE
[The degree to which distance and time become relative instead of
absolute quantities under the Special Theory of Relativity can be
stated very definitely. In the first place, we must point out that the
relativity of lengths applies with full force only to lengths that lie
parallel to the direction of relative motion. Those that lie exactly
perpendicular to that direction come out the same for both observers;
those that lie obliquely to it show an effect, depending upon the
angle, which of course becomes greater and greater as the direction
of parallelism is approached.
The magnitude of the effect is easily demonstrated, but with this
demonstration we do not need to be concerned here. It turns out
that if an observer moving with a system finds that a certain time
interval in the system is $T$ seconds and that a certain length in
the system is $L$ inches, then an observer moving parallel with $L$
and with a velocity $v$ relative to the system will find for these
the respective values $T \div K$ and $L \times K$, where
$$K = \sqrt{1 - v^2/C^2}.$$
$C$ in this expression of course represents the velocity of light. It
will be noted that the fraction $v^2/C^2$ is ordinarily very small;
that the expression under the radical is therefore less than 1 but
by a very slight margin; and that the entire expression $K$ is itself
therefore less than 1 but by an even slighter margin. This means, then,
that the observer outside the system finds the lengths in the system
to be a wee bit shorter and the time intervals a wee bit longer than
does the observer in the system. Another way of putting the matter
is based, ultimately, upon the fact that in order for the observer
in the system to get the larger value for distance and the smaller
value for time, his measuring rod must go into the distance under
measurement more times than that of the moving observer, while his
clock must beat a longer second in order that less of them shall be
recorded in a given interval between two events. So it is often said
that the measuring rod as observed from without is contracted and
the clock runs slow. This does not impress me as a happy statement,
either in form or in content.]*
[The argument that these formulae are contradicted by human experience
can be refuted by examining a concrete instance. If a train is
1,000 feet long at rest, how long will it be when running a mile a
minute?]232 [I have quoted this question exactly as it appears in the
essay from which it is taken, because it is such a capital example of
the objectionable way in which this business is customarily put. For
the statement that lengths decrease and time-intervals increase "with
velocity" is not true in just this form. The velocity, to have meaning,
must be relative to some external system; and it is the observations
from that external system that are affected. So long as we confine
ourselves to the system in which the alleged modifications of size
are stated as having taken place, there is nothing to observe that
is any different from what is usual; there is no way to establish
that we are enjoying a velocity, and in fact within the intent of
the relativity theory we are not enjoying a velocity, for we are
moving with the objects which we are observing. It is inter-systemic
observations, and these alone, that show the effect. When we travel
with the system under observation, we get the same results as any
other observer on this system; when we do not so travel, we must
conduct our observations from our own system, in relative motion to
the other, and refer our results to our system.
Now when no particular observer is specified, we must of course
assume an observer connected with the train, or with whatever the body
mentioned. To that observer it doesn't make the slightest difference
what the train does; it may stand at rest with respect to some external
system or it may move at any velocity whatsoever; its length remains
always 1,000 feet. In order for this question to have the significance
which its propounder means it to have, I must restate it as follows:
A train is 1,000 feet long as measured by an observer travelling with
it. If it passes a second observer at 60 miles per hour, what is its
length as observed by him? The answer is now easy.]* [According to the
formula the length of the moving train as seen from the ground will be
$$1,000 \times \sqrt{1 - (88)^2 / (186,000 \times 5,280)^2} =
999.999,999,999,996$$
feet, a change entirely too small for detection by the most delicate
instruments. Examination of the expression K shows that in so far
as terrestrial movements of material objects are concerned it is
equal to 1]232 [within a far smaller margin than we can ever hope to
make our observations. Even the diameter of the earth, as many of the
essayists point out, will be shortened only 2 1/2 inches for an outside
observer past whom it rushes with its orbital speed of 18.5 miles per
second. But slight as the difference may be in these familiar cases,
its scientific importance remains the same.]*
RELATIVITY AND REALITY
[A simple computation shows that this effect is exactly the amount
suggested by Lorentz and Fitzgerald to explain the Michelson-Morley
experiment.]188 [This ought not to surprise us, since both that
explanation and the present one are got up with the same purpose. If
they both achieve that purpose they must, numerically, come to the
same thing in any numerical case. It is, however, most emphatically
to be insisted that the present "shortening" of lengths]* [no longer
appears as a "physical" shortening caused by absolute motion through
the ether but is simply a result of our methods of measuring space
and time.]188 [Where Fitzgerald and Lorentz had assumed that a body in
motion has its dimensions shortened in the direction of its motion,]220
[this very form of statement ceases to possess significance under
the relativity assumption.]* [For if we cannot tell which of two
bodies is moving, which one is shortened? The answer is, both--for
the other fellow. For each frame of reference there is a scale of
length and a scale of time, and these scales for different frames
are related in a manner involving both the length and the time.]220
[But we must not yield to the temptation to say that all this is not
real; the confinement of a certain scale of length and of time to
a single observation system does not in the least make it unreal.]*
[The situation is real--as real as any other physical event.]165
[The word physical is used in two senses in the above paragraph. It
is denied that the observed variability in lengths indicates any
"physical" contraction or shrinkage; and on the heels of this it
is asserted that this observed variability is of itself an actual
"physical" event. It is difficult to express in words the distinction
between the two senses in which the term physical is employed in these
two statements, but I think this distinction ought to be clear once
its existence is emphasized. There is no material contraction; it is
not right to say that objects in motion contract or are shorter; they
are not shorter to an observer in motion with them. The whole thing is
a phenomenon of observation. The definitions which we are obliged to
lay down and the assumptions which we are obliged to make in order,
first, that we shall be able to measure at all, and second, that we
shall be able to escape the inadmissible concept of absolute motion,
are such that certain realities which we had supposed ought to be the
same for all observers turn out not to be the same for observers who
are in relative motion with respect to one another. We have found this
out, and we have found out the numerical relation which holds between
the reality of the one observer and that of the other. We have found
that this relation depends upon nothing save the relative velocity
of the two observers. As good a way of emphasizing this as any is to
point out that two observers who have the same velocity with respect
to the system under examination (and whose mutual relative velocity
is therefore zero) will always get the same results when measuring
lengths and times on that system. The object does not go through any
process of contraction; it is simply shorter because it is observed
from a station with respect to which it is moving. Similar remarks
might be made about the time effect; but the time-interval is not so
easily visualized as a concrete thing and hence does not offer such
temptation for loose statement.
The purely relative aspect of the matter is further brought out if
we consider a single example both backwards and forwards. Systems $S$
and $S'$ are in relative motion. An object in $S$ which to an observer
in $S$ is $L$ units long, is shorter for an observer in $S'$--shorter
by an amount indicated through the "correction factor" $K$. Now if
we have, in the first instance, made the objectionable statement
that objects are shorter in system $S'$ than they are in $S$, it
will be quite natural for us to infer from this that objects in $S$
must be longer than those in $S'$; and from this to assert that when
the observer in $S$ measures objects lying in $S'$, he gets for them
greater lengths than does the home observer in $S'$. But if we have,
in the first instance, avoided the objectionable statement referred
to, we shall be much better able to realize that the whole business is
quite reciprocal; that the phenomena are symmetric with respect to the
two systems, to the extent that we can interchange the systems in any
of our statements without modifying the statements in any other way.
Objects in $S$ appear shorter and times in $S$ appear longer to the
external "moving" observer in $S'$ than they do to the domestic
observer in $S$. Exactly in the same way, objects in $S'$ appear
shorter to observers in the foreign system $S$ than to the home
observer in $S'$, who remains at rest with respect to them. I think
that when we get the right angle upon this situation, it loses the
alleged startling character which has been imposed upon it by many
writers. The "apparent size" of the astronomer is an analogy in
point. Objects on the moon, by virtue of their great distance, look
smaller to observers on the earth than to observers on the moon. Do
objects on the earth, on this account, look larger to a moon observer
than they do to us? They do not; any suggestion that they do we should
receive with appropriate scorn. The variation in size introduced by
distance is reciprocal, and this reciprocity does not in the least
puzzle us. Why, then, should that introduced by relative motion
puzzle us?
TIME AND SPACE IN A SINGLE PACKAGE
Our old, accustomed concepts of time and space, which have grown
up through countless generations of our ancestors, and been handed
down to us in the form in which we are familiar with them, leave no
room for a condition where time intervals and space intervals are
not universally fixed and invariant. They leave no room for us to
say that]* [one cannot know the time until he knows where he is,
nor where he is until he knows the time,]220 [nor either time or
place until he knows something about velocity. But in this concise
formulation of the difference between what we have always believed
and what we have seen to be among the consequences of Einstein's
postulates of the universal relativity of uniform motion, we may
at once locate the assumption which, underlying all the old ideas,
is the root of all the trouble. The fact is we have always supposed
time and space to be absolutely distinct and independent entities.]*
[The concept of time has ever been one of the most absolute of all
the categories. It is true that there is much of the mysterious about
time; and philosophers have spent much effort trying to clear up
the mystery--with unsatisfactory results. However, to most persons
it has seemed possible to adopt an arbitrary measure or unit of
duration and to say that this is absolute, independent of the state
of the body or bodies on which it is used for practical purposes.]272
[Time has thus been regarded as something which of itself flows on
regularly and continuously, regardless of physical events concerning
matter.]150 [In other words, according to this view, time is not
affected by conditions or motions in space.]272 [We have deliberately
chosen to ignore the obvious fact that time can never appear to us,
be measured by us, or have the least significance for us, save as
a measure of something that is closely tied up with space and with
material space-dimensions. Not merely have we supposed that time and
space are separated in nature as in our easiest perceptions, but we
have supposed that they are of such fundamentally distinct character
that they can never be tied up together. In no way whatever, assumes
the Euclidean and Newtonian intellect, may space ever depend upon
time or time upon space. This is the assumption which we must remove
in order to attain universal relativity; and while it may come hard,
it will not come so hard as the alternative. For this alternative is
nothing other than to abandon universal relativity. This course would
leave us with logical contradictions and discrepancies that could not
be resolved by any revision of fundamental concepts or by any cleaning
out of the Augean stables of old assumptions; whereas the relativity
doctrine as built up by Einstein requires only such a cleaning out in
order to leave us with a strictly logical and consistent whole. The
rôle of Hercules is a very difficult one for us to play. Einstein
has played it for the race at large, but each of us must follow him
in playing it for himself.
SOME FURTHER CONSEQUENCES
I need not trespass upon the subject matter of those essays which
appear in full by going here into any details with regard to the
manner in which time and space are finally found to depend upon one
another and to form the parts of a single universal whole. But I
may appropriately point out that if time and space are found to be
relative, we may surely expect some of the less fundamental concepts
that depend upon them to be relative also. In this expectation we are
not disappointed. For one thing,]* [mass has always been assumed to
be a constant, independent of any motion or energy which it might
possess. Just as lengths and times depend upon relative motion,
however, it is found that mass, which is the remaining factor in
the expression for energy due to motion, also depends upon relative
velocities. The dependence is such that if a body takes up an amount
of energy $E$ with respect to a certain system, the body behaves,
to measurements made from that system, as though its mass had been
increased by an amount $E/C^2$, where $C$ is as usual the velocity
of light.]194
[This should not startle us. The key to the situation lies in the
italicized words above, which indicate that the answer to the query
whether a body has taken up energy or not depends upon the seat of
observation. If I take up my location on the system $S$, and you
on the system $S'$, and if we find that we are in relative motion,
we must make some assumption about the energy which was necessary,
initially, to get us into this condition. Suppose we are on two
passing trains.]* [The chances are that either of us will assume that
he is at rest and that it is the other train which moves, although
if sufficiently sophisticated one of us may assume that he is moving
and that the other train is at rest.]272 [Whatever our assumption,
whatever the system, the localization of the energy that is carried
in latent form by our systems depends upon this assumption. Indeed,
if our systems are of differing mass, our assumptions will even govern
our ideas of the amount of energy which is represented by our relative
motion; if your system be the more massive, more energy would have to
be localized in it than in mine to produce our relative motion. If
we did not have the universal principle of relativity to forbid,
we might make an arbitrary assumption about our motions and hence
about our respective latent energies; in the presence of this veto,
the only chance of adjustment lies in our masses, which must differ
according to whether you or I observe them.]*
[For most of the velocities with which we are familiar $E/C^2$ is,
like the difference between $K$ and unity, such an extremely small
quantity that the most delicate measurements fail to detect it. But
the electrons in a highly evacuated tube and the particles shot
out from radioactive materials attain in some cases velocities as
high as eight-tenths that of light. When we measure the mass of such
particles at different velocities we find that it actually increases
with the velocity, and in accordance with the foregoing law.]194
[This observation, in fact, antedates Einstein's explanation, which
is far more satisfactory than the earlier differentiation between
"normal mass" and "electrical mass" which was called upon to account
for the increase.]*
[But if the quantity $E/C^2$ is to be considered as an actual increase
in mass, may it not be possible that all mass is energy? This would
lead to the conclusion that the energy stored up in any mass is
$mC^2$. The value is very great, since $C$ is so large; but it is in
good agreement with the internal energy of the atom as calculated
from other considerations. It is obvious that conservation of mass
and of momentum cannot both hold good under a theory that translates
the one into the other. Mass is then not considered by Einstein as
conservative in the ordinary sense, but it is the total quantity of
mass plus energy in any closed system that remains constant. Small
amounts of energy may be transformed into mass, and vice versa.]194
[Other features of the theory which are often displayed as consequences
are really more in the nature of assumptions. It will be recalled that
when we had agreed upon the necessity of employing signals of some
sort, we selected as the means of signalling the speediest messenger
with which we happened to be acquainted. Our subsequent difficulties
were largely due to the impossibility of making a proper allowance
for this messenger's speed, even though we knew its numerical value;
and as a consequence, this speed enters into our formulae. Now we have
not said in so many words that $C$ is the greatest speed attainable,
but we have tacitly assumed that it is. We need not, therefore, be
surprised if our formulae give us absurd results for speeds higher than
$C$, and indicate the impossibility of ever attaining these. Whatever
we put into a problem the algebra is bound to give us back. If we look
at our formula for $K$, we see that in the event of $v$ equalling $C$,
lengths become zero and times infinite. The light messenger itself,
then, has no dimension; and for it time stands still.
If we suppose $v$ to be greater than $C$, we get even more bizarre
results, for then the factor $K$ is the square root of a negative
number, or as the mathematician calls it an "imaginary" quantity;
and with it, lengths and times become imaginary too.
The fact that time stops for it, and the fact that it is the limiting
velocity, give to $C$ certain of the attributes of the mathematician's
infinity. Certainly if it can never be exceeded, we must have a new
formula for the composition of velocities. Otherwise when my system
passes yours at a speed of 100,000 miles per second, while yours
passes a third in the same direction at the same velocity, I shall
be passing this third framework at the forbidden velocity of 200,000
miles per second--greater than $C$. In fact Einstein is able to show
that an old formula, which had already been found to connect the speed
of light in a material medium with the speed of that medium, will now
serve universally for the composition of velocities. When we combine
the velocities $v$ and $u$, instead of getting the resultant $v+u$
as we would have supposed, we get the resultant
$$(v + u) / (1 + (uv / C^2))\,,$$ or $$C^2(v + u) / (C^2 + uv)$$
This need not surprise us either, if we will but reflect that
the second velocity effects a second revision of length and time
measurements between the systems involved. And now, if we let either
$v$, or $u$, or even both of them, take the value $C$, the resultant
still is $C$. In another way we have found $C$ to behave like the
mathematician's infinity, to which, in the words of the blind poet,
if we add untold thousands, we effect no real increment.
ASSUMPTION AND CONSEQUENCE
A good many correspondents who have given the subject sufficient
thought to realize that the limiting character of the velocity $C$ is
really read into Einstein's system by assumption have written, in more
or less perturbed inquiry, to know whether this does not invalidate
the whole structure. The answer, of course, is yes--provided you can
show this assumption to be invalid. The same answer may be made of
any scientific doctrine whatever, and in reference to any one of the
multitudinous assumptions underlying it. If we were to discover,
tomorrow, a way of sending signals absolutely instantaneously,
Einstein's whole structure would collapse as soon as we had agreed to
use this new method. If we were to discover a signalling agent with
finite velocity greater than that of light, relativity would persist
with this velocity written in its formulae in the place of $C$.
It is a mistake to quote Einstein's theory in support of the
statement that such a velocity can never be. An assumption proves its
consequences, but never can prove itself; it must remain always an
assumption. But in the presence of long human experience supporting
Einstein's assumption that no velocity in excess of $C$ can be found,
it is fair to demand that it be disputed not with argument but with
demonstration. The one line of argument that would hold out a priori
hope of reducing the assumption to an absurdity would be one based
on the familiar idea of adding velocities; but Einstein has spiked
this argument before it is started by replacing the direct addition
of velocities with another method of combining them that fits in
with his assumption and as well with the observed facts. The burden
of proof is then on the prosecution; anyone who would contradict our
assertion that $C$ is the greatest velocity attainable may do so only
by showing us a greater one. Until this has been done, the admission
that it may properly be attempted can in no way be construed as a
confession of weakness on the part of Einstein.
It may be well to point out that in no event may analogy be drawn with
sound, as many have tried to do. In the first place sound requires
a material medium and its velocity with regard to this rather than
relative to the observer we know to be fixed; in the second place,
requiring a material medium, sound is not a universal signalling
agent; in the third place, we know definitely that its velocity can
be exceeded, and are therefore barred from making the assumption
necessary to establish the analogy. The very extraordinary behavior
of light in presenting a velocity that is the same for all observers,
and in refusing to betray the least material evidence of any medium
for its transmission, rather fortifies us in believing that Einstein's
assumption regarding the ultimate character of this velocity is in
accord with the nature of things.
RELATIVITY AND THE LAYMAN
A great deal can be said in the direction of general comment making the
Special Theory and its surprising accompaniments easier of acceptance,
and we shall conclude the present discussion by saying some of these
things.]* [It has been objected that the various effects catalogued
above are only apparent, due to the finite velocity of light--that
the real shape and size of a body or the real time of an event cannot
be affected by the point of view or the motion of an observer. This
argument would be perfectly valid, if there were real times and
distances; but there are not. These are earth-bound notions, due to
our experience on an apparently motionless platform, with slow-moving
bodies. Under these circumstances different observations of the same
thing or of the same event agree. But when we no longer have the solid
earth to stand on, and are dealing with velocities so high that the
relativity effects become appreciable, there is no standard by which
to resolve the disagreements. No one of the observations can claim
to be nearer reality than any other. To demand the real size of a
thing is to demand a stationary observer or an instantaneous means
of information. Both are impossible.
When relativity asks us to give up our earth-bound notions of absolute
space and absolute time the sensation, at first, is that we have
nothing left to stand on. So must the contemporaries of Columbus have
felt when told that the earth rested on--nothing. The remedy too is
similar. Just as they had to be taught that falling is a local affair,
that the earth is self-contained, and needs no external support--so we
must be taught that space and time standards are local affairs. Each
moving body carries its own space and time standards with it; it is
self-contained. It does not need to reach out for eternal support,
for an absolute space and time that can never quite be attained. All
we ever need to know is the relation of the other fellow's space
and time standards to our own. This is the first thing relativity
teaches us.]141
[The consequences of Einstein's assumptions have led many to reject the
theory of relativity, on the ground that its conclusions are contrary
to common sense--as they undoubtedly are. But to the contemporaries
of Copernicus and Galileo the theory that the earth rotates on its
axis and revolves around the sun was contrary to common sense; yet
this theory prevailed. There is nothing sacred about common sense;
in the last analysis its judgments are based on the accumulated
experience of the human race. From the beginning of the world up to
the present generation, no bodies were known whose velocities were
not extremely small compared with that of light. The development of
modern physics has led to discovery of very much larger velocities,
some as high as 165,000 miles per second. It is not to be wondered at
that such an enlargement of our experience requires a corresponding
enlargement or generalization of the concepts of space and time. Just
as the presupposition of primitive man that the earth was flat had
to be given up in the light of advancing knowledge, so we are now
called upon to give up our presupposition that space and time are
absolute and independent in their nature.
The reader must not expect to understand the theory of relativity in
the sense of making it fit in with his previous ideas. If the theory
be right these ideas are wrong and must be modified, a process apt
to be painful.]223 [All the reader can do is to become familiar with
the new concepts, just as a child gets used to the simple relations
and quantities he meets until he "understands" them.]221 [Mr. Francis
has said something of the utmost significance when he points out that
"understanding" really means nothing in the world except familiarity
and accustomedness.]* [The one thing about the relativity doctrine
that we can hope thus to understand at once and without pain is the
logical process used in arriving at our results.]221 [Particularly is
it hard to give a satisfactory explanation of the theory in popular
language, because the language itself is based on the old concepts;
the only language which is really adequate is that of mathematics.]223
[Unless we have, in addition to the terms of our ordinary knowledge,
a set of definitions that comes with a wide knowledge of mathematics
and a lively sense of the reality of mathematical constructions, we
are likely to view the theory of relativity through a fog of familiar
terms suddenly become self-contradictory and deceptive. Not that we
are unfamiliar with the idea that some of our habitual notions may
be wrong; but knowledge of their illusory nature arises and becomes
convincing only with time. We may now be ready to grant that the earth,
seemingly so solid, is really a whirling globe rushing through space;
but we are no more ready immediately to accept the bald assertion that
this space is not what it seems than our ancestors were to accept the
idea that the earth was round or that it moved.]156 [What we must have,
if we are to comprehend relativity with any degree of thoroughness,
is the mathematician's attitude toward his assumptions, and his
complete readiness to swap one set of assumptions for another as a
mere part of the day's work, the spirit of which I have endeavored
to convey in the chapter on non-Euclidean geometry.]*
PHYSICS VS. METAPHYSICS
[The ideas of relativity may seem, at first sight, to be giving us
a new and metaphysical theory of time and space. New, doubtless; but
certainly the theory was meant by its author to be quite the opposite
of metaphysical. Our actual perception of space is by measurement,
real and imagined, of distances between objects, just as our actual
perception of time is by measurement. Is it not less metaphysical
to accept space and time as our measurements present them to us,
than to invent hypotheses to force our perceptual space into an
absolute space that is forever hidden from us?]182 [In order not to
be metaphysical, we must eliminate our preconceived notions of space
and time and motion, and focus our attention upon the indications
of our instruments of observation, as affording the only objective
manifestations of these qualities and therefore the only attributes
which we can consider as functions of observed phenomena.]47 [Einstein
has consistently followed the teachings of experience, and completely
freed himself from metaphysics.]114 [That this is not always easy
to do is clear, I think, if we will recall the highly metaphysical
character often taken by the objections to action-at-a-distance
theories and concepts; and if we will remind ourselves that it was
on purely metaphysical grounds that Newton refused to countenance
Huyghens' wave theory of light. Whether, as in the one case, it
leads us to valid conclusions, or, as in the other, to false ones,
metaphysical reasoning is something to avoid. Einstein, I think,
has avoided it about as thoroughly as anyone ever did.]*
V
THAT PARALLEL POSTULATE
Modern Geometric Methods; the Dividing Line Between Euclidean and
Non-Euclidean; and the Significance of the Latter
BY THE EDITOR
The science of geometry has undergone a revolution of which the
outsider is not informed, but which it is necessary to understand
if we are to attain any comprehension of the geometric formulation
of Einstein's results; and especially if we are to appreciate why
it is proper and desirable to formulate these results geometrically
at all. The classical geometer regarded his science from a narrow
viewpoint, as the study of a certain set of observed phenomena--those
of the space about us, considered as an entity in itself and divorced
from everything in it. It is clear that some things about that space
are not as they appear (optical illusions), and that other things
about it are true but by no means apparent (the sum-of-squares
property of a right triangle, the formulæ for surface and volume
of a sphere, etc.). While many things about space are "obvious,"
these need in the one case disproof and in the other discovery and
proof. With all their love of mental processes for their own sake,
it is then not surprising that the Greeks should have set themselves
the task of proving by logical process the properties of space,
which a less thoughtful folk would have regarded as a subject only
for observational and experimental determination.
But, abstract or concrete, the logical structure must have a starting
point; and it is fair to demand that this consist in a statement
of the terms we are going to use and the meanings we are going to
attach to them. In other words, the first thing on the program will
be a definition, or more probably, several definitions.
Now the modern scientist has a somewhat iconoclastic viewpoint toward
definitions, and especially toward the definition of his very most
fundamental ideas.
We do not speak here in terms of dictionary definitions. These have
for object the eminent necessity of explaining the meaning and use
of a word to some one who has just met it for the first time. It
is easy enough to do this, if the doer possesses a good command
of the language. It is not even a matter of grave concern that the
words used in the definition be themselves known to the reader; if
they are not, he must make their acquaintance too. Dr. Johnson's
celebrated definition of a needle stands as perpetual evidence
that when he cannot define a simple thing in terms of things still
simpler, the lexicographer is forced to define it in terms of things
more complex. Or we might demonstrate this by noting that the best
dictionaries are driven to define such words as "and" and "but"
by using such complex notions as are embodied in "connective,"
"continuative," "adversative," and "particle."
It is otherwise with the scientist who undertakes to lay down a
definition as the basis of further procedure in building up the
tissue of his science. Here a degree of rigorous logic is called for
which would be as superfluous in the dictionary as the effort there
to attain it would be out of place. The scientist, in building up
a logical structure that will withstand every assault, must define
everything, not in terms of something which he is more or less
warranted in supposing his audience to know about, but actually in
terms of things that have already been defined. This really means that
he must explain what he is talking about in terms of simpler ideas
and simpler things, which is precisely what the lexicographer does
not have to worry about. This is why it is quite trivial to quote a
dictionary definition of time or space or matter or force or motion
in settlement of a controversy of scientific or semi-scientific nature.
TERMS WE CANNOT DEFINE
But the scientist who attempts to carry out this ideal system of
defining everything in terms of what precedes meets one obstacle
which he cannot surmount directly. Even a layman can construct a
passable definition of a complex thing like a parallelopiped, in
terms of simpler concepts like point, line, plane and parallel. But
who shall define point in terms of something simpler and something
which precedes point in the formulation of geometry? The scientist is
embarrassed, not in handling the complicated later parts of his work,
but in the very beginning, in dealing with the simplest concepts with
which he has to deal.
Suppose a dictionary were to be compiled with the definitions
arranged in logical rather than alphabetical order: every word to be
defined by the use only of words that have already been defined. The
further back toward the beginning we push this project, the harder it
gets. Obviously we can never define the first word, or the second,
save as synonymous with the first. In fact we should need a dozen
words, more or less, to start with--God-given words which we cannot
define and shall not try to define, but of which we must agree that
we know the significance. Then we have tools for further procedure;
we can start with, say, the thirteenth word and define all the rest
of the words in the language, in strictly logical fashion.
What we have said about definitions applies equally to statements
of fact, of the sort which are going to constitute the body of our
science. In the absence of simpler facts to cite as authority, we shall
never be able to prove anything, however simple this may itself be;
and in fact the simpler it be, the harder it is to find something
simpler to underlie it. If we are to have a logical structure of any
sort, we must begin by laying down certain terms which we shall not
attempt to define, and certain statements which we shall not try to
prove. Mathematics, physics, chemistry--in the large and in all their
many minor fields--all these must start somewhere. Instead of deceiving
ourselves as to the circumstances surrounding their start, we prefer to
be quite frank in recognizing that they start where we decide to start
them. If we don't like one set of undefined terms as the foundation,
by all means let us try another. But always we must have such a set.
The classical geometer sensed the difficulty of defining his first
terms. But he supposed that he had met it when he defined these in
words free of technical significance. "A point is that which has
position without size" seemed to him an adequate definition, because
"position" and "size" are words of the ordinary language with which
we may all be assumed familiar. But today we feel that "position"
and "size" represent ideas that are not necessarily more fundamental
than those of "line" and "point," and that such a definition begs
the question. We get nowhere by replacing the undefined terms "point"
and "line" and "plane," which really everybody understands, by other
undefined terms which nobody understands any better.
In handling the facts that it was inconvenient to prove, the classical
geometer came closer to modern practice. He laid down at the beginning
a few statements which he called "axioms," and which he considered to
be so self-evident that demonstration was superfluous. That the term
"self-evident" left room for a vast amount of ambiguity appears to
have escaped him altogether. His axioms were axioms solely because
they were obviously true.
LAYING THE FOUNDATION
The modern geometer falls in with Euclid when he writes an elementary
text, satisfying the beginner's demand for apparent rigor by defining
point and line in some fashion. But when he addresses to his peers an
effort to clarify the foundations of geometry to a further degree of
rigor and lucidity than has ever before been attained, he meets these
difficulties from another quarter. In the first place he is always
in search of the utmost possible generality, for he has found this to
be his most effective tool, enabling him as it does to make a single
general statement take the place and do the work of many particular
statements. The classical geometer attained generality of a sort,
for all his statements were of any point or line or plane. But the
modern geometer, confronted with a relation that holds among points
or between points and lines, at once goes to speculating whether there
are not other elements among or between which it holds. The classical
geometer isn't interested in this question at all, because he is
seeking the absolute truth about the points and lines and planes which
he sees as the elements of space; to him it is actually an object so to
circumscribe his statements that they may by no possibility refer to
anything other than these elements. Whereas the modern geometer feels
that his primary concern is with the fabric of logical propositions
that he is building up, and not at all with the elements about which
those propositions revolve.
It is of obvious value if the mathematician can lay down a proposition
true of points, lines and planes. But he would much rather lay down
a proposition true at once of these and of numerous other things;
for such a proposition will group more phenomena under a single
principle. He feels that on pure scientific grounds there is quite
as much interest in any one set of elements to which his proposition
applies as there is in any other; that if any person is to confine
his attention to the set that stands for the physicist's space,
that person ought to be the physicist, not the geometer. If he has
produced a tool which the physicist can use, the physicist is welcome
to use it; but the geometer cannot understand why, on that ground,
he should be asked to confine his attention to the materials on which
the physicist employs that tool.
It will be alleged that points and lines and planes lie in the
mathematician's domain, and that the other things to which his
propositions may apply may not so lie--and especially that if he will
not name them in advance he cannot expect that they will so lie. But
the mathematician will not admit this. If mathematics is defined on
narrow grounds as the science of number, even the point and line
and plane may be excluded from its field. If any wider definition
be sought--and of course one must be--there is just one definition
that the mathematician will accept: Dr. Keyser's statement that
"mathematics is the art or science of rigorous thinking."
The immediate concern of this science is the means of rigorous
thinking--undefined terms and definitions, axioms and propositions. Its
collateral concern is the things to which these may apply, the things
which may be thought about rigorously--everything. But now the
mathematician's domain is so vastly extended that it becomes more
than ever important for him to attain the utmost generality in all
his pronouncements.
One barrier to such generalization is the very name "geometry,"
with the restricted significance which its derivation and long usage
carry. The geometer therefore must have it distinctly understood
that for him "geometry" means simply the process of deducing a set
of propositions from a set of undefined primitive terms and axioms;
and that when he speaks of "a geometry" he means some particular set
of propositions so deduced, together with the axioms, etc., on which
they are based. If you take a new set of axioms you get a new geometry.
The geometer will, if you insist, go on calling his undefined terms by
the familiar names "point," "line," "plane." But you must distinctly
understand that this is a concession to usage, and that you are not
for a moment to restrict the application of his statements in any
way. He would much prefer, however, to be allowed new names for his
elements, to say "We start with three elements of different sorts,
which we assume to exist, and to which we attach the names A, B and
C--or if you prefer, primary, secondary and tertiary elements--or
yet again, names possessing no intrinsic significance at all, such as
ching, chang and chung." He will then lay down whatever statements he
requires to serve the purposes of the ancient axioms, all of these
referring to some one or more of his elements. Then he is ready for
the serious business of proving that, all his hypotheses being granted,
his elements A, B and C, or I, II and III, or ching, chang and chung,
are subject to this and that and the other propositions.
The objection will be urged that the mathematician who does all
this usurps the place of the logician. A little reflection will
show this not to be the case. The logician in fact occupies the same
position with reference to the geometer that the geometer occupies
with reference to the physicist, the chemist, the arithmetician,
the engineer, or anybody else whose primary interest lies with some
particular set of elements to which the geometer's system applies. The
mathematician is the tool-maker of all science, but he does not make
his own tools--these the logician supplies. The logician in turn
never descends to the actual practice of rigorous thinking, save as
he must necessarily do this in laying down the general procedures
which govern rigorous thinking. He is interested in processes, not
in their application. He tells us that if a proposition is true its
converse may be true or false or ambiguous, but its contrapositive
is always true, while its negative is always false. But he never,
from a particular proposition "If A is B then C is D," draws the
particular contrapositive inference "If C is not D then A is not
B." That is the mathematician's business.
THE RÔLE OF GEOMETRY
The mathematician is the quantity-production man of science. In
his absence, the worker in each narrower field where the elements
under discussion take particular concrete forms could work out,
for himself, the propositions of the logical structure that applies
to those elements. But it would then be found that the engineer had
duplicated the work of the physicist, and so for many other cases;
for the whole trend of modern science is toward showing that the
same background of principles lies at the root of all things. So the
mathematician develops the fabric of propositions that follows from
this, that and the other group of assumptions, and does this without
in the least concerning himself as to the nature of the elements of
which these propositions may be true. He knows only that they are true
for any elements of which his assumptions are true, and that is all he
needs to know. Whenever the worker in some particular field finds that
a certain group of the geometer's assumptions is true for his elements,
the geometry of those elements is ready at hand for him to use.
Now it is all right purposely to avoid knowing what it is that we
are talking about, so that the names of these things shall constitute
mere blank forms which may be filled in, when and if we wish, by the
names of any things in the universe of which our "axioms" turn out
to be true. But what about these axioms themselves? When we lay them
down in ignorance of the identity of the elements to which they may
eventually apply, they cannot by any possibility be "self-evident." We
may, at pleasure, accept as self-evident a statement about points and
lines and planes; or one about electrons, centimeters and seconds;
or one about integers, fractions, and irrational numbers; or one about
any other concrete thing or things whatever. But we cannot accept as
self-evident a statement about chings, changs and chungs. So we must
base our "axioms" on some other ground than this; and our modern
geometer has his ground ready and waiting. He accepts his axioms
on the ground that it pleases him to do so. To avoid all suggestion
that they are supposed to be self-evident, or even necessarily true,
he drops the term "axiom" and substitutes for it the more color-less
word "postulate." A postulate is merely something that we agreed to
accept, for the time being, as a basis of further argument. If it
turns out to be true, or if we can find circumstances under which and
elements to which it applies, any conclusions which we deduce from it
by trustworthy processes are valid within the same limitations. And
the propositions which tell us that, if our postulates are true, such
and such conclusions are true--they, too, are valid, but without any
reservation at all!
Perhaps an illustration of just what this means will not be out of
place. Let it be admitted, as a postulate, that $7+20$ is greater, by
1, than $7+19$. Let us then consider the statement: "If $7 + 19 = 65$,
then $7 + 20 = 66$." We know--at least we are quite certain--that $7 +
19$ is not equal to 65, if by "7" and "19" and "65" we mean what you
think we mean. We are equally sure, on the same grounds, that $7 +
20$ is not equal to 66. But, under the one assumption that we have
permitted ourselves, it is unquestionable that if $7 + 19$ were equal
to 65, then $7 + 20$ certainly would be equal to 66. So, while the
conclusion of the proposition which I have put in quotation marks is
altogether false, the proposition itself, under our assumption, is
entirely true. I have taken an illustration designed to be striking
rather than to possess scientific interest; I could just as easily
have shown a true proposition leading to a false conclusion, but of
such sort that it would be of decided scientific interest as telling
us one of the consequences of a certain assumption.
WHAT MAY WE TAKE FOR GRANTED?
This is all very fine; but how does the geometer know what postulates
to lay down? One is tempted to say that he is at liberty to postulate
anything he pleases, and investigate the results; and that whether or
not his postulate ever be realized, the propositions that he deduces
from it, being true, are of scientific interest. Actually, however,
it is not quite as simple as all that. If it were sufficient to make
a single postulate it would be as simple as all that; but it turns
out that this is not sufficient any more than it is sufficient to
have a single undefined term. We must have several postulates; and
they must be such, as a whole, that a geometry flows out of them. The
requirements are three.
In the first place, the system of postulates must be "categorical"
or complete--there must be enough of them, and they must cover enough
ground, for the support of a complete system of geometry. In practice
the test for this is direct. If we got to a point in the building up
of a geometry where we could not prove whether a certain thing was
one way always, or always the other way, or sometimes one way and
sometimes the other, we should conclude that we needed an additional
postulate covering this ground directly or indirectly. And we should
make that postulate--because it is precisely the things that we can't
prove which, in practical work, we agree to assume. Even Euclid had
to adopt this philosophy.
In the second place, the system of postulates must be consistent--no
one or more of them may lead, individually or collectively, to
consequences that contradict the results or any other or others. If
in the course of building up a geometry we find we have proved
two propositions that deny one another, we search out the implied
contradiction in our postulates and remedy it.
Finally, the postulates ought to be independent. It should not be
possible to prove any one of them as a consequence of the others. If
this property fails, the geometry does not fail with it; but it is
seriously disfigured by the superfluity of assumptions, and one of
them should be eliminated. If we are to assume anything unnecessarily,
we may as well assume the whole geometry and be done with it.
The geometer's business then is to draw up a set of postulates. This
he may do on any basis whatever. They may be suggested to him by
the behavior of points, lines and planes, or by some other concrete
phenomena; they may with equal propriety be the product of an
inventive imagination. On proceeding to deduce their consequences,
he will discover and remedy any lack of categoricity or consistence
or independence which his original system of postulates may have
lacked. In the end he will have so large a body of propositions without
contradiction or failure that he will conclude the propriety of his
postulates to have been established, and the geometry based on them
to be a valid one.
AND WHAT IS IT ALL ABOUT?
Is this geometry ever realized? Strictly it is not the geometer's
business to ask or answer this question. But research develops two
viewpoints. There is always the man who indulges in the pursuit of
facts for their sake alone, and equally the man who wants to see his
new facts lead to something else. One great mathematician is quoted
as enunciating a new theory of surpassing mathematical beauty with
the climacteric remark "And, thank God, no one will ever be able to
find any use for it." An equally distinguished contemporary, on being
interrogated concerning possible applications for one of his most
abstruse theorems, replied that he knew no present use for it; but that
long experience had made him confident that the mathematician would
never develop any tool, however remote from immediate utility, for
which the delvers in other fields would not presently find some use.
If we wish, however, we may inquire with perfect propriety, from the
side lines, whether a given geometry is ever realized. We may learn
that so far as has yet been discovered there are no elements for
which all its postulates are verified, and that there is therefore no
realization known. On the other hand, we may more likely find that
many different sets of elements are such that the postulates can be
interpreted as applying to them, and that we therefore have numerous
realizations of the geometry. As a human being the geometer may be
interested in all this, but as a geometer it really makes little
difference to him.
When we look at space about us, we see it, for some reason grounded
in the psychological history of the human race, as made up in the
small of points, which go to make up lines, which in turn constitute
planes. Or we can start at the other end and break space down first
into planes, then into lines, finally into points. Our perceptions
and conceptions of these points, lines and planes are very definite
indeed; it seems indeed, as the Greeks thought, that certain things
about them are self-evident. If we wish to take these self-evident
properties of point, line and plane, and combine with them enough
additional hair-splitting specifications to assure the modern geometer
that we have really a categorical system of assumptions, we shall
have the basis of a perfectly good system of geometry. This will be
what we unavoidably think of as the absolute truth with regard to
the space about us; but you mustn't say so in the presence of the
geometer. It will also be what we call the Euclidean geometry. It
has been satisfactory in the last degree, because not only space,
but pretty much every other system of two or three elements bearing
any relations to one another can be made, by employing as a means
of interpretation the Cartesian scheme of plotting, to fit into the
framework of Euclidean geometry. But it is not the only thing in the
world of conceptual possibilities, and it begins to appear that it
may not even be the only thing in the world of cold hard fact that
surrounds us. To see just how this is so we must return to Euclid,
and survey the historical development of geometry from his day to
the present time.
EUCLID'S GEOMETRY
Point, line and plane Euclid attempts to define. Modern objection
to these efforts was made clear above. Against Euclid's specific
performance we urge the further specific fault that his "definitions"
are really assumptions bestowing certain properties upon points, lines
and planes. These assumptions Euclid supplements in his axioms; and in
the process of proving propositions he unconsciously supplements them
still further. This is to be expected from one whose justification for
laying down an axiom was the alleged obvious character of the statement
made. If some things are too obvious to require demonstration, others
may be admitted as too obvious to demand explicit statement at all.
Thus, if Euclid has two points A and B in a plane, on opposite sides of
a line M, he will draw the line AB and without further formality speak
of the point C in which it intersects M. That it does so intercept M,
rather than in some way dodges it, is really an assumption as to the
nature of lines and planes. Or again, Euclid will speak of a point D
on the line AB, between or outside the points A and B, without making
the formal assumption necessary to insure that the line is "full" of
points so that such a point as D must exist. That such assumptions as
these are necessary follows from our previous remarks. If we think of
our geometry as dealing with "chings," "changs," and "chungs," or with
elements I, II and III, it is no longer in the least degree obvious
that the simplest property in the world applies to these elements. If
we wish any property to prevail we must state it explicitly.
With the postulates embodied in his definitions, those stated in his
axioms, and those which he reads into his structure by his methods of
proof, Euclid has a categorical set--enough to serve as foundation
for a geometry. We may then climb into Euclid's shoes and take the
next step with him. We follow him while he proves a number of things
about intersecting lines and about triangles. To be sure, when he
proves that two triangles are identically constituted by moving one of
them over on top of the other, we may protest on the ground that the
admission of motion, especially of motion thus imposed from without,
into a geometry of things is not beyond dispute. If Euclid has caught
our modern viewpoint, he will rejoin that if we have any doubts as to
the admissibility of motion he will lay down a postulate admitting it,
and we shall be silenced.
Having exhausted for the present the interest of intersecting lines,
our guide now passes to a consideration of lines in the same plane
that never meet. He defines such lines as parallel. If we object that
he should show the existence of a derived concept like this before
laying down a definition that calls for it to exist, he can show that
two lines drawn perpendicular to the same line never meet. He will
execute this proof by a special sort of superposition, which requires
that the plane be folded over on itself, through the third dimension
of surrounding space, rather than merely slid along upon itself.
We remain quiet while Euclid demonstrates that if two lines are cut by
any transversal in such a way as to make corresponding angles at the
two intersections equal, the lines are parallel. It is then in order
to investigate the converse: if the lines are parallel to begin with,
are the angles equal?
AXIOMS MADE TO ORDER
This sounds innocent enough; but in no way was Euclid able to devise a
proof--or, for that matter, a disproof. So he took the only way out,
and said that if the lines were parallel, obviously they extended in
the same direction and made the angles equal. The thing was so obvious,
he argued, that it was really an axiom and he didn't have to prove
it; so he stated it as an axiom and proceeded. He didn't state it
in precisely the form I have used; he apparently cast about for the
form in which it would appear most obvious, and found a statement that
suited him better than this one, and that comes to the same thing. This
statement tells us that if the transversal makes two corresponding
angles unequal, the lines that it cuts are not parallel and do meet
if sufficiently prolonged. But wisely enough, he did not transplant
this axiom, once he had arrived at it, to the beginning of the book
where the other axioms were grouped; he left it right where it was,
following the proposition that if the angles were equal the lines were
parallel. This of course was so that it might appeal back, for its
claim to obviousness, to its demonstrated converse of the proposition.
Euclid must have been dissatisfied with this cutting of the Gordian
knot; his successors were acutely so. For twenty centuries the parallel
axiom was regarded as the one blemish in an otherwise perfect work;
every respectable mathematician had his shot at removing the defect by
"proving" the objectionable axiom. The procedure was always the same:
expunge the parallel axiom, in its place write another more or less
"obvious" assumption, and from this derive the parallel statement more
or less directly. Thus if we may assume that the sum of the angles
of a triangle is always exactly 180 degrees, or that there can be
drawn only one line through a given point parallel to a given line,
we can prove Euclid's axiom. Sometimes the substitute assumption was
openly made and stated, as in the two instances cited; as often it
was admitted into the demonstration implicitly, as when it is quietly
assumed that we can draw a triangle similar to any given triangle and
with area as great as we please, or when parallels are "defined" as
everywhere equidistant. But such "proofs" never satisfied anyone other
than the man who made them; the search went merrily on for a valid
"proof" that should not in substance assume the thing to be proved.
LOCATING THE DISCREPANCY
Saccheri, an Italian Jesuit, would have struck bottom if he had had a
little more imagination. He gave an exhaustive reductio ad absurdum,
on the basis of the angle-sum theorem. This sum must be (a) greater
than or (b) equal to or (c) less than 180 degrees. Saccheri showed
that if one of these alternatives occurs in a single triangle, it must
occur in every triangle. The first case gave little trouble; admitting
the possibility of superposing in the special manner mentioned above,
which he did implicitly, he showed that this "obtuse-angled hypothesis"
contradicted itself. He pursued the "acute-angled hypothesis" for
a long time before he satisfied himself that he had caught it, too,
in an inconsistency. This left only the "right-angled hypothesis,"
proving the Euclidean angle-sum theory and through it the parallel
postulate. But Saccheri was wrong: he had found no actual contradiction
in the acute-angled hypothesis--for none exists therein.
The full facts were probably first known to Gauss, who had a finger in
every mathematical pie that had to do with the transition to modern
times. They were first published by Lobatchewsky, the Russian, who
anticipated the Hungarian John Bolyai by a narrow margin. All three
worked independently of Saccheri, whose book, though theoretically
available in Italian libraries, was actually lost to sight and had
to be rediscovered in recent years.
Like Saccheri, Lobatchewsky investigated alternative possibilities. But
he chose another point of attack: through a given point it must be
possible to draw, in the same plane with a given line (a) no lines or
(b) one line or (c) a plurality of lines, which shall not meet the
given line. The word parallel is defined only in terms of the second
of these hypotheses, so we avoid it here. These three cases correspond,
respectively, to those of Saccheri.
The first case Lobatchewsky ruled out just as did Saccheri, but
accepting consciously the proviso attached to its elimination;
the third he could not rule out. He developed the consequences of
this hypothesis as far as Euclid develops those of the second one,
sketching in a full outline for a system of geometry and trigonometry
based on a plurality of "non-cutters." This geometry constitutes a
coherent whole, without a logical flaw.
This made it plain what was the matter with Euclid's parallel
axiom. Nobody could prove it from his other assumptions because it
is not a consequence of these. True or false, it is independent of
them. Trinity Church is in New York, Faneuil Hall is in Boston, but
Faneuil Hall is not in Boston because Trinity is in New York; and we
could not prove that Faneuil Hall was in Boston if we knew nothing
about America save that Trinity is in New York. The mathematicians
of 2,000 years had been pursuing, on a gigantic scale, a delusion of
post hoc, ergo propter hoc.
WHAT THE POSTULATE REALLY DOES
Moreover, in the absence of an assumption covering the ground, we
shall not know which of the alternatives (a), (b), (c) holds. But
when one holds in a single case it holds permanently, as Saccheri and
Lobatchewsky both showed. So we cannot proceed on this indefinite
basis; we must know which one is to hold. Without the parallel
postulate or a substitute therefor that shall tell us the same thing
or tell us something different, we have not got a categorical set of
assumptions--we cannot build a geometry at all. That is why Euclid
had to have his parallel postulate before he could proceed. That is
why his successors had to have an assumption equivalent to his.
The reason why it took so long for this to percolate into the
understanding of the mathematicians was that they were thinking,
not in terms of the modern geometry and about undefined elements;
but in terms of the old geometry and about strictly defined and
circumscribed elements. If we understand what is meant by Euclidean
line and plane, of course the parallel postulate, to use the old
geometer's word, is true--of course, to adopt the modern viewpoint,
if we agree to employ an element to which that assumption applies,
the assumption is realized. The very fact of accepting the "straight"
line and the "flat" plane of Euclid constitutes acceptance of his
parallel postulate--the only thing that can separate his geometry
from other geometries. But of course we can't prove it; the prior
postulates which we would have to use in such an attempt apply where
it does not apply, and hence it cannot possibly be consequences of.
To all this the classical Euclidean rejoins that we seem to have
in mind elements of some sort to which, with one reservation, his
postulates apply. He wants to know what these elements look like. We
can, and must, produce them--else our talk about generality is mere
drivel. But we must take care that the Euclidean geometer does not
try to apply to our elements the notions of straightness and flatness
which inhere in the parallel postulate. We cannot satisfy and defy
that postulate at the same time. If we do not insist on this point, we
shall find that we are reading non-Euclidean properties into Euclidean
geometry, and interpreting the elements of the latter as straight
lines that are not straight, flat planes that are not flat. It is not
the mission of non-Euclidean geometry thus to deny the possibility
of Euclidean geometry; it merely demands a place of equal honor.
THE GEOMETRY OF SURFACES
Let us ask the Euclidean geometer whether he can recognize his plane
after we have crumpled it up like a piece of paper en route to the
waste basket. He will hesitate only long enough to recall that in
the special case of superposition he has reserved for himself the
privilege of deforming his own plane, and to realize that he can
always iron his plane out smooth again after we are through with
it. This emphasizes the true nature of the two-dimensionality which
is the fundamental characteristic of the plane (and of other things,
as we shall directly see). The plane is two-dimensional in points not
because two sets of mutually perpendicular Euclidean straight lines
can be drawn in it defining directions of north-south and east-west,
but because a point in it can be located by means of two measures. The
same statement may be made of anything whatever to which the term
"surface" is applicable; anything, however crumpled or irregular it
be, that possesses length and breadth without thickness. The surface
of a sphere, of a cylinder, of an ellipsoid, of a cone, of a doughnut
(mathematically known as a torus), of a gear wheel, of a French horn,
all these possess two-dimensionality in points; on all of them we can
draw lines and curves and derive a geometry of these figures. If we
get away from the notion that geometry of two dimensions must deal
with planes, and adopt in place of this idea the broader restriction
that it shall deal with surfaces, we shall have the generalization
which the Euclidean has demanded that we produce, and the one which
in the hands of the modern geometer has shown results.