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ETHER AND THE THEORY OF RELATIVITY

An Address delivered on May 5th, 1920, in the University of Leyden

GEOMETRY AND EXPERIENCE

An expanded form of an Address to the Prussian Academy of Sciences

in Berlin on January 27th, 1921.

ETHER AND THE THEORY OF RELATIVITY

An Address delivered on May 5th, 1920, in the University of Leyden

How does it come about that alongside of the idea of ponderable

matter, which is derived by abstraction from everyday life, the

physicists set the idea of the existence of another kind of matter,

the ether? The explanation is probably to be sought in those phenomena

which have given rise to the theory of action at a distance, and

in the properties of light which have led to the undulatory theory.

Let us devote a little while to the consideration of these two

subjects.

Outside of physics we know nothing of action at a distance. When

we try to connect cause and effect in the experiences which natural

objects afford us, it seems at first as if there were no other mutual

actions than those of immediate contact, e.g. the communication of

motion by impact, push and pull, heating or inducing combustion by

means of a flame, etc. It is true that even in everyday experience

weight, which is in a sense action at a distance, plays a very

important part. But since in daily experience the weight of bodies

meets us as something constant, something not linked to any cause

which is variable in time or place, we do not in everyday life

speculate as to the cause of gravity, and therefore do not become

conscious of its character as action at a distance. It was Newton's

theory of gravitation that first assigned a cause for gravity by

interpreting it as action at a distance, proceeding from masses.

Newton's theory is probably the greatest stride ever made in

the effort towards the causal nexus of natural phenomena. And yet

this theory evoked a lively sense of discomfort among Newton's

contemporaries, because it seemed to be in conflict with the

principle springing from the rest of experience, that there can be

reciprocal action only through contact, and not through immediate

action at a distance. It is only with reluctance that man's desire

for knowledge endures a dualism of this kind. How was unity to

be preserved in his comprehension of the forces of nature? Either

by trying to look upon contact forces as being themselves distant

forces which admittedly are observable only at a very small

distance--and this was the road which Newton's followers, who were

entirely under the spell of his doctrine, mostly preferred to

take; or by assuming that the Newtonian action at a distance is

only _apparently_ immediate action at a distance, but in truth is

conveyed by a medium permeating space, whether by movements or by

elastic deformation of this medium. Thus the endeavour toward a

unified view of the nature of forces leads to the hypothesis of an

ether. This hypothesis, to be sure, did not at first bring with it

any advance in the theory of gravitation or in physics generally,

so that it became customary to treat Newton's law of force as an

axiom not further reducible. But the ether hypothesis was bound

always to play some part in physical science, even if at first only

a latent part.

When in the first half of the nineteenth century the far-reaching

similarity was revealed which subsists between the properties of

light and those of elastic waves in ponderable bodies, the ether

hypothesis found fresh support. It appeared beyond question that

light must be interpreted as a vibratory process in an elastic, inert

medium filling up universal space. It also seemed to be a necessary

consequence of the fact that light is capable of polarisation that

this medium, the ether, must be of the nature of a solid body,

because transverse waves are not possible in a fluid, but only in

a solid. Thus the physicists were bound to arrive at the theory

of the "quasi-rigid" luminiferous ether, the parts of which can

carry out no movements relatively to one another except the small

movements of deformation which correspond to light-waves.

This theory--also called the theory of the stationary luminiferous

ether--moreover found a strong support in an experiment which is

also of fundamental importance in the special theory of relativity,

the experiment of Fizeau, from which one was obliged to infer

that the luminiferous ether does not take part in the movements of

bodies. The phenomenon of aberration also favoured the theory of

the quasi-rigid ether.

The development of the theory of electricity along the path opened

up by Maxwell and Lorentz gave the development of our ideas concerning

the ether quite a peculiar and unexpected turn. For Maxwell himself

the ether indeed still had properties which were purely mechanical,

although of a much more complicated kind than the mechanical

properties of tangible solid bodies. But neither Maxwell nor his

followers succeeded in elaborating a mechanical model for the ether

which might furnish a satisfactory mechanical interpretation of

Maxwell's laws of the electro-magnetic field. The laws were clear

and simple, the mechanical interpretations clumsy and contradictory.

Almost imperceptibly the theoretical physicists adapted themselves

to a situation which, from the standpoint of their mechanical

programme, was very depressing. They were particularly influenced

by the electro-dynamical investigations of Heinrich Hertz. For

whereas they previously had required of a conclusive theory that

it should content itself with the fundamental concepts which belong

exclusively to mechanics (e.g. densities, velocities, deformations,

stresses) they gradually accustomed themselves to admitting electric and

magnetic force as fundamental concepts side by side with those of

mechanics, without requiring a mechanical interpretation for them.

Thus the purely mechanical view of nature was gradually abandoned.

But this change led to a fundamental dualism which in the long-run

was insupportable. A way of escape was now sought in the reverse

direction, by reducing the principles of mechanics to those

of electricity, and this especially as confidence in the strict

validity of the equations of Newton's mechanics was shaken by the

experiments with beta-rays and rapid kathode rays.

This dualism still confronts us in unextenuated form in the theory

of Hertz, where matter appears not only as the bearer of velocities,

kinetic energy, and mechanical pressures, but also as the bearer of

electromagnetic fields. Since such fields also occur _in vacuo_--i.e.

in free ether--the ether also appears as bearer of electromagnetic

fields. The ether appears indistinguishable in its functions from

ordinary matter. Within matter it takes part in the motion of matter

and in empty space it has everywhere a velocity; so that the ether

has a definitely assigned velocity throughout the whole of space.

There is no fundamental difference between Hertz's ether and

ponderable matter (which in part subsists in the ether).

The Hertz theory suffered not only from the defect of ascribing

to matter and ether, on the one hand mechanical states, and on the

other hand electrical states, which do not stand in any conceivable

relation to each other; it was also at variance with the result of

Fizeau's important experiment on the velocity of the propagation

of light in moving fluids, and with other established experimental

results.

Such was the state of things when H. A. Lorentz entered upon the

scene. He brought theory into harmony with experience by means of

a wonderful simplification of theoretical principles. He achieved

this, the most important advance in the theory of electricity since

Maxwell, by taking from ether its mechanical, and from matter its

electromagnetic qualities. As in empty space, so too in the interior

of material bodies, the ether, and not matter viewed atomistically,

was exclusively the seat of electromagnetic fields. According to

Lorentz the elementary particles of matter alone are capable of

carrying out movements; their electromagnetic activity is entirely

confined to the carrying of electric charges. Thus Lorentz succeeded

in reducing all electromagnetic happenings to Maxwell's equations

for free space.

As to the mechanical nature of the Lorentzian ether, it may be said

of it, in a somewhat playful spirit, that immobility is the only

mechanical property of which it has not been deprived by H. A.

Lorentz. It may be added that the whole change in the conception

of the ether which the special theory of relativity brought about,

consisted in taking away from the ether its last mechanical quality,

namely, its immobility. How this is to be understood will forthwith

be expounded.

The space-time theory and the kinematics of the special theory

of relativity were modelled on the Maxwell-Lorentz theory of the

electromagnetic field. This theory therefore satisfies the conditions

of the special theory of relativity, but when viewed from the latter

it acquires a novel aspect. For if K be a system of co-ordinates

relatively to which the Lorentzian ether is at rest, the

Maxwell-Lorentz equations are valid primarily with reference to K.

But by the special theory of relativity the same equations without

any change of meaning also hold in relation to any new system of

co-ordinates K' which is moving in uniform translation relatively

to K. Now comes the anxious question:--Why must I in the theory

distinguish the K system above all K' systems, which are physically

equivalent to it in all respects, by assuming that the ether

is at rest relatively to the K system? For the theoretician such

an asymmetry in the theoretical structure, with no corresponding

asymmetry in the system of experience, is intolerable. If we assume

the ether to be at rest relatively to K, but in motion relatively

to K', the physical equivalence of K and K' seems to me from the

logical standpoint, not indeed downright incorrect, but nevertheless

inacceptable.

The next position which it was possible to take up in face of this

state of things appeared to be the following. The ether does not

exist at all. The electromagnetic fields are not states of a medium,

and are not bound down to any bearer, but they are independent

realities which are not reducible to anything else, exactly like

the atoms of ponderable matter. This conception suggests itself

the more readily as, according to Lorentz's theory, electromagnetic

radiation, like ponderable matter, brings impulse and energy with

it, and as, according to the special theory of relativity, both

matter and radiation are but special forms of distributed energy,

ponderable mass losing its isolation and appearing as a special

form of energy.

More careful reflection teaches us, however, that the special theory

of relativity does not compel us to deny ether. We may assume the

existence of an ether; only we must give up ascribing a definite

state of motion to it, i.e. we must by abstraction take from it the

last mechanical characteristic which Lorentz had still left it. We

shall see later that this point of view, the conceivability of which

I shall at once endeavour to make more intelligible by a somewhat

halting comparison, is justified by the results of the general

theory of relativity.

Think of waves on the surface of water. Here we can describe two

entirely different things. Either we may observe how the undulatory

surface forming the boundary between water and air alters in the course

of time; or else--with the help of small floats, for instance--we

can observe how the position of the separate particles of water

alters in the course of time. If the existence of such floats for

tracking the motion of the particles of a fluid were a fundamental

impossibility in physics--if, in fact, nothing else whatever were

observable than the shape of the space occupied by the water as it

varies in time, we should have no ground for the assumption that

water consists of movable particles. But all the same we could

characterise it as a medium.

We have something like this in the electromagnetic field. For we may

picture the field to ourselves as consisting of lines of force. If

we wish to interpret these lines of force to ourselves as something

material in the ordinary sense, we are tempted to interpret the

dynamic processes as motions of these lines of force, such that each

separate line of force is tracked through the course of time. It is

well known, however, that this way of regarding the electromagnetic

field leads to contradictions.

Generalising we must say this:--There may be supposed to be extended

physical objects to which the idea of motion cannot be applied.

They may not be thought of as consisting of particles which allow

themselves to be separately tracked through time. In Minkowski's

idiom this is expressed as follows:--Not every extended conformation

in the four-dimensional world can be regarded as composed

of world-threads. The special theory of relativity forbids us to

assume the ether to consist of particles observable through time,

but the hypothesis of ether in itself is not in conflict with the

special theory of relativity. Only we must be on our guard against

ascribing a state of motion to the ether.

Certainly, from the standpoint of the special theory of relativity,

the ether hypothesis appears at first to be an empty hypothesis. In

the equations of the electromagnetic field there occur, in addition

to the densities of the electric charge, _only_ the intensities

of the field. The career of electromagnetic processes _in vacuo_

appears to be completely determined by these equations, uninfluenced

by other physical quantities. The electromagnetic fields appear as

ultimate, irreducible realities, and at first it seems superfluous

to postulate a homogeneous, isotropic ether-medium, and to envisage

electromagnetic fields as states of this medium.

But on the other hand there is a weighty argument to be adduced

in favour of the ether hypothesis. To deny the ether is ultimately

to assume that empty space has no physical qualities whatever. The

fundamental facts of mechanics do not harmonize with this view.

For the mechanical behaviour of a corporeal system hovering freely

in empty space depends not only on relative positions (distances)

and relative velocities, but also on its state of rotation, which

physically may be taken as a characteristic not appertaining to the

system in itself. In order to be able to look upon the rotation of

the system, at least formally, as something real, Newton objectivises

space.

Since he classes his absolute space together with real things, for

him rotation relative to an absolute space is also something real.

Newton might no less well have called his absolute space "Ether";

what is essential is merely that besides observable objects, another

thing, which is not perceptible, must be looked upon as real,

to enable acceleration or rotation to be looked upon as something

real.

It is true that Mach tried to avoid having to accept as real something

which is not observable by endeavouring to substitute in mechanics

a mean acceleration with reference to the totality of the masses in

the universe in place of an acceleration with reference to absolute

space. But inertial resistance opposed to relative acceleration of

distant masses presupposes action at a distance; and as the modern

physicist does not believe that he may accept this action at

a distance, he comes back once more, if he follows Mach, to the

ether, which has to serve as medium for the effects of inertia. But

this conception of the ether to which we are led by Mach's way of

thinking differs essentially from the ether as conceived by Newton,

by Fresnel, and by Lorentz. Mach's ether not only _conditions_ the

behaviour of inert masses, but _is also conditioned_ in its state

by them.

Mach's idea finds its full development in the ether of the general

theory of relativity. According to this theory the metrical

qualities of the continuum of space-time differ in the environment

of different points of space-time, and are partly conditioned by the

matter existing outside of the territory under consideration. This

space-time variability of the reciprocal relations of the standards

of space and time, or, perhaps, the recognition of the fact that

"empty space" in its physical relation is neither homogeneous nor

isotropic, compelling us to describe its state by ten functions (the

gravitation potentials g_(mn)), has, I think, finally disposed of

the view that space is physically empty. But therewith the

conception of the ether has again acquired an intelligible content,

although this content differs widely from that of the ether of the

mechanical undulatory theory of light. The ether of the general

theory of relativity is a medium which is itself devoid of _all_

mechanical and kinematical qualities, but helps to determine

mechanical (and electromagnetic) events.

What is fundamentally new in the ether of the general theory of

relativity as opposed to the ether of Lorentz consists in this, that

the state of the former is at every place determined by connections

with the matter and the state of the ether in neighbouring places,

which are amenable to law in the form of differential equations;

whereas the state of the Lorentzian ether in the absence of

electromagnetic fields is conditioned by nothing outside itself,

and is everywhere the same. The ether of the general theory of

relativity is transmuted conceptually into the ether of Lorentz if

we substitute constants for the functions of space which describe

the former, disregarding the causes which condition its state.

Thus we may also say, I think, that the ether of the general theory

of relativity is the outcome of the Lorentzian ether, through

relativation.

As to the part which the new ether is to play in the physics of

the future we are not yet clear. We know that it determines the

metrical relations in the space-time continuum, e.g. the configurative

possibilities of solid bodies as well as the gravitational fields;

but we do not know whether it has an essential share in the structure

of the electrical elementary particles constituting matter. Nor do

we know whether it is only in the proximity of ponderable masses

that its structure differs essentially from that of the Lorentzian

ether; whether the geometry of spaces of cosmic extent is approximately

Euclidean. But we can assert by reason of the relativistic equations

of gravitation that there must be a departure from Euclidean

relations, with spaces of cosmic order of magnitude, if there exists

a positive mean density, no matter how small, of the matter in the

universe. In this case the universe must of necessity be spatially

unbounded and of finite magnitude, its magnitude being determined

by the value of that mean density.

If we consider the gravitational field and the electromagnetic field

from the stand-point of the ether hypothesis, we find a remarkable

difference between the two. There can be no space nor any part

of space without gravitational potentials; for these confer upon

space its metrical qualities, without which it cannot be imagined

at all. The existence of the gravitational field is inseparably

bound up with the existence of space. On the other hand a part of

space may very well be imagined without an electromagnetic field;

thus in contrast with the gravitational field, the electromagnetic

field seems to be only secondarily linked to the ether, the formal

nature of the electromagnetic field being as yet in no way determined

by that of gravitational ether. From the present state of theory

it looks as if the electromagnetic field, as opposed to the

gravitational field, rests upon an entirely new formal _motif_,

as though nature might just as well have endowed the gravitational

ether with fields of quite another type, for example, with fields

of a scalar potential, instead of fields of the electromagnetic

type.

Since according to our present conceptions the elementary particles

of matter are also, in their essence, nothing else than condensations

of the electromagnetic field, our present view of the universe

presents two realities which are completely separated from each other

conceptually, although connected causally, namely, gravitational ether

and electromagnetic field, or--as they might also be called--space

and matter.

Of course it would be a great advance if we could succeed in

comprehending the gravitational field and the electromagnetic field

together as one unified conformation. Then for the first time the

epoch of theoretical physics founded by Faraday and Maxwell would

reach a satisfactory conclusion. The contrast between ether and

matter would fade away, and, through the general theory of relativity,

the whole of physics would become a complete system of thought,

like geometry, kinematics, and the theory of gravitation. An

exceedingly ingenious attempt in this direction has been made by

the mathematician H. Weyl; but I do not believe that his theory will

hold its ground in relation to reality. Further, in contemplating

the immediate future of theoretical physics we ought not unconditionally

to reject the possibility that the facts comprised in the quantum

theory may set bounds to the field theory beyond which it cannot

pass.

Recapitulating, we may say that according to the general theory of

relativity space is endowed with physical qualities; in this sense,

therefore, there exists an ether. According to the general theory

of relativity space without ether is unthinkable; for in such space

there not only would be no propagation of light, but also no possibility

of existence for standards of space and time (measuring-rods and

clocks), nor therefore any space-time intervals in the physical

sense. But this ether may not be thought of as endowed with the

quality characteristic of ponderable media, as consisting of parts

which may be tracked through time. The idea of motion may not be

applied to it.

GEOMETRY AND EXPERIENCE

An expanded form of an Address to the Prussian Academy of Sciences

in Berlin on January 27th, 1921.

One reason why mathematics enjoys special esteem, above all other

sciences, is that its laws are absolutely certain and indisputable,

while those of all other sciences are to some extent debatable and

in constant danger of being overthrown by newly discovered facts.

In spite of this, the investigator in another department of science

would not need to envy the mathematician if the laws of mathematics

referred to objects of our mere imagination, and not to objects

of reality. For it cannot occasion surprise that different persons

should arrive at the same logical conclusions when they have already

agreed upon the fundamental laws (axioms), as well as the methods

by which other laws are to be deduced therefrom. But there is another

reason for the high repute of mathematics, in that it is mathematics

which affords the exact natural sciences a certain measure of

security, to which without mathematics they could not attain.

At this point an enigma presents itself which in all ages has agitated

inquiring minds. How can it be that mathematics, being after all

a product of human thought which is independent of experience, is

so admirably appropriate to the objects of reality? Is human reason,

then, without experience, merely by taking thought, able to fathom

the properties of real things.

In my opinion the answer to this question is, briefly, this:--As far

as the laws of mathematics refer to reality, they are not certain;

and as far as they are certain, they do not refer to reality.

It seems to me that complete clearness as to this state of things

first became common property through that new departure in mathematics

which is known by the name of mathematical logic or "Axiomatics."

The progress achieved by axiomatics consists in its having neatly

separated the logical-formal from its objective or intuitive

content; according to axiomatics the logical-formal alone forms

the subject-matter of mathematics, which is not concerned with the

intuitive or other content associated with the logical-formal.

Let us for a moment consider from this point of view any axiom of

geometry, for instance, the following:--Through two points in space

there always passes one and only one straight line. How is this

axiom to be interpreted in the older sense and in the more modern

sense?

The older interpretation:--Every one knows what a straight line

is, and what a point is. Whether this knowledge springs from an

ability of the human mind or from experience, from some collaboration

of the two or from some other source, is not for the mathematician

to decide. He leaves the question to the philosopher. Being based

upon this knowledge, which precedes all mathematics, the axiom

stated above is, like all other axioms, self-evident, that is, it

is the expression of a part of this _a priori_ knowledge.

The more modern interpretation:--Geometry treats of entities which

are denoted by the words straight line, point, etc. These entities

do not take for granted any knowledge or intuition whatever, but

they presuppose only the validity of the axioms, such as the one

stated above, which are to be taken in a purely formal sense, i.e.

as void of all content of intuition or experience. These axioms are

free creations of the human mind. All other propositions of geometry

are logical inferences from the axioms (which are to be taken in

the nominalistic sense only). The matter of which geometry treats

is first defined by the axioms. Schlick in his book on epistemology has

therefore characterised axioms very aptly as "implicit definitions."

This view of axioms, advocated by modern axiomatics, purges mathematics

of all extraneous elements, and thus dispels the mystic obscurity

which formerly surrounded the principles of mathematics.

But a presentation of its principles thus clarified makes it also

evident that mathematics as such cannot predicate anything about

perceptual objects or real objects. In axiomatic geometry the words

"point," "straight line," etc., stand only for empty conceptual

schemata. That which gives them substance is not relevant to

mathematics.

Yet on the other hand it is certain that mathematics generally,

and particularly geometry, owes its existence to the need which

was felt of learning something about the relations of real things

to one another. The very word geometry, which, of course, means

earth-measuring, proves this. For earth-measuring has to do with

the possibilities of the disposition of certain natural objects

with respect to one another, namely, with parts of the earth,

measuring-lines, measuring-wands, etc. It is clear that the system

of concepts of axiomatic geometry alone cannot make any assertions

as to the relations of real objects of this kind, which we will

call practically-rigid bodies. To be able to make such assertions,

geometry must be stripped of its merely logical-formal character

by the co-ordination of real objects of experience with the empty

conceptual frame-work of axiomatic geometry. To accomplish this,

we need only add the proposition:--Solid bodies are related, with

respect to their possible dispositions, as are bodies in Euclidean

geometry of three dimensions. Then the propositions of Euclid contain

affirmations as to the relations of practically-rigid bodies.

Geometry thus completed is evidently a natural science; we may in

fact regard it as the most ancient branch of physics. Its affirmations

rest essentially on induction from experience, but not on logical

inferences only. We will call this completed geometry "practical

geometry," and shall distinguish it in what follows from "purely

axiomatic geometry." The question whether the practical geometry

of the universe is Euclidean or not has a clear meaning, and its

answer can only be furnished by experience. All linear measurement

in physics is practical geometry in this sense, so too is geodetic

and astronomical linear measurement, if we call to our help the

law of experience that light is propagated in a straight line, and

indeed in a straight line in the sense of practical geometry.

I attach special importance to the view of geometry which I

have just set forth, because without it I should have been unable

to formulate the theory of relativity. Without it the following

reflection would have been impossible:--In a system of reference

rotating relatively to an inert system, the laws of disposition of

rigid bodies do not correspond to the rules of Euclidean geometry

on account of the Lorentz contraction; thus if we admit non-inert

systems we must abandon Euclidean geometry. The decisive step in

the transition to general co-variant equations would certainly not

have been taken if the above interpretation had not served as a

stepping-stone. If we deny the relation between the body of axiomatic

Euclidean geometry and the practically-rigid body of reality,

we readily arrive at the following view, which was entertained by

that acute and profound thinker, H. Poincare:--Euclidean geometry

is distinguished above all other imaginable axiomatic geometries

by its simplicity. Now since axiomatic geometry by itself contains

no assertions as to the reality which can be experienced, but can

do so only in combination with physical laws, it should be possible

and reasonable--whatever may be the nature of reality--to retain

Euclidean geometry. For if contradictions between theory and

experience manifest themselves, we should rather decide to change

physical laws than to change axiomatic Euclidean geometry. If we

deny the relation between the practically-rigid body and geometry,

we shall indeed not easily free ourselves from the convention

that Euclidean geometry is to be retained as the simplest. Why

is the equivalence of the practically-rigid body and the body of

geometry--which suggests itself so readily--denied by Poincare and

other investigators? Simply because under closer inspection the

real solid bodies in nature are not rigid, because their geometrical

behaviour, that is, their possibilities of relative disposition,

depend upon temperature, external forces, etc. Thus the original,

immediate relation between geometry and physical reality appears

destroyed, and we feel impelled toward the following more general

view, which characterizes Poincare's standpoint. Geometry (G)

predicates nothing about the relations of real things, but only

geometry together with the purport (P) of physical laws can do so.

Using symbols, we may say that only the sum of (G) + (P) is subject

to the control of experience. Thus (G) may be chosen arbitrarily,

and also parts of (P); all these laws are conventions. All that

is necessary to avoid contradictions is to choose the remainder of

(P) so that (G) and the whole of (P) are together in accord with

experience. Envisaged in this way, axiomatic geometry and the part

of natural law which has been given a conventional status appear

as epistemologically equivalent.

_Sub specie aeterni_ Poincare, in my opinion, is right. The idea

of the measuring-rod and the idea of the clock co-ordinated with it

in the theory of relativity do not find their exact correspondence

in the real world. It is also clear that the solid body and the

clock do not in the conceptual edifice of physics play the part of

irreducible elements, but that of composite structures, which may

not play any independent part in theoretical physics. But it is my

conviction that in the present stage of development of theoretical

physics these ideas must still be employed as independent ideas;

for we are still far from possessing such certain knowledge

of theoretical principles as to be able to give exact theoretical

constructions of solid bodies and clocks.

Further, as to the objection that there are no really rigid bodies

in nature, and that therefore the properties predicated of rigid

bodies do not apply to physical reality,--this objection is by

no means so radical as might appear from a hasty examination. For

it is not a difficult task to determine the physical state of a

measuring-rod so accurately that its behaviour relatively to other

measuring-bodies shall be sufficiently free from ambiguity to allow

it to be substituted for the "rigid" body. It is to measuring-bodies

of this kind that statements as to rigid bodies must be referred.

All practical geometry is based upon a principle which is accessible

to experience, and which we will now try to realise. We will

call that which is enclosed between two boundaries, marked upon a

practically-rigid body, a tract. We imagine two practically-rigid

bodies, each with a tract marked out on it. These two tracts are

said to be "equal to one another" if the boundaries of the one tract

can be brought to coincide permanently with the boundaries of the

other. We now assume that:

If two tracts are found to be equal once and anywhere, they are

equal always and everywhere.

Not only the practical geometry of Euclid, but also its nearest

generalisation, the practical geometry of Riemann, and therewith

the general theory of relativity, rest upon this assumption. Of the

experimental reasons which warrant this assumption I will mention

only one. The phenomenon of the propagation of light in empty space

assigns a tract, namely, the appropriate path of light, to each

interval of local time, and conversely. Thence it follows that

the above assumption for tracts must also hold good for intervals

of clock-time in the theory of relativity. Consequently it may be

formulated as follows:--If two ideal clocks are going at the same

rate at any time and at any place (being then in immediate proximity

to each other), they will always go at the same rate, no matter where

and when they are again compared with each other at one place.--If

this law were not valid for real clocks, the proper frequencies

for the separate atoms of the same chemical element would not be

in such exact agreement as experience demonstrates. The existence

of sharp spectral lines is a convincing experimental proof of the

above-mentioned principle of practical geometry. This is the ultimate

foundation in fact which enables us to speak with meaning of the

mensuration, in Riemann's sense of the word, of the four-dimensional

continuum of space-time.

The question whether the structure of this continuum is Euclidean,

or in accordance with Riemann's general scheme, or otherwise,

is, according to the view which is here being advocated, properly

speaking a physical question which must be answered by experience,

and not a question of a mere convention to be selected on practical

grounds. Riemann's geometry will be the right thing if the laws

of disposition of practically-rigid bodies are transformable into

those of the bodies of Euclid's geometry with an exactitude which

increases in proportion as the dimensions of the part of space-time

under consideration are diminished.

It is true that this proposed physical interpretation of geometry

breaks down when applied immediately to spaces of sub-molecular

order of magnitude. But nevertheless, even in questions as

to the constitution of elementary particles, it retains part of

its importance. For even when it is a question of describing the

electrical elementary particles constituting matter, the attempt

may still be made to ascribe physical importance to those ideas

of fields which have been physically defined for the purpose

of describing the geometrical behaviour of bodies which are large

as compared with the molecule. Success alone can decide as to the

justification of such an attempt, which postulates physical reality

for the fundamental principles of Riemann's geometry outside of the

domain of their physical definitions. It might possibly turn out

that this extrapolation has no better warrant than the extrapolation

of the idea of temperature to parts of a body of molecular order

of magnitude.

It appears less problematical to extend the ideas of practical

geometry to spaces of cosmic order of magnitude. It might, of course,

be objected that a construction composed of solid rods departs more

and more from ideal rigidity in proportion as its spatial extent

becomes greater. But it will hardly be possible, I think, to assign

fundamental significance to this objection. Therefore the question

whether the universe is spatially finite or not seems to me

decidedly a pregnant question in the sense of practical geometry.

I do not even consider it impossible that this question will be

answered before long by astronomy. Let us call to mind what the

general theory of relativity teaches in this respect. It offers

two possibilities:--

1. The universe is spatially infinite. This can be so only if the

average spatial density of the matter in universal space, concentrated

in the stars, vanishes, i.e. if the ratio of the total mass of the

stars to the magnitude of the space through which they are scattered

approximates indefinitely to the value zero when the spaces taken

into consideration are constantly greater and greater.

2. The universe is spatially finite. This must be so, if there is

a mean density of the ponderable matter in universal space differing

from zero. The smaller that mean density, the greater is the volume

of universal space.

I must not fail to mention that a theoretical argument can be adduced in

favour of the hypothesis of a finite universe. The general theory

of relativity teaches that the inertia of a given body is greater as

there are more ponderable masses in proximity to it; thus it seems

very natural to reduce the total effect of inertia of a body to

action and reaction between it and the other bodies in the universe,

as indeed, ever since Newton's time, gravity has been completely

reduced to action and reaction between bodies. From the equations

of the general theory of relativity it can be deduced that this

total reduction of inertia to reciprocal action between masses--as

required by E. Mach, for example--is possible only if the universe

is spatially finite.

On many physicists and astronomers this argument makes no impression.

Experience alone can finally decide which of the two possibilities

is realised in nature. How can experience furnish an answer? At first

it might seem possible to determine the mean density of matter by

observation of that part of the universe which is accessible to our

perception. This hope is illusory. The distribution of the visible

stars is extremely irregular, so that we on no account may venture

to set down the mean density of star-matter in the universe as

equal, let us say, to the mean density in the Milky Way. In any

case, however great the space examined may be, we could not feel

convinced that there were no more stars beyond that space. So it

seems impossible to estimate the mean density. But there is another

road, which seems to me more practicable, although it also presents

great difficulties. For if we inquire into the deviations shown

by the consequences of the general theory of relativity which are

accessible to experience, when these are compared with the consequences

of the Newtonian theory, we first of all find a deviation which

shows itself in close proximity to gravitating mass, and has been

confirmed in the case of the planet Mercury. But if the universe

is spatially finite there is a second deviation from the Newtonian

theory, which, in the language of the Newtonian theory, may be

expressed thus:--The gravitational field is in its nature such as

if it were produced, not only by the ponderable masses, but also by

a mass-density of negative sign, distributed uniformly throughout

space. Since this factitious mass-density would have to be enormously

small, it could make its presence felt only in gravitating systems

of very great extent.

Assuming that we know, let us say, the statistical distribution

of the stars in the Milky Way, as well as their masses, then by

Newton's law we can calculate the gravitational field and the mean

velocities which the stars must have, so that the Milky Way should

not collapse under the mutual attraction of its stars, but should

maintain its actual extent. Now if the actual velocities of the stars,

which can, of course, be measured, were smaller than the calculated

velocities, we should have a proof that the actual attractions

at great distances are smaller than by Newton's law. From such a

deviation it could be proved indirectly that the universe is finite.

It would even be possible to estimate its spatial magnitude.

Can we picture to ourselves a three-dimensional universe which is

finite, yet unbounded?

The usual answer to this question is "No," but that is not the right

answer. The purpose of the following remarks is to show that the

answer should be "Yes." I want to show that without any extraordinary

difficulty we can illustrate the theory of a finite universe by

means of a mental image to which, with some practice, we shall soon

grow accustomed.

First of all, an observation of epistemological nature. A

geometrical-physical theory as such is incapable of being directly

pictured, being merely a system of concepts. But these concepts

serve the purpose of bringing a multiplicity of real or imaginary

sensory experiences into connection in the mind. To "visualise"

a theory, or bring it home to one's mind, therefore means to give

a representation to that abundance of experiences for which the

theory supplies the schematic arrangement. In the present case we

have to ask ourselves how we can represent that relation of solid

bodies with respect to their reciprocal disposition (contact) which

corresponds to the theory of a finite universe. There is really

nothing new in what I have to say about this; but innumerable

questions addressed to me prove that the requirements of those who

thirst for knowledge of these matters have not yet been completely

satisfied.

So, will the initiated please pardon me, if part of what I shall

bring forward has long been known?

What do we wish to express when we say that our space is infinite?

Nothing more than that we might lay any number whatever of bodies

of equal sizes side by side without ever filling space. Suppose

that we are provided with a great many wooden cubes all of the

same size. In accordance with Euclidean geometry we can place them

above, beside, and behind one another so as to fill a part of space

of any dimensions; but this construction would never be finished;

we could go on adding more and more cubes without ever finding

that there was no more room. That is what we wish to express when

we say that space is infinite. It would be better to say that space

is infinite in relation to practically-rigid bodies, assuming that

the laws of disposition for these bodies are given by Euclidean

geometry.

Another example of an infinite continuum is the plane. On a plane

surface we may lay squares of cardboard so that each side of any

square has the side of another square adjacent to it. The construction

is never finished; we can always go on laying squares--if their laws

of disposition correspond to those of plane figures of Euclidean

geometry. The plane is therefore infinite in relation to the

cardboard squares. Accordingly we say that the plane is an infinite

continuum of two dimensions, and space an infinite continuum of

three dimensions. What is here meant by the number of dimensions,

I think I may assume to be known.

Now we take an example of a two-dimensional continuum which is

finite, but unbounded. We imagine the surface of a large globe and

a quantity of small paper discs, all of the same size. We place

one of the discs anywhere on the surface of the globe. If we move

the disc about, anywhere we like, on the surface of the globe,

we do not come upon a limit or boundary anywhere on the journey.

Therefore we say that the spherical surface of the globe is an

unbounded continuum. Moreover, the spherical surface is a finite

continuum. For if we stick the paper discs on the globe, so that

no disc overlaps another, the surface of the globe will finally

become so full that there is no room for another disc. This simply

means that the spherical surface of the globe is finite in relation

to the paper discs. Further, the spherical surface is a non-Euclidean

continuum of two dimensions, that is to say, the laws of disposition

for the rigid figures lying in it do not agree with those of the

Euclidean plane. This can be shown in the following way. Place

a paper disc on the spherical surface, and around it in a circle

place six more discs, each of which is to be surrounded in turn

by six discs, and so on. If this construction is made on a plane

surface, we have an uninterrupted disposition in which there are

six discs touching every disc except those which lie on the outside.

[Figure 1: Discs maximally packed on a plane]

On the spherical surface the construction also seems to promise

success at the outset, and the smaller the radius of the disc

in proportion to that of the sphere, the more promising it seems.

But as the construction progresses it becomes more and more patent

that the disposition of the discs in the manner indicated, without

interruption, is not possible, as it should be possible by Euclidean

geometry of the the plane surface. In this way creatures which

cannot leave the spherical surface, and cannot even peep out from

the spherical surface into three-dimensional space, might discover,

merely by experimenting with discs, that their two-dimensional

"space" is not Euclidean, but spherical space.

From the latest results of the theory of relativity it is probable

that our three-dimensional space is also approximately spherical,

that is, that the laws of disposition of rigid bodies in it are

not given by Euclidean geometry, but approximately by spherical

geometry, if only we consider parts of space which are sufficiently

great. Now this is the place where the reader's imagination boggles.

"Nobody can imagine this thing," he cries indignantly. "It can be

said, but cannot be thought. I can represent to myself a spherical

surface well enough, but nothing analogous to it in three dimensions."

[Figure 2: A circle projected from a sphere onto a plane]

We must try to surmount this barrier in the mind, and the patient

reader will see that it is by no means a particularly difficult

task. For this purpose we will first give our attention once more to

the geometry of two-dimensional spherical surfaces. In the adjoining

figure let _K_ be the spherical surface, touched at _S_ by a plane,

_E_, which, for facility of presentation, is shown in the drawing as

a bounded surface. Let _L_ be a disc on the spherical surface. Now

let us imagine that at the point _N_ of the spherical surface,

diametrically opposite to _S_, there is a luminous point, throwing a

shadow _L'_ of the disc _L_ upon the plane _E_. Every point on the

sphere has its shadow on the plane. If the disc on the sphere _K_ is

moved, its shadow _L'_ on the plane _E_ also moves. When the disc

_L_ is at _S_, it almost exactly coincides with its shadow. If it

moves on the spherical surface away from _S_ upwards, the disc

shadow _L'_ on the plane also moves away from _S_ on the plane

outwards, growing bigger and bigger. As the disc _L_ approaches the

luminous point _N_, the shadow moves off to infinity, and becomes

infinitely great.

Now we put the question, What are the laws of disposition of the

disc-shadows _L'_ on the plane _E_? Evidently they are exactly the

same as the laws of disposition of the discs _L_ on the spherical

surface. For to each original figure on _K_ there is a corresponding

shadow figure on _E_. If two discs on _K_ are touching, their

shadows on _E_ also touch. The shadow-geometry on the plane agrees

with the the disc-geometry on the sphere. If we call the disc-shadows

rigid figures, then spherical geometry holds good on the plane _E_

with respect to these rigid figures. Moreover, the plane is finite

with respect to the disc-shadows, since only a finite number of

the shadows can find room on the plane.

At this point somebody will say, "That is nonsense. The disc-shadows

are _not_ rigid figures. We have only to move a two-foot rule about

on the plane _E_ to convince ourselves that the shadows constantly

increase in size as they move away from _S_ on the plane towards

infinity." But what if the two-foot rule were to behave on the

plane _E_ in the same way as the disc-shadows _L'_? It would then

be impossible to show that the shadows increase in size as they

move away from _S_; such an assertion would then no longer have

any meaning whatever. In fact the only objective assertion that can

be made about the disc-shadows is just this, that they are related

in exactly the same way as are the rigid discs on the spherical

surface in the sense of Euclidean geometry.

We must carefully bear in mind that our statement as to the growth

of the disc-shadows, as they move away from _S_ towards infinity,

has in itself no objective meaning, as long as we are unable to

employ Euclidean rigid bodies which can be moved about on the plane

_E_ for the purpose of comparing the size of the disc-shadows. In

respect of the laws of disposition of the shadows _L'_, the point

_S_ has no special privileges on the plane any more than on the

spherical surface.

The representation given above of spherical geometry on the

plane is important for us, because it readily allows itself to be

transferred to the three-dimensional case.

Let us imagine a point _S_ of our space, and a great number

of small spheres, _L'_, which can all be brought to coincide with

one another. But these spheres are not to be rigid in the sense

of Euclidean geometry; their radius is to increase (in the sense

of Euclidean geometry) when they are moved away from _S_ towards

infinity, and this increase is to take place in exact accordance

with the same law as applies to the increase of the radii of the

disc-shadows _L'_ on the plane.

After having gained a vivid mental image of the geometrical

behaviour of our _L'_ spheres, let us assume that in our space there

are no rigid bodies at all in the sense of Euclidean geometry, but

only bodies having the behaviour of our _L'_ spheres. Then we shall

have a vivid representation of three-dimensional spherical space,

or, rather of three-dimensional spherical geometry. Here our spheres

must be called "rigid" spheres. Their increase in size as they

depart from _S_ is not to be detected by measuring with

measuring-rods, any more than in the case of the disc-shadows on

_E_, because the standards of measurement behave in the same way as

the spheres. Space is homogeneous, that is to say, the same

spherical configurations are possible in the environment of all

points.* Our space is finite, because, in consequence of the

"growth" of the spheres, only a finite number of them can find room

in space.

* This is intelligible without calculation--but only for the

two-dimensional case--if we revert once more to the case of the disc

on the surface of the sphere.

In this way, by using as stepping-stones the practice in thinking

and visualisation which Euclidean geometry gives us, we have acquired

a mental picture of spherical geometry. We may without difficulty

impart more depth and vigour to these ideas by carrying out special

imaginary constructions. Nor would it be difficult to represent the

case of what is called elliptical geometry in an analogous manner.

My only aim to-day has been to show that the human faculty of

visualisation is by no means bound to capitulate to non-Euclidean

geometry.

ETHER AND THE THEORY OF RELATIVITY

An Address delivered on May 5th, 1920, in the University of Leyden

GEOMETRY AND EXPERIENCE

An expanded form of an Address to the Prussian Academy of Sciences

in Berlin on January 27th, 1921.

ETHER AND THE THEORY OF RELATIVITY

An Address delivered on May 5th, 1920, in the University of Leyden

How does it come about that alongside of the idea of ponderable

matter, which is derived by abstraction from everyday life, the

physicists set the idea of the existence of another kind of matter,

the ether? The explanation is probably to be sought in those phenomena

which have given rise to the theory of action at a distance, and

in the properties of light which have led to the undulatory theory.

Let us devote a little while to the consideration of these two

subjects.

Outside of physics we know nothing of action at a distance. When

we try to connect cause and effect in the experiences which natural

objects afford us, it seems at first as if there were no other mutual

actions than those of immediate contact, e.g. the communication of

motion by impact, push and pull, heating or inducing combustion by

means of a flame, etc. It is true that even in everyday experience

weight, which is in a sense action at a distance, plays a very

important part. But since in daily experience the weight of bodies

meets us as something constant, something not linked to any cause

which is variable in time or place, we do not in everyday life

speculate as to the cause of gravity, and therefore do not become

conscious of its character as action at a distance. It was Newton's

theory of gravitation that first assigned a cause for gravity by

interpreting it as action at a distance, proceeding from masses.

Newton's theory is probably the greatest stride ever made in

the effort towards the causal nexus of natural phenomena. And yet

this theory evoked a lively sense of discomfort among Newton's

contemporaries, because it seemed to be in conflict with the

principle springing from the rest of experience, that there can be

reciprocal action only through contact, and not through immediate

action at a distance. It is only with reluctance that man's desire

for knowledge endures a dualism of this kind. How was unity to

be preserved in his comprehension of the forces of nature? Either

by trying to look upon contact forces as being themselves distant

forces which admittedly are observable only at a very small

distance--and this was the road which Newton's followers, who were

entirely under the spell of his doctrine, mostly preferred to

take; or by assuming that the Newtonian action at a distance is

only _apparently_ immediate action at a distance, but in truth is

conveyed by a medium permeating space, whether by movements or by

elastic deformation of this medium. Thus the endeavour toward a

unified view of the nature of forces leads to the hypothesis of an

ether. This hypothesis, to be sure, did not at first bring with it

any advance in the theory of gravitation or in physics generally,

so that it became customary to treat Newton's law of force as an

axiom not further reducible. But the ether hypothesis was bound

always to play some part in physical science, even if at first only

a latent part.

When in the first half of the nineteenth century the far-reaching

similarity was revealed which subsists between the properties of

light and those of elastic waves in ponderable bodies, the ether

hypothesis found fresh support. It appeared beyond question that

light must be interpreted as a vibratory process in an elastic, inert

medium filling up universal space. It also seemed to be a necessary

consequence of the fact that light is capable of polarisation that

this medium, the ether, must be of the nature of a solid body,

because transverse waves are not possible in a fluid, but only in

a solid. Thus the physicists were bound to arrive at the theory

of the "quasi-rigid" luminiferous ether, the parts of which can

carry out no movements relatively to one another except the small

movements of deformation which correspond to light-waves.

This theory--also called the theory of the stationary luminiferous

ether--moreover found a strong support in an experiment which is

also of fundamental importance in the special theory of relativity,

the experiment of Fizeau, from which one was obliged to infer

that the luminiferous ether does not take part in the movements of

bodies. The phenomenon of aberration also favoured the theory of

the quasi-rigid ether.

The development of the theory of electricity along the path opened

up by Maxwell and Lorentz gave the development of our ideas concerning

the ether quite a peculiar and unexpected turn. For Maxwell himself

the ether indeed still had properties which were purely mechanical,

although of a much more complicated kind than the mechanical

properties of tangible solid bodies. But neither Maxwell nor his

followers succeeded in elaborating a mechanical model for the ether

which might furnish a satisfactory mechanical interpretation of

Maxwell's laws of the electro-magnetic field. The laws were clear

and simple, the mechanical interpretations clumsy and contradictory.

Almost imperceptibly the theoretical physicists adapted themselves

to a situation which, from the standpoint of their mechanical

programme, was very depressing. They were particularly influenced

by the electro-dynamical investigations of Heinrich Hertz. For

whereas they previously had required of a conclusive theory that

it should content itself with the fundamental concepts which belong

exclusively to mechanics (e.g. densities, velocities, deformations,

stresses) they gradually accustomed themselves to admitting electric and

magnetic force as fundamental concepts side by side with those of

mechanics, without requiring a mechanical interpretation for them.

Thus the purely mechanical view of nature was gradually abandoned.

But this change led to a fundamental dualism which in the long-run

was insupportable. A way of escape was now sought in the reverse

direction, by reducing the principles of mechanics to those

of electricity, and this especially as confidence in the strict

validity of the equations of Newton's mechanics was shaken by the

experiments with beta-rays and rapid kathode rays.

This dualism still confronts us in unextenuated form in the theory

of Hertz, where matter appears not only as the bearer of velocities,

kinetic energy, and mechanical pressures, but also as the bearer of

electromagnetic fields. Since such fields also occur _in vacuo_--i.e.

in free ether--the ether also appears as bearer of electromagnetic

fields. The ether appears indistinguishable in its functions from

ordinary matter. Within matter it takes part in the motion of matter

and in empty space it has everywhere a velocity; so that the ether

has a definitely assigned velocity throughout the whole of space.

There is no fundamental difference between Hertz's ether and

ponderable matter (which in part subsists in the ether).

The Hertz theory suffered not only from the defect of ascribing

to matter and ether, on the one hand mechanical states, and on the

other hand electrical states, which do not stand in any conceivable

relation to each other; it was also at variance with the result of

Fizeau's important experiment on the velocity of the propagation

of light in moving fluids, and with other established experimental

results.

Such was the state of things when H. A. Lorentz entered upon the

scene. He brought theory into harmony with experience by means of

a wonderful simplification of theoretical principles. He achieved

this, the most important advance in the theory of electricity since

Maxwell, by taking from ether its mechanical, and from matter its

electromagnetic qualities. As in empty space, so too in the interior

of material bodies, the ether, and not matter viewed atomistically,

was exclusively the seat of electromagnetic fields. According to

Lorentz the elementary particles of matter alone are capable of

carrying out movements; their electromagnetic activity is entirely

confined to the carrying of electric charges. Thus Lorentz succeeded

in reducing all electromagnetic happenings to Maxwell's equations

for free space.

As to the mechanical nature of the Lorentzian ether, it may be said

of it, in a somewhat playful spirit, that immobility is the only

mechanical property of which it has not been deprived by H. A.

Lorentz. It may be added that the whole change in the conception

of the ether which the special theory of relativity brought about,

consisted in taking away from the ether its last mechanical quality,

namely, its immobility. How this is to be understood will forthwith

be expounded.

The space-time theory and the kinematics of the special theory

of relativity were modelled on the Maxwell-Lorentz theory of the

electromagnetic field. This theory therefore satisfies the conditions

of the special theory of relativity, but when viewed from the latter

it acquires a novel aspect. For if K be a system of co-ordinates

relatively to which the Lorentzian ether is at rest, the

Maxwell-Lorentz equations are valid primarily with reference to K.

But by the special theory of relativity the same equations without

any change of meaning also hold in relation to any new system of

co-ordinates K' which is moving in uniform translation relatively

to K. Now comes the anxious question:--Why must I in the theory

distinguish the K system above all K' systems, which are physically

equivalent to it in all respects, by assuming that the ether

is at rest relatively to the K system? For the theoretician such

an asymmetry in the theoretical structure, with no corresponding

asymmetry in the system of experience, is intolerable. If we assume

the ether to be at rest relatively to K, but in motion relatively

to K', the physical equivalence of K and K' seems to me from the

logical standpoint, not indeed downright incorrect, but nevertheless

inacceptable.

The next position which it was possible to take up in face of this

state of things appeared to be the following. The ether does not

exist at all. The electromagnetic fields are not states of a medium,

and are not bound down to any bearer, but they are independent

realities which are not reducible to anything else, exactly like

the atoms of ponderable matter. This conception suggests itself

the more readily as, according to Lorentz's theory, electromagnetic

radiation, like ponderable matter, brings impulse and energy with

it, and as, according to the special theory of relativity, both

matter and radiation are but special forms of distributed energy,

ponderable mass losing its isolation and appearing as a special

form of energy.

More careful reflection teaches us, however, that the special theory

of relativity does not compel us to deny ether. We may assume the

existence of an ether; only we must give up ascribing a definite

state of motion to it, i.e. we must by abstraction take from it the

last mechanical characteristic which Lorentz had still left it. We

shall see later that this point of view, the conceivability of which

I shall at once endeavour to make more intelligible by a somewhat

halting comparison, is justified by the results of the general

theory of relativity.

Think of waves on the surface of water. Here we can describe two

entirely different things. Either we may observe how the undulatory

surface forming the boundary between water and air alters in the course

of time; or else--with the help of small floats, for instance--we

can observe how the position of the separate particles of water

alters in the course of time. If the existence of such floats for

tracking the motion of the particles of a fluid were a fundamental

impossibility in physics--if, in fact, nothing else whatever were

observable than the shape of the space occupied by the water as it

varies in time, we should have no ground for the assumption that

water consists of movable particles. But all the same we could

characterise it as a medium.

We have something like this in the electromagnetic field. For we may

picture the field to ourselves as consisting of lines of force. If

we wish to interpret these lines of force to ourselves as something

material in the ordinary sense, we are tempted to interpret the

dynamic processes as motions of these lines of force, such that each

separate line of force is tracked through the course of time. It is

well known, however, that this way of regarding the electromagnetic

field leads to contradictions.

Generalising we must say this:--There may be supposed to be extended

physical objects to which the idea of motion cannot be applied.

They may not be thought of as consisting of particles which allow

themselves to be separately tracked through time. In Minkowski's

idiom this is expressed as follows:--Not every extended conformation

in the four-dimensional world can be regarded as composed

of world-threads. The special theory of relativity forbids us to

assume the ether to consist of particles observable through time,

but the hypothesis of ether in itself is not in conflict with the

special theory of relativity. Only we must be on our guard against

ascribing a state of motion to the ether.

Certainly, from the standpoint of the special theory of relativity,

the ether hypothesis appears at first to be an empty hypothesis. In

the equations of the electromagnetic field there occur, in addition

to the densities of the electric charge, _only_ the intensities

of the field. The career of electromagnetic processes _in vacuo_

appears to be completely determined by these equations, uninfluenced

by other physical quantities. The electromagnetic fields appear as

ultimate, irreducible realities, and at first it seems superfluous

to postulate a homogeneous, isotropic ether-medium, and to envisage

electromagnetic fields as states of this medium.

But on the other hand there is a weighty argument to be adduced

in favour of the ether hypothesis. To deny the ether is ultimately

to assume that empty space has no physical qualities whatever. The

fundamental facts of mechanics do not harmonize with this view.

For the mechanical behaviour of a corporeal system hovering freely

in empty space depends not only on relative positions (distances)

and relative velocities, but also on its state of rotation, which

physically may be taken as a characteristic not appertaining to the

system in itself. In order to be able to look upon the rotation of

the system, at least formally, as something real, Newton objectivises

space.

Since he classes his absolute space together with real things, for

him rotation relative to an absolute space is also something real.

Newton might no less well have called his absolute space "Ether";

what is essential is merely that besides observable objects, another

thing, which is not perceptible, must be looked upon as real,

to enable acceleration or rotation to be looked upon as something

real.

It is true that Mach tried to avoid having to accept as real something

which is not observable by endeavouring to substitute in mechanics

a mean acceleration with reference to the totality of the masses in

the universe in place of an acceleration with reference to absolute

space. But inertial resistance opposed to relative acceleration of

distant masses presupposes action at a distance; and as the modern

physicist does not believe that he may accept this action at

a distance, he comes back once more, if he follows Mach, to the

ether, which has to serve as medium for the effects of inertia. But

this conception of the ether to which we are led by Mach's way of

thinking differs essentially from the ether as conceived by Newton,

by Fresnel, and by Lorentz. Mach's ether not only _conditions_ the

behaviour of inert masses, but _is also conditioned_ in its state

by them.

Mach's idea finds its full development in the ether of the general

theory of relativity. According to this theory the metrical

qualities of the continuum of space-time differ in the environment

of different points of space-time, and are partly conditioned by the

matter existing outside of the territory under consideration. This

space-time variability of the reciprocal relations of the standards

of space and time, or, perhaps, the recognition of the fact that

"empty space" in its physical relation is neither homogeneous nor

isotropic, compelling us to describe its state by ten functions (the

gravitation potentials g_(mn)), has, I think, finally disposed of

the view that space is physically empty. But therewith the

conception of the ether has again acquired an intelligible content,

although this content differs widely from that of the ether of the

mechanical undulatory theory of light. The ether of the general

theory of relativity is a medium which is itself devoid of _all_

mechanical and kinematical qualities, but helps to determine

mechanical (and electromagnetic) events.

What is fundamentally new in the ether of the general theory of

relativity as opposed to the ether of Lorentz consists in this, that

the state of the former is at every place determined by connections

with the matter and the state of the ether in neighbouring places,

which are amenable to law in the form of differential equations;

whereas the state of the Lorentzian ether in the absence of

electromagnetic fields is conditioned by nothing outside itself,

and is everywhere the same. The ether of the general theory of

relativity is transmuted conceptually into the ether of Lorentz if

we substitute constants for the functions of space which describe

the former, disregarding the causes which condition its state.

Thus we may also say, I think, that the ether of the general theory

of relativity is the outcome of the Lorentzian ether, through

relativation.

As to the part which the new ether is to play in the physics of

the future we are not yet clear. We know that it determines the

metrical relations in the space-time continuum, e.g. the configurative

possibilities of solid bodies as well as the gravitational fields;

but we do not know whether it has an essential share in the structure

of the electrical elementary particles constituting matter. Nor do

we know whether it is only in the proximity of ponderable masses

that its structure differs essentially from that of the Lorentzian

ether; whether the geometry of spaces of cosmic extent is approximately

Euclidean. But we can assert by reason of the relativistic equations

of gravitation that there must be a departure from Euclidean

relations, with spaces of cosmic order of magnitude, if there exists

a positive mean density, no matter how small, of the matter in the

universe. In this case the universe must of necessity be spatially

unbounded and of finite magnitude, its magnitude being determined

by the value of that mean density.

If we consider the gravitational field and the electromagnetic field

from the stand-point of the ether hypothesis, we find a remarkable

difference between the two. There can be no space nor any part

of space without gravitational potentials; for these confer upon

space its metrical qualities, without which it cannot be imagined

at all. The existence of the gravitational field is inseparably

bound up with the existence of space. On the other hand a part of

space may very well be imagined without an electromagnetic field;

thus in contrast with the gravitational field, the electromagnetic

field seems to be only secondarily linked to the ether, the formal

nature of the electromagnetic field being as yet in no way determined

by that of gravitational ether. From the present state of theory

it looks as if the electromagnetic field, as opposed to the

gravitational field, rests upon an entirely new formal _motif_,

as though nature might just as well have endowed the gravitational

ether with fields of quite another type, for example, with fields

of a scalar potential, instead of fields of the electromagnetic

type.

Since according to our present conceptions the elementary particles

of matter are also, in their essence, nothing else than condensations

of the electromagnetic field, our present view of the universe

presents two realities which are completely separated from each other

conceptually, although connected causally, namely, gravitational ether

and electromagnetic field, or--as they might also be called--space

and matter.

Of course it would be a great advance if we could succeed in

comprehending the gravitational field and the electromagnetic field

together as one unified conformation. Then for the first time the

epoch of theoretical physics founded by Faraday and Maxwell would

reach a satisfactory conclusion. The contrast between ether and

matter would fade away, and, through the general theory of relativity,

the whole of physics would become a complete system of thought,

like geometry, kinematics, and the theory of gravitation. An

exceedingly ingenious attempt in this direction has been made by

the mathematician H. Weyl; but I do not believe that his theory will

hold its ground in relation to reality. Further, in contemplating

the immediate future of theoretical physics we ought not unconditionally

to reject the possibility that the facts comprised in the quantum

theory may set bounds to the field theory beyond which it cannot

pass.

Recapitulating, we may say that according to the general theory of

relativity space is endowed with physical qualities; in this sense,

therefore, there exists an ether. According to the general theory

of relativity space without ether is unthinkable; for in such space

there not only would be no propagation of light, but also no possibility

of existence for standards of space and time (measuring-rods and

clocks), nor therefore any space-time intervals in the physical

sense. But this ether may not be thought of as endowed with the

quality characteristic of ponderable media, as consisting of parts

which may be tracked through time. The idea of motion may not be

applied to it.

GEOMETRY AND EXPERIENCE

An expanded form of an Address to the Prussian Academy of Sciences

in Berlin on January 27th, 1921.

One reason why mathematics enjoys special esteem, above all other

sciences, is that its laws are absolutely certain and indisputable,

while those of all other sciences are to some extent debatable and

in constant danger of being overthrown by newly discovered facts.

In spite of this, the investigator in another department of science

would not need to envy the mathematician if the laws of mathematics

referred to objects of our mere imagination, and not to objects

of reality. For it cannot occasion surprise that different persons

should arrive at the same logical conclusions when they have already

agreed upon the fundamental laws (axioms), as well as the methods

by which other laws are to be deduced therefrom. But there is another

reason for the high repute of mathematics, in that it is mathematics

which affords the exact natural sciences a certain measure of

security, to which without mathematics they could not attain.

At this point an enigma presents itself which in all ages has agitated

inquiring minds. How can it be that mathematics, being after all

a product of human thought which is independent of experience, is

so admirably appropriate to the objects of reality? Is human reason,

then, without experience, merely by taking thought, able to fathom

the properties of real things.

In my opinion the answer to this question is, briefly, this:--As far

as the laws of mathematics refer to reality, they are not certain;

and as far as they are certain, they do not refer to reality.

It seems to me that complete clearness as to this state of things

first became common property through that new departure in mathematics

which is known by the name of mathematical logic or "Axiomatics."

The progress achieved by axiomatics consists in its having neatly

separated the logical-formal from its objective or intuitive

content; according to axiomatics the logical-formal alone forms

the subject-matter of mathematics, which is not concerned with the

intuitive or other content associated with the logical-formal.

Let us for a moment consider from this point of view any axiom of

geometry, for instance, the following:--Through two points in space

there always passes one and only one straight line. How is this

axiom to be interpreted in the older sense and in the more modern

sense?

The older interpretation:--Every one knows what a straight line

is, and what a point is. Whether this knowledge springs from an

ability of the human mind or from experience, from some collaboration

of the two or from some other source, is not for the mathematician

to decide. He leaves the question to the philosopher. Being based

upon this knowledge, which precedes all mathematics, the axiom

stated above is, like all other axioms, self-evident, that is, it

is the expression of a part of this _a priori_ knowledge.

The more modern interpretation:--Geometry treats of entities which

are denoted by the words straight line, point, etc. These entities

do not take for granted any knowledge or intuition whatever, but

they presuppose only the validity of the axioms, such as the one

stated above, which are to be taken in a purely formal sense, i.e.

as void of all content of intuition or experience. These axioms are

free creations of the human mind. All other propositions of geometry

are logical inferences from the axioms (which are to be taken in

the nominalistic sense only). The matter of which geometry treats

is first defined by the axioms. Schlick in his book on epistemology has

therefore characterised axioms very aptly as "implicit definitions."

This view of axioms, advocated by modern axiomatics, purges mathematics

of all extraneous elements, and thus dispels the mystic obscurity

which formerly surrounded the principles of mathematics.

But a presentation of its principles thus clarified makes it also

evident that mathematics as such cannot predicate anything about

perceptual objects or real objects. In axiomatic geometry the words

"point," "straight line," etc., stand only for empty conceptual

schemata. That which gives them substance is not relevant to

mathematics.

Yet on the other hand it is certain that mathematics generally,

and particularly geometry, owes its existence to the need which

was felt of learning something about the relations of real things

to one another. The very word geometry, which, of course, means

earth-measuring, proves this. For earth-measuring has to do with

the possibilities of the disposition of certain natural objects

with respect to one another, namely, with parts of the earth,

measuring-lines, measuring-wands, etc. It is clear that the system

of concepts of axiomatic geometry alone cannot make any assertions

as to the relations of real objects of this kind, which we will

call practically-rigid bodies. To be able to make such assertions,

geometry must be stripped of its merely logical-formal character

by the co-ordination of real objects of experience with the empty

conceptual frame-work of axiomatic geometry. To accomplish this,

we need only add the proposition:--Solid bodies are related, with

respect to their possible dispositions, as are bodies in Euclidean

geometry of three dimensions. Then the propositions of Euclid contain

affirmations as to the relations of practically-rigid bodies.

Geometry thus completed is evidently a natural science; we may in

fact regard it as the most ancient branch of physics. Its affirmations

rest essentially on induction from experience, but not on logical

inferences only. We will call this completed geometry "practical

geometry," and shall distinguish it in what follows from "purely

axiomatic geometry." The question whether the practical geometry

of the universe is Euclidean or not has a clear meaning, and its

answer can only be furnished by experience. All linear measurement

in physics is practical geometry in this sense, so too is geodetic

and astronomical linear measurement, if we call to our help the

law of experience that light is propagated in a straight line, and

indeed in a straight line in the sense of practical geometry.

I attach special importance to the view of geometry which I

have just set forth, because without it I should have been unable

to formulate the theory of relativity. Without it the following

reflection would have been impossible:--In a system of reference

rotating relatively to an inert system, the laws of disposition of

rigid bodies do not correspond to the rules of Euclidean geometry

on account of the Lorentz contraction; thus if we admit non-inert

systems we must abandon Euclidean geometry. The decisive step in

the transition to general co-variant equations would certainly not

have been taken if the above interpretation had not served as a

stepping-stone. If we deny the relation between the body of axiomatic

Euclidean geometry and the practically-rigid body of reality,

we readily arrive at the following view, which was entertained by

that acute and profound thinker, H. Poincare:--Euclidean geometry

is distinguished above all other imaginable axiomatic geometries

by its simplicity. Now since axiomatic geometry by itself contains

no assertions as to the reality which can be experienced, but can

do so only in combination with physical laws, it should be possible

and reasonable--whatever may be the nature of reality--to retain

Euclidean geometry. For if contradictions between theory and

experience manifest themselves, we should rather decide to change

physical laws than to change axiomatic Euclidean geometry. If we

deny the relation between the practically-rigid body and geometry,

we shall indeed not easily free ourselves from the convention

that Euclidean geometry is to be retained as the simplest. Why

is the equivalence of the practically-rigid body and the body of

geometry--which suggests itself so readily--denied by Poincare and

other investigators? Simply because under closer inspection the

real solid bodies in nature are not rigid, because their geometrical

behaviour, that is, their possibilities of relative disposition,

depend upon temperature, external forces, etc. Thus the original,

immediate relation between geometry and physical reality appears

destroyed, and we feel impelled toward the following more general

view, which characterizes Poincare's standpoint. Geometry (G)

predicates nothing about the relations of real things, but only

geometry together with the purport (P) of physical laws can do so.

Using symbols, we may say that only the sum of (G) + (P) is subject

to the control of experience. Thus (G) may be chosen arbitrarily,

and also parts of (P); all these laws are conventions. All that

is necessary to avoid contradictions is to choose the remainder of

(P) so that (G) and the whole of (P) are together in accord with

experience. Envisaged in this way, axiomatic geometry and the part

of natural law which has been given a conventional status appear

as epistemologically equivalent.

_Sub specie aeterni_ Poincare, in my opinion, is right. The idea

of the measuring-rod and the idea of the clock co-ordinated with it

in the theory of relativity do not find their exact correspondence

in the real world. It is also clear that the solid body and the

clock do not in the conceptual edifice of physics play the part of

irreducible elements, but that of composite structures, which may

not play any independent part in theoretical physics. But it is my

conviction that in the present stage of development of theoretical

physics these ideas must still be employed as independent ideas;

for we are still far from possessing such certain knowledge

of theoretical principles as to be able to give exact theoretical

constructions of solid bodies and clocks.

Further, as to the objection that there are no really rigid bodies

in nature, and that therefore the properties predicated of rigid

bodies do not apply to physical reality,--this objection is by

no means so radical as might appear from a hasty examination. For

it is not a difficult task to determine the physical state of a

measuring-rod so accurately that its behaviour relatively to other

measuring-bodies shall be sufficiently free from ambiguity to allow

it to be substituted for the "rigid" body. It is to measuring-bodies

of this kind that statements as to rigid bodies must be referred.

All practical geometry is based upon a principle which is accessible

to experience, and which we will now try to realise. We will

call that which is enclosed between two boundaries, marked upon a

practically-rigid body, a tract. We imagine two practically-rigid

bodies, each with a tract marked out on it. These two tracts are

said to be "equal to one another" if the boundaries of the one tract

can be brought to coincide permanently with the boundaries of the

other. We now assume that:

If two tracts are found to be equal once and anywhere, they are

equal always and everywhere.

Not only the practical geometry of Euclid, but also its nearest

generalisation, the practical geometry of Riemann, and therewith

the general theory of relativity, rest upon this assumption. Of the

experimental reasons which warrant this assumption I will mention

only one. The phenomenon of the propagation of light in empty space

assigns a tract, namely, the appropriate path of light, to each

interval of local time, and conversely. Thence it follows that

the above assumption for tracts must also hold good for intervals

of clock-time in the theory of relativity. Consequently it may be

formulated as follows:--If two ideal clocks are going at the same

rate at any time and at any place (being then in immediate proximity

to each other), they will always go at the same rate, no matter where

and when they are again compared with each other at one place.--If

this law were not valid for real clocks, the proper frequencies

for the separate atoms of the same chemical element would not be

in such exact agreement as experience demonstrates. The existence

of sharp spectral lines is a convincing experimental proof of the

above-mentioned principle of practical geometry. This is the ultimate

foundation in fact which enables us to speak with meaning of the

mensuration, in Riemann's sense of the word, of the four-dimensional

continuum of space-time.

The question whether the structure of this continuum is Euclidean,

or in accordance with Riemann's general scheme, or otherwise,

is, according to the view which is here being advocated, properly

speaking a physical question which must be answered by experience,

and not a question of a mere convention to be selected on practical

grounds. Riemann's geometry will be the right thing if the laws

of disposition of practically-rigid bodies are transformable into

those of the bodies of Euclid's geometry with an exactitude which

increases in proportion as the dimensions of the part of space-time

under consideration are diminished.

It is true that this proposed physical interpretation of geometry

breaks down when applied immediately to spaces of sub-molecular

order of magnitude. But nevertheless, even in questions as

to the constitution of elementary particles, it retains part of

its importance. For even when it is a question of describing the

electrical elementary particles constituting matter, the attempt

may still be made to ascribe physical importance to those ideas

of fields which have been physically defined for the purpose

of describing the geometrical behaviour of bodies which are large

as compared with the molecule. Success alone can decide as to the

justification of such an attempt, which postulates physical reality

for the fundamental principles of Riemann's geometry outside of the

domain of their physical definitions. It might possibly turn out

that this extrapolation has no better warrant than the extrapolation

of the idea of temperature to parts of a body of molecular order

of magnitude.

It appears less problematical to extend the ideas of practical

geometry to spaces of cosmic order of magnitude. It might, of course,

be objected that a construction composed of solid rods departs more

and more from ideal rigidity in proportion as its spatial extent

becomes greater. But it will hardly be possible, I think, to assign

fundamental significance to this objection. Therefore the question

whether the universe is spatially finite or not seems to me

decidedly a pregnant question in the sense of practical geometry.

I do not even consider it impossible that this question will be

answered before long by astronomy. Let us call to mind what the

general theory of relativity teaches in this respect. It offers

two possibilities:--

1. The universe is spatially infinite. This can be so only if the

average spatial density of the matter in universal space, concentrated

in the stars, vanishes, i.e. if the ratio of the total mass of the

stars to the magnitude of the space through which they are scattered

approximates indefinitely to the value zero when the spaces taken

into consideration are constantly greater and greater.

2. The universe is spatially finite. This must be so, if there is

a mean density of the ponderable matter in universal space differing

from zero. The smaller that mean density, the greater is the volume

of universal space.

I must not fail to mention that a theoretical argument can be adduced in

favour of the hypothesis of a finite universe. The general theory

of relativity teaches that the inertia of a given body is greater as

there are more ponderable masses in proximity to it; thus it seems

very natural to reduce the total effect of inertia of a body to

action and reaction between it and the other bodies in the universe,

as indeed, ever since Newton's time, gravity has been completely

reduced to action and reaction between bodies. From the equations

of the general theory of relativity it can be deduced that this

total reduction of inertia to reciprocal action between masses--as

required by E. Mach, for example--is possible only if the universe

is spatially finite.

On many physicists and astronomers this argument makes no impression.

Experience alone can finally decide which of the two possibilities

is realised in nature. How can experience furnish an answer? At first

it might seem possible to determine the mean density of matter by

observation of that part of the universe which is accessible to our

perception. This hope is illusory. The distribution of the visible

stars is extremely irregular, so that we on no account may venture

to set down the mean density of star-matter in the universe as

equal, let us say, to the mean density in the Milky Way. In any

case, however great the space examined may be, we could not feel

convinced that there were no more stars beyond that space. So it

seems impossible to estimate the mean density. But there is another

road, which seems to me more practicable, although it also presents

great difficulties. For if we inquire into the deviations shown

by the consequences of the general theory of relativity which are

accessible to experience, when these are compared with the consequences

of the Newtonian theory, we first of all find a deviation which

shows itself in close proximity to gravitating mass, and has been

confirmed in the case of the planet Mercury. But if the universe

is spatially finite there is a second deviation from the Newtonian

theory, which, in the language of the Newtonian theory, may be

expressed thus:--The gravitational field is in its nature such as

if it were produced, not only by the ponderable masses, but also by

a mass-density of negative sign, distributed uniformly throughout

space. Since this factitious mass-density would have to be enormously

small, it could make its presence felt only in gravitating systems

of very great extent.

Assuming that we know, let us say, the statistical distribution

of the stars in the Milky Way, as well as their masses, then by

Newton's law we can calculate the gravitational field and the mean

velocities which the stars must have, so that the Milky Way should

not collapse under the mutual attraction of its stars, but should

maintain its actual extent. Now if the actual velocities of the stars,

which can, of course, be measured, were smaller than the calculated

velocities, we should have a proof that the actual attractions

at great distances are smaller than by Newton's law. From such a

deviation it could be proved indirectly that the universe is finite.

It would even be possible to estimate its spatial magnitude.

Can we picture to ourselves a three-dimensional universe which is

finite, yet unbounded?

The usual answer to this question is "No," but that is not the right

answer. The purpose of the following remarks is to show that the

answer should be "Yes." I want to show that without any extraordinary

difficulty we can illustrate the theory of a finite universe by

means of a mental image to which, with some practice, we shall soon

grow accustomed.

First of all, an observation of epistemological nature. A

geometrical-physical theory as such is incapable of being directly

pictured, being merely a system of concepts. But these concepts

serve the purpose of bringing a multiplicity of real or imaginary

sensory experiences into connection in the mind. To "visualise"

a theory, or bring it home to one's mind, therefore means to give

a representation to that abundance of experiences for which the

theory supplies the schematic arrangement. In the present case we

have to ask ourselves how we can represent that relation of solid

bodies with respect to their reciprocal disposition (contact) which

corresponds to the theory of a finite universe. There is really

nothing new in what I have to say about this; but innumerable

questions addressed to me prove that the requirements of those who

thirst for knowledge of these matters have not yet been completely

satisfied.

So, will the initiated please pardon me, if part of what I shall

bring forward has long been known?

What do we wish to express when we say that our space is infinite?

Nothing more than that we might lay any number whatever of bodies

of equal sizes side by side without ever filling space. Suppose

that we are provided with a great many wooden cubes all of the

same size. In accordance with Euclidean geometry we can place them

above, beside, and behind one another so as to fill a part of space

of any dimensions; but this construction would never be finished;

we could go on adding more and more cubes without ever finding

that there was no more room. That is what we wish to express when

we say that space is infinite. It would be better to say that space

is infinite in relation to practically-rigid bodies, assuming that

the laws of disposition for these bodies are given by Euclidean

geometry.

Another example of an infinite continuum is the plane. On a plane

surface we may lay squares of cardboard so that each side of any

square has the side of another square adjacent to it. The construction

is never finished; we can always go on laying squares--if their laws

of disposition correspond to those of plane figures of Euclidean

geometry. The plane is therefore infinite in relation to the

cardboard squares. Accordingly we say that the plane is an infinite

continuum of two dimensions, and space an infinite continuum of

three dimensions. What is here meant by the number of dimensions,

I think I may assume to be known.

Now we take an example of a two-dimensional continuum which is

finite, but unbounded. We imagine the surface of a large globe and

a quantity of small paper discs, all of the same size. We place

one of the discs anywhere on the surface of the globe. If we move

the disc about, anywhere we like, on the surface of the globe,

we do not come upon a limit or boundary anywhere on the journey.

Therefore we say that the spherical surface of the globe is an

unbounded continuum. Moreover, the spherical surface is a finite

continuum. For if we stick the paper discs on the globe, so that

no disc overlaps another, the surface of the globe will finally

become so full that there is no room for another disc. This simply

means that the spherical surface of the globe is finite in relation

to the paper discs. Further, the spherical surface is a non-Euclidean

continuum of two dimensions, that is to say, the laws of disposition

for the rigid figures lying in it do not agree with those of the

Euclidean plane. This can be shown in the following way. Place

a paper disc on the spherical surface, and around it in a circle

place six more discs, each of which is to be surrounded in turn

by six discs, and so on. If this construction is made on a plane

surface, we have an uninterrupted disposition in which there are

six discs touching every disc except those which lie on the outside.

[Figure 1: Discs maximally packed on a plane]

On the spherical surface the construction also seems to promise

success at the outset, and the smaller the radius of the disc

in proportion to that of the sphere, the more promising it seems.

But as the construction progresses it becomes more and more patent

that the disposition of the discs in the manner indicated, without

interruption, is not possible, as it should be possible by Euclidean

geometry of the the plane surface. In this way creatures which

cannot leave the spherical surface, and cannot even peep out from

the spherical surface into three-dimensional space, might discover,

merely by experimenting with discs, that their two-dimensional

"space" is not Euclidean, but spherical space.

From the latest results of the theory of relativity it is probable

that our three-dimensional space is also approximately spherical,

that is, that the laws of disposition of rigid bodies in it are

not given by Euclidean geometry, but approximately by spherical

geometry, if only we consider parts of space which are sufficiently

great. Now this is the place where the reader's imagination boggles.

"Nobody can imagine this thing," he cries indignantly. "It can be

said, but cannot be thought. I can represent to myself a spherical

surface well enough, but nothing analogous to it in three dimensions."

[Figure 2: A circle projected from a sphere onto a plane]

We must try to surmount this barrier in the mind, and the patient

reader will see that it is by no means a particularly difficult

task. For this purpose we will first give our attention once more to

the geometry of two-dimensional spherical surfaces. In the adjoining

figure let _K_ be the spherical surface, touched at _S_ by a plane,

_E_, which, for facility of presentation, is shown in the drawing as

a bounded surface. Let _L_ be a disc on the spherical surface. Now

let us imagine that at the point _N_ of the spherical surface,

diametrically opposite to _S_, there is a luminous point, throwing a

shadow _L'_ of the disc _L_ upon the plane _E_. Every point on the

sphere has its shadow on the plane. If the disc on the sphere _K_ is

moved, its shadow _L'_ on the plane _E_ also moves. When the disc

_L_ is at _S_, it almost exactly coincides with its shadow. If it

moves on the spherical surface away from _S_ upwards, the disc

shadow _L'_ on the plane also moves away from _S_ on the plane

outwards, growing bigger and bigger. As the disc _L_ approaches the

luminous point _N_, the shadow moves off to infinity, and becomes

infinitely great.

Now we put the question, What are the laws of disposition of the

disc-shadows _L'_ on the plane _E_? Evidently they are exactly the

same as the laws of disposition of the discs _L_ on the spherical

surface. For to each original figure on _K_ there is a corresponding

shadow figure on _E_. If two discs on _K_ are touching, their

shadows on _E_ also touch. The shadow-geometry on the plane agrees

with the the disc-geometry on the sphere. If we call the disc-shadows

rigid figures, then spherical geometry holds good on the plane _E_

with respect to these rigid figures. Moreover, the plane is finite

with respect to the disc-shadows, since only a finite number of

the shadows can find room on the plane.

At this point somebody will say, "That is nonsense. The disc-shadows

are _not_ rigid figures. We have only to move a two-foot rule about

on the plane _E_ to convince ourselves that the shadows constantly

increase in size as they move away from _S_ on the plane towards

infinity." But what if the two-foot rule were to behave on the

plane _E_ in the same way as the disc-shadows _L'_? It would then

be impossible to show that the shadows increase in size as they

move away from _S_; such an assertion would then no longer have

any meaning whatever. In fact the only objective assertion that can

be made about the disc-shadows is just this, that they are related

in exactly the same way as are the rigid discs on the spherical

surface in the sense of Euclidean geometry.

We must carefully bear in mind that our statement as to the growth

of the disc-shadows, as they move away from _S_ towards infinity,

has in itself no objective meaning, as long as we are unable to

employ Euclidean rigid bodies which can be moved about on the plane

_E_ for the purpose of comparing the size of the disc-shadows. In

respect of the laws of disposition of the shadows _L'_, the point

_S_ has no special privileges on the plane any more than on the

spherical surface.

The representation given above of spherical geometry on the

plane is important for us, because it readily allows itself to be

transferred to the three-dimensional case.

Let us imagine a point _S_ of our space, and a great number

of small spheres, _L'_, which can all be brought to coincide with

one another. But these spheres are not to be rigid in the sense

of Euclidean geometry; their radius is to increase (in the sense

of Euclidean geometry) when they are moved away from _S_ towards

infinity, and this increase is to take place in exact accordance

with the same law as applies to the increase of the radii of the

disc-shadows _L'_ on the plane.

After having gained a vivid mental image of the geometrical

behaviour of our _L'_ spheres, let us assume that in our space there

are no rigid bodies at all in the sense of Euclidean geometry, but

only bodies having the behaviour of our _L'_ spheres. Then we shall

have a vivid representation of three-dimensional spherical space,

or, rather of three-dimensional spherical geometry. Here our spheres

must be called "rigid" spheres. Their increase in size as they

depart from _S_ is not to be detected by measuring with

measuring-rods, any more than in the case of the disc-shadows on

_E_, because the standards of measurement behave in the same way as

the spheres. Space is homogeneous, that is to say, the same

spherical configurations are possible in the environment of all

points.* Our space is finite, because, in consequence of the

"growth" of the spheres, only a finite number of them can find room

in space.

* This is intelligible without calculation--but only for the

two-dimensional case--if we revert once more to the case of the disc

on the surface of the sphere.

In this way, by using as stepping-stones the practice in thinking

and visualisation which Euclidean geometry gives us, we have acquired

a mental picture of spherical geometry. We may without difficulty

impart more depth and vigour to these ideas by carrying out special

imaginary constructions. Nor would it be difficult to represent the

case of what is called elliptical geometry in an analogous manner.

My only aim to-day has been to show that the human faculty of

visualisation is by no means bound to capitulate to non-Euclidean

geometry.