,,sidelights ,,on ,,relativity  ,by     
,albert ,einstein  ,contents  ,,ether   
,,and ,,the ,,theory ,,of ,,relativity  
,an ,address delivered on ,may #eth1    
#aibj1 in the ,university of ,leyden    
,,geometry ,,and ,,experience  ,an      
expanded form of an ,address to the     
,prussian ,academy of ,sciences in      
,berlin on ,january #bgth1 #aiba4       
,,ether ,,and ,,the ,,theory ,,of       
,,relativity  ,an ,address delivered on 
,may #eth1 #aibj1 in the ,university of 
,leyden    ,how does it come about that 
alongside of the idea of ponderable     
matter1 which is derived by abstraction 
from everyday life1 the physicists set  
the idea of the existence of another    
kind of matter1 the ether8 ,the         
explanation is probably to be sought in 
those phenomena which have given rise to
 the theory of action at a distance1 and
 in the properties of light which have  
led to the undulatory theory4 ,let us   
devote a little while to the            
consideration of these two subjects4    

,outside of physics we know nothing of  
action at a distance4 ,when we try to   
connect cause and effect in the         
experiences which natural objects afford
 us1 it seems at first as if there were 
no other mutual actions than those of   
immediate contact1 e4g4 the             
communication of motion by impact1 push 
and pull1 heating or inducing combustion
 by means of a flame1 etc4 ,it is true  
that even in everyday experience weight1
 which is in a sense action at a        
distance1 plays a very important part4  
,but since in daily experience the      
weight of bodies meets us as something  
constant1 something not linked to any   
cause which is variable in time or      
place1 we do not in everyday life       
speculate as to the cause of gravity1   
and therefore do not become conscious of
 its character as action at a distance4 
,it was ,newton's theory of gravitation 
that first assigned a cause for gravity 
by interpreting it as action at a       
distance1 proceeding from masses4       

,newton's theory is probably the        
greatest stride ever made in the effort 
towards the causal nexus of natural     
phenomena4 ,and yet this theory evoked a
 lively sense of discomfort among       
,newton's contemporaries1 because it    
seemed to be in conflict with the       
principle springing from the rest of    
experience1 that there can be reciprocal
 action only through contact1 and not   
through immediate action at a distance4 
,it is only with reluctance that man's  
desire for knowledge endures a dualism  
of this kind4 ,how was unity to be      
preserved in his comprehension of the   
forces of nature8 ,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 followers1 who were     
entirely under the spell of his         
doctrine1 mostly preferred to take2 or  
by assuming that the ,newtonian action  
at a distance is only apparently        

immediate action at a distance1 but in  
truth is conveyed by a medium permeating
 space1 whether by movements or by      
elastic deformation of this medium4     
,thus the endeavour toward a unified    
view of the nature of forces leads to   
the hypothesis of an ether4 ,this       
hypothesis1 to be sure1 did not at first
 bring with it any advance in the theory
 of gravitation or in physics generally1
 so that it became customary to treat   
,newton's law of force as an axiom not  
further reducible4 ,but the ether       
hypothesis was bound always to play some
 part in physical science1 even if at   
first only a latent part4  ,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 bodies1 the ether hypothesis 
found fresh support4 ,it appeared beyond
 question that light must be interpreted
 as a vibratory process in an elastic1  
inert medium filling up universal space4

 ,it also seemed to be a necessary      
consequence of the fact that light is   
capable of polarisation that this       
medium1 the ether1 must be of the nature
 of a solid body1 because transverse    
waves are not possible in a fluid1 but  
only in a solid4 ,thus the physicists   
were bound to arrive at the theory of   
the quasi-rigid luminiferous ether1 the 
parts of which can carry out no         
movements relatively to one another     
except the small movements of           
deformation which correspond to         
light-waves4  ,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 relativity1 the       
experiment of ,fizeau1 from which one   
was obliged to infer that the           
luminiferous ether does not take part in
 the movements of bodies4 ,the          
phenomenon of aberration also favoured  
the theory of the quasi-rigid ether4    

,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   
turn4 ,for ,maxwell himself the ether   
indeed still had properties which were  
purely mechanical1 although of a much   
more complicated kind than the          
mechanical properties of tangible solid 
bodies4 ,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 field4 ,the laws were 
clear and simple1 the mechanical        
interpretations clumsy and              
contradictory4 ,almost imperceptibly the
 theoretical physicists adapted         
themselves to a situation which1 from   
the standpoint of their mechanical      
programme1 was very depressing4 ,they   
were particularly influenced by the     
electro-dynamical investigations of     

,heinrich ,hertz4 ,for whereas they     
previously had required of a conclusive 
theory that it should content itself    
with the fundamental concepts which     
belong exclusively to mechanics 7e4g4   
densities1 velocities1 deformations1    
stresses7 they gradually accustomed     
themselves to admitting electric and    
magnetic force as fundamental concepts  
side by side with those of mechanics1   
without requiring a mechanical          
interpretation for them4 ,thus the      
purely mechanical view of nature was    
gradually abandoned4 ,but this change   
led to a fundamental dualism which in   
the long-run was insupportable4 ,a way  
of escape was now sought in the reverse 
direction1 by reducing the principles of
 mechanics to those of electricity1 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 rays4  ,this dualism still      
confronts us in unextenuated form in the

 theory of ,hertz1 where matter appears 
not only as the bearer of velocities1   
kinetic energy1 and mechanical          
pressures1 but also as the bearer of    
electromagnetic fields4 ,since such     
fields also occur in vacuo--i4e4 in free
 ether--the ether also appears as bearer
 of electromagnetic fields4 ,the ether  
appears indistinguishable in its        
functions from ordinary matter4 ,within 
matter it takes part in the motion of   
matter and in empty space it has        
everywhere a velocity2 so that the ether
 has a definitely assigned velocity     
throughout the whole of space4 ,there is
 no fundamental difference between      
,hertz's ether and ponderable matter    
7which in part subsists in the ether74  
,the ,hertz theory suffered not only    
from the defect of ascribing to matter  
and ether1 on the one hand mechanical   
states1 and on the other hand electrical
 states1 which do not stand in any      
conceivable relation to each other2 it  
was also at variance with the result of 

,fizeau's important experiment on the   
velocity of the propagation of light in 
moving fluids1 and with other           
established experimental results4  ,such
 was the state of things when ,h4 ,a4   
,lorentz entered upon the scene4 ,he    
brought theory into harmony with        
experience by means of a wonderful      
simplification of theoretical           
principles4 ,he achieved this1 the most 
important advance in the theory of      
electricity since ,maxwell1 by taking   
from ether its mechanical1 and from     
matter its electromagnetic qualities4   
,as in empty space1 so too in the       
interior of material bodies1 the ether1 
and not matter viewed atomistically1 was
 exclusively the seat of electromagnetic
 fields4 ,according to ,lorentz the     
elementary particles of matter alone are
 capable of carrying out movements2     
their electromagnetic activity is       
entirely confined to the carrying of    
electric charges4 ,thus ,lorentz        
succeeded in reducing all               

electromagnetic happenings to ,maxwell's
 equations for free space4  ,as to the  
mechanical nature of the ,lorentzian    
ether1 it may be said of it1 in a       
somewhat playful spirit1 that immobility
 is the only mechanical property of     
which it has not been deprived by ,h4   
,a4 ,lorentz4 ,it may be added that the 
whole change in the conception of the   
ether which the special theory of       
relativity brought about1 consisted in  
taking away from the ether its last     
mechanical quality1 namely1 its         
immobility4 ,how this is to be          
understood will forthwith be expounded4 
,the space-time theory and the          
kinematics of the special theory of     
relativity were modelled on the         
,maxwell-,lorentz theory of the         
electromagnetic field4 ,this theory     
therefore satisfies the conditions of   
the special theory of relativity1 but   
when viewed from the latter it acquires 
a novel aspect4 ,for if ,k be a system  
of co-ordinates relatively to which the 

,lorentzian ether is at rest1 the       
,maxwell-,lorentz equations are valid   
primarily with reference to ,k4 ,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 ,k4 ,now comes the anxious          
question3--,why must ,i in the theory   
distinguish the ,k system above all ,k' 
systems1 which are physically equivalent
 to it in all respects1 by assuming that
 the ether is at rest relatively to the 
,k system8 ,for the theoretician such an
 asymmetry in the theoretical structure1
 with no corresponding asymmetry in the 
system of experience1 is intolerable4   
,if we assume the ether to be at rest   
relatively to ,k1 but in motion         
relatively to ,k'1 the physical         
equivalence of ,k and ,k' seems to me   
from the logical standpoint1 not indeed 
downright incorrect1 but nevertheless   
inacceptable4  ,the next position which 

it was possible to take up in face of   
this state of things appeared to be the 
following4 ,the ether does not exist at 
all4 ,the electromagnetic fields are not
 states of a medium1 and are not bound  
down to any bearer1 but they are        
independent realities which are not     
reducible to anything else1 exactly like
 the atoms of ponderable matter4 ,this  
conception suggests itself the more     
readily as1 according to ,lorentz's     
theory1 electromagnetic radiation1 like 
ponderable matter1 brings impulse and   
energy with it1 and as1 according to the
 special theory of relativity1 both     
matter and radiation are but special    
forms of distributed energy1 ponderable 
mass losing its isolation and appearing 
as a special form of energy4  ,more     
careful reflection teaches us1 however1 
that the special theory of relativity   
does not compel us to deny ether4 ,we   
may assume the existence of an ether2   
only we must give up ascribing a        
definite state of motion to it1 i4e4 we 

must by abstraction take from it the    
last mechanical characteristic which    
,lorentz had still left it4 ,we shall   
see later that this point of view1 the  
conceivability of which ,i shall at once
 endeavour to make more intelligible by 
a somewhat halting comparison1 is       
justified by the results of the general 
theory of relativity4  ,think of waves  
on the surface of water4 ,here we can   
describe two entirely different things4 
,either we may observe how the          
undulatory surface forming the boundary 
between water and air alters in the     
course of time2 or else--with the help  
of small floats1 for instance--we can   
observe how the position of the separate
 particles of water alters in the course
 of time4 ,if the existence of such     
floats for tracking the motion of the   
particles of a fluid were a fundamental 
impossibility in physics--if1 in fact1  
nothing else whatever were observable   
than the shape of the space occupied by 
the water as it varies in time1 we      

should have no ground for the assumption
 that water consists of movable         
particles4 ,but all the same we could   
characterise it as a medium4  ,we have  
something like this in the              
electromagnetic field4 ,for we may      
picture the field to ourselves as       
consisting of lines of force4  ,if we   
wish to interpret these lines of force  
to ourselves as something material in   
the ordinary sense1 we are tempted to   
interpret the dynamic processes as      
motions of these lines of force1 such   
that each separate line of force is     
tracked through the course of time4 ,it 
is well known1 however1 that this way of
 regarding the electromagnetic field    
leads to contradictions4  ,generalising 
we must say this3--,there may be        
supposed to be extended physical objects
 to which the idea of motion cannot be  
applied4 ,they may not be thought of as 
consisting of particles which allow     
themselves to be separately tracked     
through time4 ,in ,minkowski's idiom    

this is expressed as follows3--,not     
every extended conformation in the      
four-dimensional world can be regarded  
as composed of world-threads4 ,the      
special theory of relativity forbids us 
to assume the ether to consist of       
particles observable through time1 but  
the hypothesis of ether in itself is not
 in conflict with the special theory of 
relativity4 ,only we must be on our     
guard against ascribing a state of      
motion to the ether4  ,certainly1 from  
the standpoint of the special theory of 
relativity1 the ether hypothesis appears
 at first to be an empty hypothesis4 ,in
 the equations of the electromagnetic   
field there occur1 in addition to the   
densities of the electric charge1 only  
the intensities of the field4 ,the      
career of electromagnetic processes in  
vacuo appears to be completely          
determined by these equations1          
uninfluenced by other physical          
quantities4 ,the electromagnetic fields 
appear as ultimate1 irreducible         

realities1 and at first it seems        
superfluous to postulate a homogeneous1 
isotropic ether-medium1 and to envisage 
electromagnetic fields as states of this
 medium4  ,but on the other hand there  
is a weighty argument to be adduced in  
favour of the ether hypothesis4 ,to deny
 the ether is ultimately to assume that 
empty space has no physical qualities   
whatever4 ,the fundamental facts of     
mechanics do not harmonize with this    
view4 ,for the mechanical behaviour of a
 corporeal system hovering freely in    
empty space depends not only on relative
 positions 7distances7 and relative     
velocities1 but also on its state of    
rotation1 which physically may be taken 
as a characteristic not appertaining to 
the system in itself4 ,in order to be   
able to look upon the rotation of the   
system1 at least formally1 as something 
real1 ,newton objectivises space4       
,since he classes his absolute space    
together with real things1 for him      
rotation relative to an absolute space  

is also something real4 ,newton might no
 less well have called his absolute     
space ,ether2 what is essential is      
merely that besides observable objects1 
another thing1 which is not perceptible1
 must be looked upon as real1 to enable 
acceleration or rotation to be looked   
upon as something real4  ,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 
space4 ,but inertial resistance opposed 
to relative acceleration of distant     
masses presupposes action at a distance2
 and as the modern physicist does not   
believe that he may accept this action  
at a distance1 he comes back once more1 
if he follows ,mach1 to the ether1 which
 has to serve as medium for the effects 
of inertia4 ,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 ,newton1 by   
,fresnel1 and by ,lorentz4 ,mach's ether
 not only conditions the behaviour of   
inert masses1 but is also conditioned in
 its state by them4  ,mach's idea finds 
its full development in the ether of the
 general theory of relativity4          
,according to this theory the metrical  
qualities of the continuum of space-time
 differ in the environment of different 
points of space-time1 and are partly    
conditioned by the matter existing      
outside of the territory under          
consideration4  ,this space-time        
variability of the reciprocal relations 
of the standards of space and time1 or1 
perhaps1 the recognition of the fact    
that empty space in its physical        
relation is neither homogeneous nor     
isotropic1 compelling us to describe its
 state by ten functions 7the gravitation
 potentials g7mn771 has1 ,i think1      
finally disposed of the view that space 
is physically empty4 ,but therewith the 

conception of the ether has again       
acquired an intelligible content1       
although this content differs widely    
from that of the ether of the mechanical
 undulatory theory of light4  ,the ether
 of the general theory of relativity is 
a medium which is itself devoid of all  
mechanical and kinematical qualities1   
but helps to determine mechanical 7and  
electromagnetic7 events4  ,what is      
fundamentally new in the ether of the   
general theory of relativity as opposed 
to the ether of ,lorentz consists in    
this1 that the state of the former is at
 every place determined by connections  
with the matter and the state of the    
ether in neighbouring places1 which are 
amenable to law in the form of          
differential equations2 whereas the     
state of the ,lorentzian ether in the   
absence of electromagnetic fields is    
conditioned by nothing outside itself1  
and is everywhere the same4 ,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 former1 disregarding the   
causes which condition its state4 ,thus 
we may also say1 ,i think1 that the     
ether of the general theory of          
relativity is the outcome of the        
,lorentzian ether1 through relativation4
  ,as to the part which the new ether is
 to play in the physics of the future we
 are not yet clear4 ,we know that it    
determines the metrical relations in the
 space-time continuum1 e4g4 the         
configurative possibilities of solid    
bodies as well as the gravitational     
fields2 but we do not know whether it   
has an essential share in the structure 
of the electrical elementary particles  
constituting matter4 ,nor do we know    
whether it is only in the proximity of  
ponderable masses that its structure    
differs essentially from that of the    
,lorentzian ether2 whether the geometry 
of spaces of cosmic extent is           
approximately ,euclidean4 ,but we can   

assert by reason of the relativistic    
equations of gravitation that there must
 be a departure from ,euclidean         
relations1 with spaces of cosmic order  
of magnitude1 if there exists a positive
 mean density1 no matter how small1 of  
the matter in the universe4 ,in this    
case the universe must of necessity be  
spatially unbounded and of finite       
magnitude1 its magnitude being          
determined by the value of that mean    
density4  ,if we consider the           
gravitational field and the             
electromagnetic field from the          
stand-point of the ether hypothesis1 we 
find a remarkable difference between the
 two4 ,there can be no space nor any    
part of space without gravitational     
potentials2 for these confer upon space 
its metrical qualities1 without which it
 cannot be imagined at all4 ,the        
existence of the gravitational field is 
inseparably bound up with the existence 
of space4 ,on the other hand a part of  
space may very well be imagined without 

an electromagnetic field2 thus in       
contrast with the gravitational field1  
the electromagnetic field seems to be   
only secondarily linked to the ether1   
the formal nature of the electromagnetic
 field being as yet in no way determined
 by that of gravitational ether4 ,from  
the present state of theory it looks as 
if the electromagnetic field1 as opposed
 to the gravitational field1 rests upon 
an entirely new formal motif1 as though 
nature might just as well have endowed  
the gravitational ether with fields of  
quite another type1 for example1 with   
fields of a scalar potential1 instead of
 fields of the electromagnetic type4    
,since according to our present         
conceptions the elementary particles of 
matter are also1 in their essence1      
nothing else than condensations of the  
electromagnetic field1 our present view 
of the universe presents two realities  
which are completely separated from each
 other conceptually1 although connected 
causally1 namely1 gravitational ether   

and electromagnetic field1 or--as they  
might also be called--space and matter4 
,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 conformation4 ,then for the     
first time the epoch of theoretical     
physics founded by ,faraday and ,maxwell
 would reach a satisfactory conclusion4 
,the contrast between ether and matter  
would fade away1 and1 through the       
general theory of relativity1 the whole 
of physics would become a complete      
system of thought1 like geometry1       
kinematics1 and the theory of           
gravitation4  ,an exceedingly ingenious 
attempt in this direction has been made 
by the mathematician ,h4 ,weyl2 but ,i  
do not believe that his theory will hold
 its ground in relation to reality4     
,further1 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    
pass4  ,recapitulating1 we may say that 
according to the general theory of      
relativity space is endowed with        
physical qualities2 in this sense1      
therefore1 there exists an ether4       
,according to the general theory of     
relativity space without ether is       
unthinkable2 for in such space there not
 only would be no propagation of light1 
but also no possibility of existence for
 standards of space and time            
7measuring-rods and clocks71 nor        
therefore any space-time intervals in   
the physical sense4 ,but this ether may 
not be thought of as endowed with the   
quality characteristic of ponderable    
media1 as consisting of parts which may 
be tracked through time4 ,the idea of   
motion may not be applied to it4        
,,geometry ,,and ,,experience  ,an      
expanded form of an ,address to the     
,prussian ,academy of ,sciences in      
,berlin on ,january #bgth1 #aiba4       

,one reason why mathematics enjoys      
special esteem1 above all other         
sciences1 is that its laws are          
absolutely certain and indisputable1    
while those of all other sciences are to
 some extent debatable and in constant  
danger of being overthrown by newly     
discovered facts4 ,in spite of this1 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        
imagination1 and not to objects of      
reality4 ,for it cannot occasion        
surprise that different persons should  
arrive at the same logical conclusions  
when they have already agreed upon the  
fundamental laws 7axioms71 as well as   
the methods by which other laws are to  
be deduced therefrom4 ,but there is     
another reason for the high repute of   
mathematics1 in that it is mathematics  
which affords the exact natural sciences
 a certain measure of security1 to which
 without mathematics they could not     

attain4  ,at this point an enigma       
presents itself which in all ages has   
agitated inquiring minds4 ,how can it be
 that mathematics1 being after all a    
product of human thought which is       
independent of experience1 is so        
admirably appropriate to the objects of 
reality8 ,is human reason1 then1 without
 experience1 merely by taking thought1  
able to fathom the properties of real   
things4  ,in my opinion the answer to   
this question is1 briefly1 this3--,as   
far as the laws of mathematics refer to 
reality1 they are not certain2 and as   
far as they are certain1 they do not    
refer to reality4 ,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 ,axiomatics4   
,the progress achieved by axiomatics    
consists in its having neatly separated 
the logical-formal from its objective or
 intuitive content2 according to        

axiomatics the logical-formal alone     
forms the subject-matter of mathematics1
 which is not concerned with the        
intuitive or other content associated   
with the logical-formal4  ,let us for a 
moment consider from this point of view 
any axiom of geometry1 for instance1 the
 following3--,through two points in     
space there always passes one and only  
one straight line4 ,how is this axiom to
 be interpreted in the older sense and  
in the more modern sense8  ,the older   
interpretation3--,every one knows what a
 straight line is1 and what a point is4 
,whether this knowledge springs from an 
ability of the human mind or from       
experience1 from some collaboration of  
the two or from some other source1 is   
not for the mathematician to decide4 ,he
 leaves the question to the philosopher4
 ,being based upon this knowledge1 which
 precedes all mathematics1 the axiom    
stated above is1 like all other axioms1 
self-evident1 that is1 it is the        
expression of a part of this a priori   

knowledge4  ,the more modern            
interpretation3--,geometry treats of    
entities which are denoted by the words 
straight line1 point1 etc4 ,these       
entities do not take for granted any    
knowledge or intuition whatever1 but    
they presuppose only the validity of the
 axioms1 such as the one stated above1  
which are to be taken in a purely formal
 sense1 i4e4 as void of all content of  
intuition or experience4 ,these axioms  
are free creations of the human mind4   
,all other propositions of geometry are 
logical inferences from the axioms      
7which are to be taken in the           
nominalistic sense only74 ,the matter of
 which geometry treats is first defined 
by the axioms4 ,schlick in his book on  
epistemology has therefore characterised
 axioms very aptly as implicit          
definitions4  ,this view of axioms1     
advocated by modern axiomatics1 purges  
mathematics of all extraneous elements1 
and thus dispels the mystic obscurity   
which formerly surrounded the principles

 of mathematics4  ,but a presentation of
 its principles thus clarified makes it 
also evident that mathematics as such   
cannot predicate anything about         
perceptual objects or real objects4 ,in 
axiomatic geometry the words point1     
straight line1 etc41 stand only for     
empty conceptual schemata4 ,that which  
gives them substance is not relevant to 
mathematics4  ,yet on the other hand it 
is certain that mathematics generally1  
and particularly geometry1 owes its     
existence to the need which was felt of 
learning something about the relations  
of real things to one another4 ,the very
 word geometry1 which1 of course1 means 
earth-measuring1 proves this4 ,for      
earth-measuring has to do with the      
possibilities of the disposition of     
certain natural objects with respect to 
one another1 namely1 with parts of the  
earth1 measuring-lines1 measuring-wands1
 etc4 ,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 kind1 
which we will call practically-rigid    
bodies4 ,to be able to make such        
assertions1 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 geometry4 ,to   
accomplish this1 we need only add the   
proposition3--,solid bodies are related1
 with respect to their possible         
dispositions1 as are bodies in          
,euclidean geometry of three dimensions4
 ,then the propositions of ,euclid      
contain affirmations as to the relations
 of practically-rigid bodies4  ,geometry
 thus completed is evidently a natural  
science2 we may in fact regard it as the
 most ancient branch of physics4 ,its   
affirmations rest essentially on        
induction from experience1 but not on   
logical inferences only4 ,we will call  
this completed geometry practical       
geometry1 and shall distinguish it in   
what follows from purely axiomatic      

geometry4 ,the question whether the     
practical geometry of the universe is   
,euclidean or not has a clear meaning1  
and its answer can only be furnished by 
experience4 ,all linear measurement in  
physics is practical geometry in this   
sense1 so too is geodetic and           
astronomical linear measurement1 if we  
call to our help the law of experience  
that light is propagated in a straight  
line1 and indeed in a straight line in  
the sense of practical geometry4  ,i    
attach special importance to the view of
 geometry which ,i have just set forth1 
because without it ,i should have been  
unable to formulate the theory of       
relativity4 ,without it the following   
reflection would have been              
impossible3--,in a system of reference  
rotating relatively to an inert system1 
the laws of disposition of rigid bodies 
do not correspond to the rules of       
,euclidean geometry on account of the   
,lorentz contraction2 thus if we admit  
non-inert systems we must abandon       

,euclidean geometry4 ,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-stone4 ,if we  
deny the relation between the body of   
axiomatic ,euclidean geometry and the   
practically-rigid body of reality1 we   
readily arrive at the following view1   
which was entertained by that acute and 
profound thinker1 ,h4                   
,poincare3--,euclidean geometry is      
distinguished above all other imaginable
 axiomatic geometries by its simplicity4
 ,now since axiomatic geometry by itself
 contains no assertions as to the       
reality which can be experienced1 but   
can do so only in combination with      
physical laws1 it should be possible and
 reasonable--whatever may be the nature 
of reality--to retain ,euclidean        
geometry4 ,for if contradictions between
 theory and experience manifest         
themselves1 we should rather decide to  
change physical laws than to change     

axiomatic ,euclidean geometry4 ,if we   
deny the relation between the           
practically-rigid body and geometry1 we 
shall indeed not easily free ourselves  
from the convention that ,euclidean     
geometry is to be retained as the       
simplest4 ,why is the equivalence of the
 practically-rigid body and the body of 
geometry--which suggests itself so      
readily--denied by ,poincare and other  
investigators8 ,simply because under    
closer inspection the real solid bodies 
in nature are not rigid1 because their  
geometrical behaviour1 that is1 their   
possibilities of relative disposition1  
depend upon temperature1 external       
forces1 etc4 ,thus the original1        
immediate relation between geometry and 
physical reality appears destroyed1 and 
we feel impelled toward the following   
more general view1 which characterizes  
,poincare's standpoint4 ,geometry 7,g7  
predicates nothing about the relations  
of real things1 but only geometry       
together with the purport 7,p7 of       

physical laws can do so4 ,using symbols1
 we may say that only the sum of 7,g7   
7,p7 is subject to the control of       
experience4 ,thus 7,g7 may be chosen    
arbitrarily1 and also parts of 7,p72 all
 these laws are conventions4 ,all that  
is necessary to avoid contradictions is 
to choose the remainder of 7,p7 so that 
7,g7 and the whole of 7,p7 are together 
in accord with experience4 ,envisaged in
 this way1 axiomatic geometry and the   
part of natural law which has been given
 a conventional status appear as        
epistemologically equivalent4  ,sub     
specie aeterni ,poincare1 in my opinion1
 is right4 ,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 world4 ,it is
 also clear that the solid body and the 
clock do not in the conceptual edifice  
of physics play the part of irreducible 
elements1 but that of composite         
structures1 which may not play any      

independent part in theoretical physics4
 ,but it is my conviction that in the   
present stage of development of         
theoretical physics these ideas must    
still be employed as independent ideas2 
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 clocks4  ,further1 as to the 
objection that there are no really rigid
 bodies in nature1 and that therefore   
the properties predicated of rigid      
bodies do not apply to physical         
reality1--this objection is by no means 
so radical as might appear from a hasty 
examination4 ,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 body4 ,it is  
to measuring-bodies of this kind that   
statements as to rigid bodies must be   

referred4  ,all practical geometry is   
based upon a principle which is         
accessible to experience1 and which we  
will now try to realise4 ,we will call  
that which is enclosed between two      
boundaries1 marked upon a               
practically-rigid body1 a tract4 ,we    
imagine two practically-rigid bodies1   
each with a tract marked out on it4     
,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  
other4 ,we now assume that3  ,if two    
tracts are found to be equal once and   
anywhere1 they are equal always and     
everywhere4  ,not only the practical    
geometry of ,euclid1 but also its       
nearest generalisation1 the practical   
geometry of ,riemann1 and therewith the 
general theory of relativity1 rest upon 
this assumption4 ,of the experimental   
reasons which warrant this assumption ,i
 will mention only one4 ,the phenomenon 
of the propagation of light in empty    

space assigns a tract1 namely1 the      
appropriate path of light1 to each      
interval of local time1 and conversely4 
,thence it follows that the above       
assumption for tracts must also hold    
good for intervals of clock-time in the 
theory of relativity4 ,consequently it  
may be formulated as follows3--,if two  
ideal clocks are going at the same rate 
at any time and at any place 7being then
 in immediate proximity to each other71 
they will always go at the same rate1 no
 matter where and when they are again   
compared with each other at one         
place4--,if this law were not valid for 
real clocks1 the proper frequencies for 
the separate atoms of the same chemical 
element would not be in such exact      
agreement as experience demonstrates4   
,the existence of sharp spectral lines  
is a convincing experimental proof of   
the above-mentioned principle of        
practical geometry4 ,this is the        
ultimate foundation in fact which       
enables us to speak with meaning of the 

mensuration1 in ,riemann's sense of the 
word1 of the four-dimensional continuum 
of space-time4  ,the question whether   
the structure of this continuum is      
,euclidean1 or in accordance with       
,riemann's general scheme1 or otherwise1
 is1 according to the view which is here
 being advocated1 properly speaking a   
physical question which must be answered
 by experience1 and not a question of a 
mere convention to be selected on       
practical grounds4 ,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      
diminished4  ,it is true that this      
proposed physical interpretation of     
geometry breaks down when applied       
immediately to spaces of sub-molecular  
order of magnitude4 ,but nevertheless1  
even in questions as to the constitution

 of elementary particles1 it retains    
part of its importance4 ,for even when  
it is a question of describing the      
electrical elementary particles         
constituting matter1 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 molecule4
 ,success alone can decide as to the    
justification of such an attempt1 which 
postulates physical reality for the     
fundamental principles of ,riemann's    
geometry outside of the domain of their 
physical definitions4 ,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 magnitude4  
,it appears less problematical to extend
 the ideas of practical geometry to     
spaces of cosmic order of magnitude4 ,it
 might1 of course1 be objected that a   

construction composed of solid rods     
departs more and more from ideal        
rigidity in proportion as its spatial   
extent becomes greater4 ,but it will    
hardly be possible1 ,i think1 to assign 
fundamental significance to this        
objection4 ,therefore the question      
whether the universe is spatially finite
 or not seems to me decidedly a pregnant
 question in the sense of practical     
geometry4 ,i do not even consider it    
impossible that this question will be   
answered before long by astronomy4 ,let 
us call to mind what the general theory 
of relativity teaches in this respect4  
,it offers two possibilities3--  #a4    
,the universe is spatially infinite4    
,this can be so only if the average     
spatial density of the matter in        
universal space1 concentrated in the    
stars1 vanishes1 i4e4 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 greater4  #b4    
,the universe is spatially finite4 ,this
 must be so1 if there is a mean density 
of the ponderable matter in universal   
space differing from zero4 ,the smaller 
that mean density1 the greater is the   
volume of universal space4  ,i must not 
fail to mention that a theoretical      
argument can be adduced in favour of the
 hypothesis of a finite universe4 ,the  
general theory of relativity teaches    
that the inertia of a given body is     
greater as there are more ponderable    
masses in proximity to it2 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 universe1 as indeed1 ever since 
,newton's time1 gravity has been        
completely reduced to action and        
reaction between bodies4  ,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  
,e4 ,mach1 for example--is possible only
 if the universe is spatially finite4   
,on many physicists and astronomers this
 argument makes no impression4          
,experience alone can finally decide    
which of the two possibilities is       
realised in nature4 ,how can experience 
furnish an answer8 ,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 perception4 ,this hope
 is illusory4 ,the distribution of the  
visible stars is extremely irregular1 so
 that we on no account may venture to   
set down the mean density of star-matter
 in the universe as equal1 let us say1  
to the mean density in the ,milky ,way4 
,in any case1 however great the space   
examined may be1 we could not feel      
convinced that there were no more stars 
beyond that space4 ,so it seems         
impossible to estimate the mean density4
 ,but there is another road1 which seems

 to me more practicable1 although it    
also presents great difficulties4 ,for  
if we inquire into the deviations shown 
by the consequences of the general      
theory of relativity which are          
accessible to experience1 when these are
 compared with the consequences of the  
,newtonian theory1 we first of all find 
a deviation which shows itself in close 
proximity to gravitating mass1 and has  
been confirmed in the case of the planet
 ,mercury4 ,but if the universe is      
spatially finite there is a second      
deviation from the ,newtonian theory1   
which1 in the language of the ,newtonian
 theory1 may be expressed thus3--,the   
gravitational field is in its nature    
such as if it were produced1 not only by
 the ponderable masses1 but also by a   
mass-density of negative sign1          
distributed uniformly throughout space4 
,since this factitious mass-density     
would have to be enormously small1 it   
could make its presence felt only in    
gravitating systems of very great       

extent4  ,assuming that we know1 let us 
say1 the statistical distribution of the
 stars in the ,milky ,way1 as well as   
their masses1 then by ,newton's law we  
can calculate the gravitational field   
and the mean velocities which the stars 
must have1 so that the ,milky ,way      
should not collapse under the mutual    
attraction of its stars1 but should     
maintain its actual extent4 ,now if the 
actual velocities of the stars1 which   
can1 of course1 be measured1 were       
smaller than the calculated velocities1 
we should have a proof that the actual  
attractions at great distances are      
smaller than by ,newton's law4 ,from    
such a deviation it could be proved     
indirectly that the universe is finite4 
,it would even be possible to estimate  
its spatial magnitude4  ,can we picture 
to ourselves a three-dimensional        
universe which is finite1 yet unbounded8
  ,the usual answer to this question is 
,no1 but that is not the right answer4  
,the purpose of the following remarks is

 to show that the answer should be ,yes4
 ,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  
which1 with some practice1 we shall soon
 grow accustomed4  ,first of all1 an    
observation of epistemological nature4  
,a geometrical-physical theory as such  
is incapable of being directly pictured1
 being merely a system of concepts4 ,but
 these concepts serve the purpose of    
bringing a multiplicity of real or      
imaginary sensory experiences into      
connection in the mind4 ,to visualise a 
theory1 or bring it home to one's mind1 
therefore means to give a representation
 to that abundance of experiences for   
which the theory supplies the schematic 
arrangement4 ,in the present case we    
have to ask ourselves how we can        
represent that relation of solid bodies 
with respect to their reciprocal        
disposition 7contact7 which corresponds 
to the theory of a finite universe4     

,there is really nothing new in what ,i 
have to say about this2 but innumerable 
questions addressed to me prove that the
 requirements of those who thirst for   
knowledge of these matters have not yet 
been completely satisfied4  ,so1 will   
the initiated please pardon me1 if part 
of what ,i shall bring forward has long 
been known8  ,what do we wish to express
 when we say that our space is infinite8
 ,nothing more than that we might lay   
any number whatever of bodies of equal  
sizes side by side without ever filling 
space4 ,suppose that we are provided    
with a great many wooden cubes all of   
the same size4 ,in accordance with      
,euclidean geometry we can place them   
above1 beside1 and behind one another so
 as to fill a part of space of any      
dimensions2 but this construction would 
never be finished2 we could go on adding
 more and more cubes without ever       
finding that there was no more room4    
,that is what we wish to express when we
 say that space is infinite4 ,it would  

be better to say that space is infinite 
in relation to practically-rigid bodies1
 assuming that the laws of disposition  
for these bodies are given by ,euclidean
 geometry4  ,another example of an      
infinite continuum is the plane4 ,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 it4 ,the construction is    
never finished2 we can always go on     
laying squares--if their laws of        
disposition correspond to those of plane
 figures of ,euclidean geometry4 ,the   
plane is therefore infinite in relation 
to the cardboard squares4 ,accordingly  
we say that the plane is an infinite    
continuum of two dimensions1 and space  
an infinite continuum of three          
dimensions4 ,what is here meant by the  
number of dimensions1 ,i think ,i may   
assume to be known4  ,now we take an    
example of a two-dimensional continuum  
which is finite1 but unbounded4 ,we     
imagine the surface of a large globe and

 a quantity of small paper discs1 all of
 the same size4 ,we place one of the    
discs anywhere on the surface of the    
globe4 ,if we move the disc about1      
anywhere we like1 on the surface of the 
globe1 we do not come upon a limit or   
boundary anywhere on the journey4       
,therefore we say that the spherical    
surface of the globe is an unbounded    
continuum4 ,moreover1 the spherical     
surface is a finite continuum4 ,for if  
we stick the paper discs on the globe1  
so that no disc overlaps another1 the   
surface of the globe will finally become
 so full that there is no room for      
another disc4 ,this simply means that   
the spherical surface of the globe is   
finite in relation to the paper discs4  
,further1 the spherical surface is a    
non-,euclidean continuum of two         
dimensions1 that is to say1 the laws of 
disposition for the rigid figures lying 
in it do not agree with those of the    
,euclidean plane4 ,this can be shown in 
the following way4 ,place a paper disc  

on the spherical surface1 and around it 
in a circle place six more discs1 each  
of which is to be surrounded in turn by 
six discs1 and so on4 ,if this          
construction is made on a plane surface1
 we have an uninterrupted disposition in
 which there are six discs touching     
every disc except those which lie on the
 outside4  ,figure #a3 ,discs maximally 
packed on a plane  ,on the spherical    
surface the construction also seems to  
promise success at the outset1 and the  
smaller the radius of the disc in       
proportion to that of the sphere1 the   
more promising it seems4 ,but as the    
construction progresses it becomes more 
and more patent that the disposition of 
the discs in the manner indicated1      
without interruption1 is not possible1  
as it should be possible by ,euclidean  
geometry of the the plane surface4 ,in  
this way creatures which cannot leave   
the spherical surface1 and cannot even  
peep out from the spherical surface into
 three-dimensional space1 might         

discover1 merely by experimenting with  
discs1 that their two-dimensional space 
is not ,euclidean1 but spherical space4 
,from the latest results of the theory  
of relativity it is probable that our   
three-dimensional space is also         
approximately spherical1 that is1 that  
the laws of disposition of rigid bodies 
in it are not given by ,euclidean       
geometry1 but approximately by spherical
 geometry1 if only we consider parts of 
space which are sufficiently great4 ,now
 this is the place where the reader's   
imagination boggles4 ,nobody can imagine
 this thing1 he cries indignantly4 ,it  
can be said1 but cannot be thought4 ,i  
can represent to myself a spherical     
surface well enough1 but nothing        
analogous to it in three dimensions4    
,figure #b3 ,a circle projected from a  
sphere onto a plane  ,we must try to    
surmount this barrier in the mind1 and  
the patient reader will see that it is  
by no means a particularly difficult    
task4 ,for this purpose we will first   

give our attention once more to the     
geometry of two-dimensional spherical   
surfaces4 ,in the adjoining figure let  
,k be the spherical surface1 touched at 
,s by a plane1 ,e1 which1 for facility  
of presentation1 is shown in the drawing
 as a bounded surface4 ,let ,l be a disc
 on the spherical surface4  ,now let us 
imagine that at the point ,n of the     
spherical surface1 diametrically        
opposite to ,s1 there is a luminous     
point1 throwing a shadow ,l' of the disc
 ,l upon the plane ,e4 ,every point on  
the sphere has its shadow on the plane4 
,if the disc on the sphere ,k is moved1 
its shadow ,l' on the plane ,e also     
moves4 ,when the disc ,l is at ,s1 it   
almost exactly coincides with its       
shadow4  ,if it moves on the spherical  
surface away from ,s upwards1 the disc  
shadow ,l' on the plane also moves away 
from ,s on the plane outwards1 growing  
bigger and bigger4 ,as the disc ,l      
approaches the luminous point ,n1 the   
shadow moves off to infinity1 and       

becomes infinitely great4  ,now we put  
the question1 ,what are the laws of     
disposition of the disc-shadows ,l' on  
the plane ,e8 ,evidently they are       
exactly the same as the laws of         
disposition of the discs ,l on the      
spherical surface4 ,for to each original
 figure on ,k there is a corresponding  
shadow figure on ,e4 ,if two discs on ,k
 are touching1 their shadows on ,e also 
touch4 ,the shadow-geometry on the plane
 agrees with the the disc-geometry on   
the sphere4 ,if we call the disc-shadows
 rigid figures1 then spherical geometry 
holds good on the plane ,e with respect 
to these rigid figures4 ,moreover1 the  
plane is finite with respect to the     
disc-shadows1 since only a finite number
 of the shadows can find room on the    
plane4  ,at this point somebody will    
say1 ,that is nonsense4 ,the            
disc-shadows are not rigid figures4 ,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 infinity4 ,but what if the
 two-foot rule were to behave on the    
plane ,e in the same way as the         
disc-shadows ,l'8 ,it would then be     
impossible to show that the shadows     
increase in size as they move away from 
,s2 such an assertion would then no     
longer have any meaning whatever4 ,in   
fact the only objective assertion that  
can be made about the disc-shadows is   
just this1 that they are related in     
exactly the same way as are the rigid   
discs on the spherical surface in the   
sense of ,euclidean geometry4  ,we must 
carefully bear in mind that our         
statement as to the growth of the       
disc-shadows1 as they move away from ,s 
towards infinity1 has in itself no      
objective meaning1 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-shadows4 ,in respect of the
 laws of disposition of the shadows ,l'1

 the point ,s has no special privileges 
on the plane any more than on the       
spherical surface4  ,the representation 
given above of spherical geometry on the
 plane is important for us1 because it  
readily allows itself to be transferred 
to the three-dimensional case4  ,let us 
imagine a point ,s of our space1 and a  
great number of small spheres1 ,l'1     
which can all be brought to coincide    
with one another4 ,but these spheres are
 not to be rigid in the sense of        
,euclidean geometry2 their radius is to 
increase 7in the sense of ,euclidean    
geometry7 when they are moved away from 
,s towards infinity1 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 plane4  ,after  
having gained a vivid mental image of   
the geometrical behaviour of our ,l'    
spheres1 let us assume that in our space
 there are no rigid bodies at all in the
 sense of ,euclidean geometry1 but only 

bodies having the behaviour of our ,l'  
spheres4 ,then we shall have a vivid    
representation of three-dimensional     
spherical space1 or1 rather of          
three-dimensional spherical geometry4   
,here our spheres must be called rigid  
spheres4 ,their increase in size as they
 depart from ,s is not to be detected by
 measuring with measuring-rods1 any more
 than in the case of the disc-shadows on
 ,e1 because the standards of           
measurement behave in the same way as   
the spheres4 ,space is homogeneous1 that
 is to say1 the same spherical          
configurations are possible in the      
environment of all points499 ,our space 
is finite1 because1 in consequence of   
the growth of the spheres1 only a finite
 number of them can find room in space4 
99 ,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 sphere4  ,in this way1 by
 using as stepping-stones the practice  

in thinking and visualisation which     
,euclidean geometry gives us1 we have   
acquired a mental picture of spherical  
geometry4 ,we may without difficulty    
impart more depth and vigour to these   
ideas by carrying out special imaginary 
constructions4 ,nor would it be         
difficult to represent the case of what 
is called elliptical geometry in an     
analogous manner4 ,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 geometry4