,,the ,,romanes ,,lecture               
#aibb                                   
,,the                                   
,,theory ,,of ,,relativity              
and its                      ,influence 
on ,scientific ,thought                 
,,by                           ,,arthur 
,,stanley ,,eddington                   
,m4,a41 ,f4,r4,s4               ,plumian
 ,professor of ,astronomy1 ,cambridge   
,president of the ,royal ,astronomical  
,society                                
,,delivered                        ,,in 
,,the ,,sheldonian ,,theatre            
#bd ,,may1 #aibb                        
,,oxford                          ,,at  
,,the ,,clarendon ,,press               
#aibb          ,has not a deeper        
meditation taught certain of every      
climate and  age1 that the where and the
 when so mysteriously inseparable  from 
all our thoughts1 are but superficial   
terrestrial adhesions to  thought8      
,,carlyle1 ,sartor ,resartus4           
,oxford ,university ,press              

,,the ,,theory ,,of ,,relativity        
,in the days before ,copernicus the     
earth was1 so it seemed1 an immovable   
foundation on which the whole structure 
of the heavens was reared4 ,man1        
favourably situated at the hub of the   
universe1 might well expect that  to him
 the scheme of nature would unfold      
itself in its simplest aspect4  ,but the
 behaviour of the heavenly bodies was   
not at all simple2 and the  planets     
literally looped the loop in fantastic  
curves called epicycles4  ,the          
cosmogonist had to fill the skies with  
spheres revolving upon  spheres to bear 
the planets in their appointed orbits2  
and wheels were  added to wheels until  
the music of the spheres seemed         
well-nigh drowned  in a discord of      
whirling machinery4 ,then came one of   
the great  revolutions of scientific    
thought1 which swept aside the          
,ptolemaic  system of spheres and       
epicycles1 and revealed the simple plan 
of the  solar system which has endured  

to this day4    ,the revolution         
consisted in changing the view-point    
from which the  phenomena were regarded4
 ,as presented to the earth the track of
 a planet  is an elaborate epicycle2 but
 ,copernicus bade us transfer ourselves 
to  the sun and look again4 ,instead of 
a path with loops and nodes1 the  orbit 
is now seen to be one of the most       
elementary curves--an ellipse4  ,we have
 to realize that the little planet on   
which we stand is of no  great account  
in the general scheme of nature2 to     
unravel that scheme we  must first      
disembarrass nature of the distortions  
arising from the local  point of view   
from which we observe it4 ,the sun1 not 
the earth1 is the  real centre of the   
scheme of things--at least of those     
things in which  astronomers at that    
time had interested themselves--and by  
transferring  our view-point to the sun 
the simplicity of the planetary system  
becomes  apparent4 ,the need for a      
cumbrous machinery of spheres and wheels

 has  disappeared4    ,every one now    
admits that the ,ptolemaic system1 which
 regarded the earth  as the centre of   
all things1 belongs to the dark ages4   
,but to our dismay  we have discovered  
that the same geocentric outlook still  
permeates  modern physics through and   
through1 unsuspected until recently4 ,it
 has  been left to ,einstein to carry   
forward the revolution begun by         
,copernicus--to free our conception of  
nature from the terrestrial bias        
imported into it by the limitations of  
our earthbound experience4 ,to  achieve 
a more neutral point of view we have to 
imagine a visit to some  other heavenly 
body4 ,that is a theme which has        
attracted the popular  novelist1 and we 
often smile at his mistakes when sooner 
or later he  forgets where he is        
supposed to be and endows his voyagers  
with some  purely terrestrial appanage  
impossible on the star they are         
visiting4  ,but scientific men1 who have
 not the novelist's licence1 have made  

the  same blunder4 ,when1 following     
,copernicus1 they station themselves on 
the  sun1 they do not realize that they 
must leave behind a certain purely      
terrestrial appanage1 namely1 the frame 
of space and time in which men  on this 
earth are accustomed to locate the      
events that happen4 ,it is  true that   
the observer on the sun will still      
locate his experiences in a  frame of   
space and time1 if he uses the same     
faculties of perception and  the same   
methods of scientific measurement as on 
the earth2 but the  solar frame of space
 and time is not precisely the same as  
the  terrestrial frame1 as we shall     
presently see4    ,i think you will     
readily understand what is meant by a   
frame of space  and time4 ,it is the    
system of location to which we appeal   
when we state1  for example1 that one   
event is #ajj miles distant from and #aj
 hours later  than another4 ,the terms  
space and time have not only a vague    
descriptive  reference to a boundless   

void and an ever-rolling stream1 but    
denote an  exact quantitative system of 
reckoning distances and time-intervals4 
,einstein's first great discovery was   
that there are many such systems of     
reckoning--many possible frames of space
 and time--exactly on all fours  with   
one another4 ,no one of these can be    
distinguished as more  fundamental than 
the rest2 no one frame rather than      
another can be  identified as the       
scaffolding used in the construction of 
the world4 ,and  yet one of them does   
present itself to us as being the actual
 space and  time of our experience2 and 
we recoil from the other equivalent     
frames  which seem to us artificial     
systems in which distance and duration  
are  mixed up in an extraordinary way4  
,what is the cause of this invidious    
selection8 ,it is not determined by     
anything distinctive in the frame2 it   
is determined by something distinctive  
in us--by the fact that our  existence  
is bound to a particular planet and our 

motion is the motion  of that planet4   
,nature offers an infinite choice of    
frames2 we select  the one in which we  
and our petty terrestrial concerns take 
the most  distinguished position4 ,our  
mischievous geocentric outlook has      
cropped  out again unsuspected1         
persuading us to insist on this         
terrestrial  space-time frame which in  
the general scheme of nature is in no   
way  superior to other frames4    ,the  
more closely we examine the processes by
 which events are assigned  to their    
positions in space and time1 the more   
clearly do we see that  our local       
circumstances play a considerable part  
in it4 ,we have no more  right to expect
 that the space-time frame on the sun   
will be identical  with our frame on the
 earth than to expect that the force of 
gravity  will be the same there as here4
 ,if there were no experimental evidence
  in support of ,einstein's theory1 it  
would nevertheless have made a  notable 
advance by exposing a fallacy underlying

 the older mode of  thought--the fallacy
 of attributing unquestioningly a more  
than local  significance to our         
terrestrial reckoning of space and time4
 ,but there  is abundant experimental   
evidence for detecting and determining  
the  difference between the frames of   
differently circumstanced observers4    
,much of the evidence is too technical  
to be discussed here1 and ,i can  only  
refer to the ,michelson-,morley         
experiment4 ,i fear that some of you    
must be getting rather tired of the     
,michelson-,morley experiment2 but      
those who go to a performance of ,hamlet
 have to put up with the ,prince  of    
,denmark4    ,this famous experiment is 
a simple test whether light travels at  
the  same speed in two different        
directions4 ,for this purpose an        
apparatus is  constructed with two equal
 arms at right angles1 providing two    
equal  tracks for the light4 ,a beam of 
light is divided into two parts so that 
one part travels along one arm and back1

 and the other along the other  arm and 
back4 ,the two rays then re-unite1 and  
by delicate interference  tests it is   
possible to tell if one has been delayed
 more than the  other2 a delay of less  
than a thousand-billionth of a second   
could be  detected4 ,the experiment is  
simply a race between two light-rays    
with  equal tracks1 but pointing in     
different directions2 the result turns  
out  to be a dead-heat4 ,at first sight 
this is just what would be expected2    
and one almost wonders why it should    
have been thought worth while to  try   
the experiment4 ,but ,michelson1 like a 
good ,copernican1 had stationed  himself
 on the sun to watch the race2          
accordingly he realized that the        
apparatus was being borne along by the  
earth's orbital motion with a  speed of 
#bj miles a second4 ,consequently the   
light does not travel  exactly the      
double length of the arm2 starting at   
one end it has to go  to the            
turning-mark at the other end which has 

moved on a little in the  meantime2 then
 it returns to the place which the      
starting-mark has  travelled to whilst  
the race is in progress4 ,that does not 
add up to  exactly the double-length of 
the arm4 ,making the calculations we    
easily  find that1 although the two arms
 are equal1 the two light-journeys are  
unequal2 the competitor whose track lies
 in the line of the earth's  motion has 
the longer journey1 and is at a         
disadvantage4 ,and yet  according to the
 experiment he does not suffer the      
expected delay4 ,from  our standpoint on
 the sun1 the experiment seems to have  
gone wrong2  ,copernicus has met with a 
rebuff1 and ,ptolemy is triumphant4     
,but that is because we have not        
admitted the full consequences of       
transferring our standpoint to the sun4 
,we have all the while been  keeping one
 foot on earth4 ,of course1 the whole   
experiment turns on the  two arms having
 been first adjusted to perfect         
equality4 ,this could only  be          

ascertained by experiment2 and the test 
applied was to rotate the  apparatus    
through a right angle1 so that if1 for  
example1 the journey in  the line of the
 earth's motion had had the advantage of
 the shorter arm  on one occasion1 the  
transverse journey would have had it on 
the  repetition4 ,that is a perfectly   
satisfactory test for a terrestrial     
observer2 to turn a rod from one        
direction to another is the simple and  
direct way of marking out equal lengths4
 ,but the test is not  satisfactory to  
an observer on the sun2 he would not    
think of attempting  to partition equal 
lengths of space by means of rods       
travelling at #bj  miles a second4 ,his 
frame of space--the space not only of   
refined  measurement1 but also of the   
cruder measurements made with the       
sense-organs of his body which determine
 his perception of space--is            
partitioned by appliances at rest       
relatively to him1 e4g4 his own eyes    
and limbs4 ,lengths of objects carried  

on the earth must be judged by him      
according to the room they occupy in his
 own frame4 ,in the space of the        
terrestrial observer the two arms of the
 apparatus were adjusted to  equal      
length2 but in the re-partitioned space 
of the solar observer they  may quite   
well occupy unequal lengths1 and when we
 take the view-point  of an observer on 
the sun we must not overlook this       
inequality4 ,this  inequality is not so 
much an hypothesis proposed to account  
for  ,michelson's result as a direct    
deduction from it4 ,the two             
light-journeys  were found to occupy    
equal times2 this clearly shows that the
 arm in the  less favoured direction is 
shorter than the other so as to         
counterbalance the handicap to which ,i 
have referred4#a    ,when the apparatus 
is turned through a right angle1 the    
experiment still  gives the same result4
 ,it does not matter which of the two   
arms we place  in the line of the       
earth's motion2 that arm must be shorter

 than the  other4 ,in other words each  
arm must automatically contract when it 
is  turned from the transverse to the   
longitudinal position with respect to   
its line of motion4 ,this is the famous 
,fitz,gerald contraction of a  moving   
rod4 ,it is of the same amount whatever 
the material of the rod1  and depends   
only on the speed of its motion4 ,for   
the earth's orbital  motion the         
contraction amounts to one part in #bjj 
million2 in fact the  earth's diameter  
in the direction of its motion is always
 shortened by  #b inches1 the transverse
 diameter being unaffected4    ,this    
contraction of a moving material object 
was first revealed to us by  the        
,michelson-,morley experiment2 but it is
 not at all disagreeable to  theoretical
 anticipations4 ,we have to remember    
that a rod consists of a  large number  
of molecules kept in position by their  
mutual forces4 ,the  chief force is the 
force of cohesion1 and there is little  
doubt that  this is of electrical       

nature4 ,but when the rod is set in     
motion1 the  electrical forces inside it
 must change4 ,for example1 each        
electric  charge when put in motion     
becomes an electric current2 and the    
currents will exert magnetic attractions
 on each other which did not  occur in  
the system at rest4 ,under the new      
system of forces the  molecules will    
have to find new positions of           
equilibrium2 they become  differently   
spaced2 and it is therefore not         
surprising that the form of  the rod    
changes4 ,without going beyond the      
classical laws of ,maxwell we  can      
anticipate theoretically what will be   
the new equilibrium state of  the rod1  
and it turns out to be contracted to the
 exact amount required  by the          
,michelson-,morley result4    ,the      
contraction of the moving rod ought not 
to surprise us2 it would be  much more  
surprising if the rod were to maintain  
the same form in spite  of the          
alteration of the electrical forces     

which determine the spacing  of the     
molecules4 ,but the remarkable thing is 
that the contraction is  only apparent  
according to the outlook of the solar   
observer2 and we on  the earth1 who     
travel with the rod1 cannot appreciate  
it4 ,the fact that  the contraction     
happens to be very small is irrelevant4 
,for convenience  suppose that the      
earth's velocity is #h1jjj times faster1
 so that the  contraction amounts to    
something like a half the original1     
length4 ,we  should still fail to notice
 it in everyday life4 ,let us say that  
the  direction of the earth's motion is 
vertically upwards4    ,i turn my arm   
from horizontal to vertical and it      
contracts to half its  length4 ,no1 you 
cannot convince me ,i am wrong2 ,i am   
not afraid of a  yard-measure4 ,bring   
one and measure my arm2 first           
horizontally1 the  result is #cj inches2
 now vertically1 the result is          
#cj--;half-inches6  ,because you must   
remember that you have turned the scale 

into the line  of the earth's motion so 
that each inch-division contracts to    
half an  inch4 ',but we can see that    
your arm does not contract4 ,are we not 
to  trust our eyes8' ,certainly not1    
unless you first correct your visual    
impressions for the contraction of the  
retina in the vertical direction1  and  
for the effect of our rapid motion on   
the apparent direction of  propagation  
of the waves of light4 ,you will find1  
when you calculate  these corrections1  
that they just conceal the contraction4 
',but if the  contraction takes place1  
ought one not to feel it happening to   
the arm8'  ,not necessarily2 ,i am an   
observer on the earth1 and my feelings  
like  other sense-impressions belong to 
the geocentric outlook on nature1  which
 ,copernicus has persuaded us to        
abandon4    ,take a pair of compasses   
and twiddle them on a sheet of paper4   
,is the  resulting curve a circle or an 
ellipse8 ,copernicus from his standpoint
  on the sun declares that owing to the 

,fitz,gerald contraction the two  points
 drew nearer together when turned in the
 direction of the earth's  orbital      
motion2 hence the curve is flattened    
into an ellipse4 ,but here ,i  think    
,ptolemy has a right to be heard2 he    
points out that from the  beginning of  
geometry circles have always been drawn 
with compasses in  this way1 and that   
when the word 'circle' is mentioned     
every intelligent  person understands   
that this is the curve meant4 ,the same 
pencil line is  in fact a circle in the 
space of the terrestrial observer and an
 ellipse  in the space of a solar       
observer4 ,it is at the same time a     
moving  ellipse and a stationary circle4
 ,i think that illustrates as well as   
possible what we mean by the relativity 
of space4    ,it is sometimes complained
 that ,einstein's conclusion that the   
frame of  space and time is different   
for observers with different motions    
tends  to make a mystery of a phenomenon
 which is not after all intrinsically   

strange4 ,we have seen that it depends  
on a contraction of moving objects      
which turns out to be quite in          
accordance with ,maxwell's classical    
theory4 ,but even if we have succeeded  
in explaining it to ourselves           
intelligibly1 that does not make the    
statement any the less true6 ,a new     
result may often be expressed in various
 ways2 one mode of statement may  sound 
less mysterious2 but another mode may   
show more clearly what will  be the     
consequences in amending and extending  
our knowledge4 ,it is for  the latter   
reason that we emphasize the relativity 
of space--that  lengths and distances   
differ according to the observer        
implied4 ,distance  and duration are the
 most fundamental terms in physics2     
velocity1  acceleration1 force1 energy1 
and so on1 all depend on them2 and we   
can  scarcely make any statement in     
physics without direct or indirect      
reference to them4 ,surely then we can  
best indicate the revolutionary         

consequences of what we have learnt by  
the statement that distance and         
duration1 and all the physical          
quantities derived from them1 do not as 
hitherto supposed refer to anything     
absolute in the external world1 but  are
 relative quantities which alter when we
 pass from one observer to  another with
 different motion4 ,the consequence in  
physics of the  discovery that a yard is
 not an absolute chunk of space1 and    
that what  is a yard for one observer   
may be eighteen inches for another      
observer1  may be compared with the     
consequences in economics of the        
discovery that  a pound sterling is not 
an absolute quantity of wealth1 and in  
certain  circumstances may 'really' be  
seven and sixpence4 ,the theorist may   
complain that this last statement tends 
to make a mystery of phenomena  of      
currency which have really an           
intelligible explanation2 but it is a   
statement which commends itself to the  
man who has an eye to the  practical    

applications of currency4    ,ptolemy on
 the earth and ,copernicus on the sun   
are both contemplating  the same        
external universe4 ,but their           
experiences are different1 and it  is in
 the process of experiencing events that
 they become fitted into  the frame of  
space and time--the frame being         
different according to the  local       
circumstances of the observer who is    
experiencing them4 ,that1 ,i  take it1  
is ,kant's doctrine1 ',space and time   
are forms of experience4'  ,the frame   
then is not in the world2 it is supplied
 by the observer and  depends on him4   
,and those relations of simplicity1     
which we seek when we  try to obtain a  
comprehension of how the universe       
functions1 must lie in  the events      
themselves before they have been        
arbitrarily fitted into the  frame4 ,the
 most we can hope for from any frame is 
that it will not have  distorted the    
simplicity which was originally present2
 whilst an  ill-chosen frame may play   

havoc with the natural simplicity of    
things4  ,we have seen that the         
simplicity of planetary motions was     
obscured in  ,ptolemy's frame1 and      
became apparent in ,copernicus's frame4 
,but for  ordinary terrestrial phenomena
 the position is reversed and ,ptolemy's
  frame allows their natural simplicity 
to become apparent4 ,in  ,copernicus's  
frame the most simple phenomena are     
brought about by highly  complicated    
processes which mutually cancel one     
another4 ,ordinary  objects contract and
 expand as they are moved about1 and the
 changes are  concealed by an elaborate 
conspiracy in which all the quantities  
of  nature--electrical1 optical1        
mechanical1 gravitational--have joined4 
,in  ,copernicus's frame we have a great
 complication of description which has  
no counterpart in anything occurring in 
the external world2 because the  terms  
of our description refer to the         
irrelevant process of fitting into  the 
selected frame of space and time4 ,this 

elaborate ,copernican scheme  rather    
reminds one of the schemes of the ,white
 ,knight--          ,but ,i was thinking
 of a plan        ,to dye one's whiskers
 green1      ,and always use so large a 
fan        ,that they could not be seen4
      ,we do not deny the subtlety and  
the remarkable efficiency of the plan2  
but we may be allowed to question       
whether it is the simplest              
interpretation of the drab monotony of  
the face of nature presented to  us4    
,the simple fact is that a terrestrial  
or ,ptolemaic frame fits  naturally the 
terrestrial phenomena1 and a solar or   
,copernican frame  fits the phenomena of
 the solar system2 but we cannot make   
one frame  serve for both without       
introducing irrelevant1 complications4  
,we go beyond ,copernicus nowadays1 and 
are not content with a visit to  the    
sun4 ,why choose the sun rather than    
some other star in order to  obtain an  
undistorted view of things8 ,the        
astronomer now places himself  so as to 

travel with the centre of gravity of the
 stellar universe1 and  is not even then
 quite satisfied4 ,the physicist dreams 
of a land of  ,weissnichtwo1 which shall
 be truly at rest in the ether4 ,we     
realize the  distortion imported into   
the world of nature by the parochial    
standpoint  from which we observe it1   
and we try to place ourselves so as to  
eliminate this distortion--so as to     
observe that which actually is4 ,but  it
 is a vain pursuit4 ,wherever we pitch  
our camera1 the photograph is           
necessarily a two-dimensional picture   
distorted according to the laws of      
perspective2 it is never a true         
semblance of the building itself4    ,we
 must try another plan4 ,i do not think 
we can ever eliminate  altogether the   
human element in our conception of      
nature2 but we can  eliminate a         
particular human element1 namely1 this  
framework of space  and time4 ,if our   
thought must be anthropocentric1 it need
 not be  geocentric4 ,nor are we        

permanently better off if we merely     
substitute  the space-time frame of some
 other star or standard of motion4 ,we  
must  leave the frame entirely          
indeterminate4 ,when we do that1 we find
 that  the world common to all          
observers--in which each observer traces
 a  different space-time frame according
 to his own outlook--is a world of  four
 dimensions4 ,when we look at any       
object1 say a chair1 the impression  on 
our eyes is a two-dimensional picture   
depending on the position from  which we
 are looking2 but we have no difficulty 
in conceiving of the  chair as a solid  
object1 not to be identified with any   
one of our  two-dimensional pictures of 
it1 but giving rise them all as the     
position  of the observer is varied4 ,we
 must now realize that this solid chair 
in  three dimensions is itself only an  
appearance1 which changes according  to 
the motion of the observer1 and that    
there is a super-object in              
four-dimensions1 not to be identified   

with the three-dimensional chair  in    
,ptolemy's scheme1 or the same chair in 
,copernicus's scheme1 but  giving rise  
to both these appearances4 ,the         
synthesis of a  three-dimensional chair 
from a number of flat pictures is easy  
to us  because we are accustomed to     
assume different positions in rapid     
succession2 indeed our two eyes give us 
slightly different points of  view      
simultaneously4 ,by sheer necessity our 
brains have been forced to  construct   
the conception of the solid chair to    
combine these changing  appearances4    
,but we do not vary our motion to any   
appreciable extent and  our brains have 
not hitherto been called upon to combine
 the appearances  for different motions2
 thus the effort which we now ask the   
brain to  make is a novel one4 ,that    
explains why the result seems to        
transcend our  ordinary mode of thought4
    ,the discovery1 or one should rather
 say the rediscovery1 of the world of   
four dimensions is due to ,minkowski4   

,einstein had worked out fully the      
relations between the frames of space   
and time for observers with  different  
motions4 ,to the genius of ,minkowski we
 owe the realization  that these frames 
are merely systems of partitions        
arbitrarily drawn  across a             
four-dimensional world which is common  
to all observers4    ,there is a strange
 delusion that the fourth dimension must
 be something  wholly beyond the        
conception of the ordinary man1 and that
 only the  mathematician can be         
initiated into its mysteries4 ,it is    
true that the  mathematician has the    
advantage of understanding the technical
 machinery  for solving the problems    
which may arise in studying the world of
 four  dimensions2 but as regards the   
conception of the four dimensions of the
  world his point of view is the same as
 that of anybody else4 ,is it  supposed 
that by intense thought he throws       
himself into some state of  trance in   
which he perceives some hitherto        

unsuspected direction  stretching away  
at right angles to length1 breadth1 and 
thickness8 ,that  would not be much use4
 ,the world of four dimensions1 of which
 we are now  speaking1 is perfectly     
familiar to everybody4 ,it is obvious to
 every  one--even to the                
mathematician--that the world of solid  
and permanent  objects has three        
dimensions and no more2 that objects are
 arranged in  a threefold order1 which  
for any particular individual may be    
analysed  into right-and-left1          
backwards-and-forwards1 up-and-down4    
,but it is no  less obvious to every one
 that the world of events is of four    
dimensions2 that events are arranged in 
a fourfold order1 which in the          
experience of any particular individual 
will be analysed into  right-and-left1  
backwards-and-forwards1 up-and-down1    
sooner-and-later4  ,the subject of our  
study is external nature1 which is a    
world of events1  common to all         
observers but represented by them       

differently in their  parochial frames  
of space and time2 it is obvious to the 
most  commonplace experience that this  
absolute world contains a fourfold      
order4#b    ,the news that the events   
around us form a world of four          
dimensions is as  stale as the news that
 ,queen ,anne is dead4 ,the reason why  
the relativist  resurrects this ancient 
truism is because it is only in this    
undissected  combination of four        
dimensions that the experiences of all  
observers  meet4 ,in our own experience 
one dimension is sharply separated from 
the  other three and is distinguished as
 time2 but our experience is solely     
terrestrial1 and if we insist on        
building the scheme of nature on purely 
terrestrial experience we are limiting  
ourselves to the mediaeval  geocentric  
system of the world4    ,we have been   
accustomed to regard the enduring world 
as composed of a  continuous succession 
of instantaneous states1 as though the  
world of  events were stratified4 ,each 

event is supposed to lie in a definite  
instant or stratum1 and the orderly     
succession of these strata makes up  the
 whole of reality4 ,the instant 'now'   
represents one such stratum  running    
throughout the universe4 ,indeed we are 
accustomed to extend it  beyond the     
universe1 and we even use the word 'now'
 with reference to  the existence of    
those who have passed away from the     
material world4 ,the  investigations of 
the relativity theory show              
incontrovertibly that this  supposed    
stratification is an illusion2 there is 
not the slightest  evidence for such a  
view of world-structure4 ,the           
instantaneous state1  which we have     
hitherto taken to be a natural stratum  
in the  four-dimensional world of       
events1 is merely an arbitrary partition
  created by ourselves to correspond    
with our geocentric outlook4 ,we can    
take a differently inclined partition1#c
 that is to say1 a section  which       
includes on the one side of us events   

which happened a little  while ago and  
on the other side of us events which    
have not yet  happened2 such a farcical 
combination is in every way equivalent  
to our  so-called instantaneous state1  
and indeed it is an instantaneous state 
according to the outlook of some        
non-terrestrial observer with suitably  
assigned motion4    ,it is so contrary  
to our natural prejudices to recognize  
that the  world-wide instant now is     
created by ourselves and has no         
existence  apart from our geocentric    
outlook1 that ,i will spend a few       
moments  trying to show its             
artificiality4 ,when ,i say that ,i am  
conscious of an  instant now1 ,i am only
 conscious of it in so far as it is     
,,here--inside  me4 ,what then has led  
me to imagine that there exists a       
continuation of  it outside me8 ,it is  
because ,i look out on the world and see
 various  events happening 'now'1 so    
that ,i jump to the conclusion that this
  instant of which ,i am conscious has  

to be extended to include  them4 ,but   
that idea is another inheritance from   
the dark ages1  overthrown by ,rmer in  
#afge4 ,it is not the events themselves 
but the  sense-impressions to which they
 give rise which are happening in the   
instant now4 ,so my justification for   
placing the events outside me in  the   
instants of which ,i am conscious has   
entirely disappeared4  ,unfortunately1  
however1 the crude outlook was not      
abolished1 but patched  up2 it was found
 that the immediate difficulties could  
be met by  locating the external events 
not in the instant of our visual        
perception  of them but in an instant   
which we had experienced a little time  
back--allowing1 as we say1 for the time 
of propagation of light4 ,thus  our     
instants were still made to extend      
through space2 but they were  carried   
like partitions among the events by an  
artificial process of  computation1 and 
no longer by immediate intuition4 ,the  
relativity theory  recognizes these     

worldwide instants for what they        
are--artificial  partitions constructed 
for purposes of calculation4 ,i may add 
that it in  no way tampers with the     
local instants which form the stream of 
our  consciousness2 it fully recognizes 
that the chain of events in such a      
time-succession is a series of an       
entirely distinctive character from  the
 succession of points along a line in   
space4 ,those who suspect that          
,einstein's theory is playing           
unjustifiable tricks with time should   
realize that it leaves entirely         
untouched that time-succession of which 
we have intuitive knowledge1 and        
confines itself to overhauling the      
artificial scheme of time which ,rmer   
first introduced into physics4    ,the  
study of the four-dimensional world of  
events gives us a new insight  into the 
processes of nature because it removes  
the irrelevant  stratification in a     
particular direction--the instantaneous 
states--which we have so unnecessarily  

introduced in our customary  outlook4   
,when this stratification is ignored we 
are enabled to see the  processes in    
their simplest aspect1 though not1 of   
course1 in their most  familiar aspect4 
,we must distinguish between simplicity 
and familiarity2  a pig may be most     
familiar to us in the form of rashers1  
but the  unstratified pig is a simpler  
object of study to the biologist who    
wishes to understand how the animal     
functions4    ,i will conclude this part
 of the argument with an experimental   
application which illustrates the power 
of ,einstein's method4 ,much study  has 
of late been given to electrons moving  
with very high speeds2 for  example1 the
  particles shot off from radioactive   
substances are  negative electrons which
 sometimes attain speeds of #ajj1jjj    
miles a  second4 ,it is found by        
experiment that the rapid motion        
produces an  increase of mass of these  
particles4 ,i want to show that the     
theory of  relativity gives a very      

simple explanation of just how this     
increase of  mass occurs4 ,but ,i must  
first remark that an explanation had    
been  previously given which had        
generally been accepted as satisfactory4
 ,the  phenomenon was actually predicted
 by ,j4 ,j4 ,thomson before relativity  
was  thought of2 because1 assuming that 
the mass of a particle is of  electrical
 origin1 an application of ,maxwell's   
equations shows that it  ought to       
increase with velocity4 ,but the precise
 law of increase cannot  be predicted on
 this basis1 since various plausible    
assumptions lead to  slightly different 
results4 ,moreover1 ,maxwell's equations
 are after all  only empirical laws1    
with a mystery of their own2 it was a   
notable  advance to connect the change  
of mass at high speeds with other       
phenomena whose strangeness has         
disappeared by long familiarity1 but    
there is still scope for a more         
far-reaching explanation4 ,einstein     
takes  us straight to the root of the   

mystery1 and he clears up one point     
which  was misleading1 if not actually  
wrong1 in the older explanation4 ,the   
change of mass does not in any way      
depend on whether the mass is of        
electrical origin or not2 it arises     
simply from the fact that mass is a     
relative quantity1 depending by its     
definition on the relative  quantities  
length and time4 ,let us look at the    
particle from its own  point of view2 it
 is just an ordinary electron in no way 
different from  any other4 ',but it is  
travelling unusually rapidly8' ',that'1 
says the  electron1 'is a matter of     
opinion4 ,so far as ,i am aware ,i am at
 rest1 if  the word rest has any        
meaning4 ,in fact ,i was just           
contemplating with  amazement your      
extraordinary speed of #ajj1jjj miles a 
second with  which you are shooting past
 me4' ,of course our motion is of no    
particular concern to the electron1 and 
it will not modify its  constitution on 
our account2 so it keeps its mass1      

radius1 electric  field1 c41 equal to   
the standard constants applying to      
electrons in  general4 ,these terms are 
relative1 and refer therefore to some   
particular frame of space and           
time--clearly the frame appropriate to  
an  electron in self-contemplation1 viz4
 the one with1 respect to which it  is  
at rest4 ,but this frame is not the     
usual geocentric frame to which  we     
refer quantities such as length1 time1  
and mass2 there is a  difference of     
#ajj1jjj miles a second between our     
station of observation  and that of the 
particle in self-contemplation4 ,it is a
 mere matter  of geometry to discover   
what the  particle's lengths and times  
become  when referred to the partitions 
which we have drawn across the world4   
,but when we calculate the consequential
 change of mass resulting from  the     
changes of length and time1 we find that
 it should be increased in  precisely   
the proportion indicated by the most    
refined experiments4    ,the point is   

that every electron1 at rest or in      
motion1 is a perfectly  constant        
structure2 but we distort it by fitting 
it into the space-time  frame           
appropriate to our own motion with which
 the electron has no  concern4 ,the     
greater our motion with respect to the  
electron1 the  greater will be the      
distortion4 ,the distortion is not      
produced by any  physical agency at work
 in the electron2 it is a purely        
subjective  distortion depending on our 
transformation of the reference frame of
  space and time4 ,this distortion      
involves a change in our physical       
description of the electron in terms of 
mass1 shape1 size2 and in  particular   
the change of mass agrees precisely with
 that found  experimentally4    ,you see
 that it is not altogether idle         
discussing the natural space-time       
frames for observers moving with huge   
velocities4 ,we know of no animate      
observers with these speeds2 but we do  
know of inanimate material  objects4    

,their common resemblance is obscured   
when we refer them  indiscriminately to 
our irrelevant geocentric frame2 we     
think they have  altered their          
properties1 varied in mass1 and so on2  
but the resemblance  is restored when we
 refer each individual to the frame     
appropriate to  it1 and so describe them
 all in comparable terms4    ,our       
measurements of distance in space are   
found to be subject to certain          
laws--the laws of geometry4 ,but it has 
now become impossible to regard  the    
subject of space-geometry as complete in
 itself4 ,consider a triangle  formed by
 three points 7or events7 in the        
four-dimensional world2 if we  happen to
 have drawn our instantaneous strata so 
that the three points  lie in one       
stratum1 then the triangle is a         
space-triangle and its  properties fall 
within the scope of our classical       
geometry4 ,but another  observer will   
draw his strata in a different          
direction1 and for him the  triangle    

would be partly in space and partly in  
time1 so that it would  not be a fit    
subject for space-geometry4 ,the subject
 of geometry is in a  desperate         
condition1 because ,copernicus and      
,ptolemy not merely disagree  as to the 
geometry of a configuration2 they even  
disagree as to whether  a given         
configuration is one to which           
space-geometry is applicable4 ,it  is   
clear that to save it we must extend our
 geometry so as to include  time as well
 as space4 ,let me give an illustration 
of this extension4  ,the terrestrial    
observer can have a space-triangle      
7formed by three  points or events at   
the same instant7 whose sides he can    
measure with  scales2 he can also have a
 'time-triangle'1 formed by three events
 on  different dates1 whose sides he    
must measure with clocks4#d ,you all    
know the law of the space-triangle--that
 if you measure with a scale  from ,a to
 ,b and from ,b to ,c the sum of the    
readings is always  greater than the    

measure from ,a to ,c4 ,it is not so    
well known that  there is a precisely   
analogous law for the                   
time-triangle--that if you  measure with
 a clock from ,a to ,b and from ,b to ,c
 the sum of the  readings is always less
 than the reading of a clock measuring  
directly  from ,a to ,c4 ,in the        
space-triangle any two sides are        
together  greater than the third side2  
in the time-triangle two sides are      
together less than the third side4#e    
,both these laws must be combined  in   
our general geometry of four dimensions1
 so that it will not be quite  so simple
 a geometry as that to which we are     
accustomed4#f    ,but the point to which
 ,i would especially direct attention is
 this4  ,evidently the proposition which
 ,i have given you about time-triangles 
cannot be dissociated from the          
corresponding proposition about         
space-triangles4 ,when we give up the   
mediaeval geocentric standpoint1 we     
must recognize that they belong to one  

geometry1 of which our ordinary         
space-geometry is only a part or        
projection4 ,but if you examine the     
proposition about time-triangles1 you   
will see that it is a statement  about  
the behaviour of clocks when they move  
about1 a subject which  obviously comes 
under the heading of mechanics4 ,when we
 deal with the  four-dimensional world  
we can no longer distinguish between    
geometry and  mechanics4 ,they become   
the same subject4 ,when we have         
completely  mastered the geometry of the
 world of events1 we shall have         
inevitably  learnt the mechanics of it4 
,that is why ,einstein1 studying the    
geometry  of the world and discovering  
that it was strictly non-,euclidean1    
found  that he was at the same time     
studying the mechanical force of        
gravitation4 ,and when he had made up   
his mind which of the possible          
varieties of non-,euclidean geometry was
 obeyed1 and so settled the laws  of the
 new geometry1 the same decision settled

 the law of gravitation--a  law         
approximating to1 but not identical     
with1 the law which ,newton had  given4 
,here a wide vista opens before us4 ,we 
see that two great divisions of         
mathematical physics1 viz4 geometry and 
mechanics1 have met in the              
four-dimensional world4 ,it is not      
merely that mechanical problems can be  
treated by formulae originally belonging
 to pure geometry2 that device  has long
 been in use4 ,experimental geometry and
 mechanics actually  relate to the same 
subject-matter2 and the young student   
who discovers  experimental laws with   
ruler and compasses and cardboard       
figures1 and  later goes on to pendulums
 and spring-balances1 is developing a   
single  subject which cannot be divided 
any more than the subject of magnetism  
can be divided from electricity4    ,it 
is through this unification of geometry 
and mechanics that ,i should  like to   
approach the problem of gravitation1    
showing that a field of  force is a     

manifestation of the geometry of space  
and time4 ,but ,i fear  that that would 
be too technical2 so we will approach it
 from a  different angle4    ,we have   
shown that the contemplation of the     
world from the standpoint  of a single  
observer is liable to distort its       
simplicity1 and we have  tried to obtain
 a juster idea by taking into account   
and combining  other points of view4    
,the more standpoints the better4 ,let  
us now  consider another point of view1 
which we have not previously thought    
about--the point of view of an observer 
who has tumbled out of an  aeroplane and
 is falling headlong4 ,in many respects 
his is an ideal  situation--temporarily4
 ,unfortunately on terra firma we are   
continually subjected to a very         
disturbing influence2 we undergo a      
terrific bombardment by the molecules of
 the ground1 which are hammering  on the
 soles of our boots with a total force  
of some ten stone weight  pressing us   
upwards4 ,now our bodies are the        

scientific appliances which  we use to  
make our common observations of the     
world4 ,i am sure that no  physicist    
would permit any one to enter his       
laboratory and hammer on his  clocks and
 galvanometers whilst he was observing  
with them2 at any rate  he would think  
it necessary to apply some corrections  
for the effect of  the disturbance4 ,let
 us then allow ourselves to fall freely 
in vacuo2  then we shall be free from   
this disturbing bombardment and able to 
take  a much more natural view of what  
is going on around us4    ,whilst       
falling1 we perform the experiment of   
letting go an apple held in  the hand4  
,the apple is now free1 but it cannot   
fall any more than it was  falling      
already2 consequently it remains poised 
in contact with our  hand4 ,in our new  
outlook--in our new frame of space and  
time--an apple  does not drop4 ,there is
 no mysterious force accelerating it4   
,and  remember that this new frame of   
space and time is the natural frame of a

  free observer2 whereas the old frame1 
in which the mysterious  accelerating   
force occurred1 was the frame of a very 
much disturbed  observer4 ,it is true   
that when we look down at the earth we  
see trees  and houses rushing up to meet
 us2 but there is no mystery about that4
  ,there is an obvious cause for it2    
plainly they are being propelled        
upwards from below by that molecular    
bombardment which ,i have mentioned4    
,you see that the apple's view of things
 is simpler than ,newton's4 ,newton  had
 to invent a mysterious force dragging  
the apple down2 the apple  observes only
 a familiar physical agency propelling  
,newton up4    ,it is not my purpose to 
emphasize unduly the superiority of the 
apple's  view over ,newton's1 but rather
 to regard both on an equal footing4 ,i 
have perhaps been a little unfair to    
,newton4 ,his position on the surface   
of the earth was unfortunate1 but he    
would have been perfectly content  to be
 at the centre of the earth1 where he   

could have remained without  support1   
i4e4 without disturbance by molecular   
bombardment4 ,from there  he would still
 have observed the well-known           
acceleration of the apple2  and the     
apple would have observed a             
corresponding acceleration of ,newton   
without any molecular bombardment       
causing it4 ,from either point of view  
there is a mysterious agent at work4    
,how shall we picture to ourselves  this
 agent8 ,shall we picture it as a       
force--a tug of some kind8 ,but if  so1 
to which of them is the tug applied8 ,if
 we take the standpoint of  ,newton the 
tug is applied to the apple1 if the     
standpoint of the apple  the tug is     
applied to ,newton2 so that in our      
synthesis of all  standpoints we cannot 
decide which is being tugged1 and the   
picture of  gravitation as a tugging    
agent becomes impossible4 ,einstein     
replaces it  by a different picture1    
which we shall perhaps better understand
 if we  compare it with a very similar  

revolution of scientific thought which  
occurred long ago4    ,the ancients     
believed that the earth was flat4 ,the  
small portion of its  surface with which
 they were chiefly concerned could be   
represented  without serious distortion 
on a flat map4 ,as more distant         
countries were  added1 it would be      
natural to think that they also could be
 included in  the flat map4 ,you have   
all seen such maps of the world1 e4 g4  
,mercator's  projection1 and you will   
remember how ,greenland appears         
enormously  exaggerated in size4 ,now   
those who adhered to the flat-earth     
theory must  hold that the flat map     
gives the true size of ,greenland4 ,how 
then  would they explain that travellers
 in that country reported that the      
distances were much shorter8 ,they      
would1 ,i suppose1 invent a theory that 
a demon resided in that country who     
helped travellers on their way1  making 
the journeys appear much shorter than   
they 'really' were4 ,no  doubt the      

scientists would preserve their         
self-respect by using some              
,graeco-,latin polysyllable instead of  
the word 'demon'1 but that must not     
disguise from us the fact that they were
 appealing to a deus ex  machina4 ,the  
name demon is rather suitable1 however1 
because he has the  impish              
characteristic that we cannot pin him   
down to any particular  locality4 ,we   
might equally well start our flat map   
with its centre in  ,greenland2 then it 
would be found that journeys there were 
quite normal1  and that the activities  
of the demon were disturbing travellers 
in  ,europe4 ,we now recognize that the 
true explanation is that the earth's    
surface is curved2 and the demoniacal   
complications appeared because we  were 
forcing the earth's surface into an     
inappropriate flat frame which  distorts
 the simplicity of things4    ,what has 
happened in the case of the earth has   
happened also in the case  of the world1
 and a similar revolution of thought is 

needed4 ,an  observer1 say at the centre
 of the earth1 finds that there is a    
frame of  space and time--a flat or     
,euclidean frame--in which he can locate
 things  happening in his neighbourhood 
without distorting their natural        
simplicity4 ,there is no gravitation1 no
 tendency of bodies to fall1 so  long as
 the observer confines his observations 
to his immediate  neighbourhood4 ,he    
extends this frame of space and time to 
greater  distances1 and ultimately to   
the earth's surface where he encounters 
the  phenomenon of falling apples4 ,this
 new phenomenon must be accounted for1  
so he invents a deus ex machina which he
 calls gravitation to whose  activities 
the disturbance is attributed4 ,but we  
have seen that we may  just as well     
start with the falling apple4 ,it has a 
flat frame of space  and time into which
 phenomena in its neighbourhood fit     
without  distortion2 and from its point 
of view bodies near it do not undergo   
any  acceleration4 ,but when it extends 

this frame farther afield1 the          
simplicity is lost2 and it too has to   
postulate the demon force of            
gravitation existing in distant parts1  
and for example causing  undisturbed    
objects at the centre of the earth to   
fall towards it4 ,as we  change from one
 observer to another--from one flat     
space-time frame to  another--so we have
 to change the region of activity of    
this demon4 ,is  not the solution now   
apparent8 ,the demon is simply the      
complication  which arises when we force
 the world into a flat ,euclidean       
space-time  frame into which it does not
 fit without distortion4 ,it does not   
fit the  frame1 because it is not a     
,euclidean or flat world4 ,admit a      
curvature  of the world and the         
mysterious disturbance disappears4      
,einstein has  exorcized the demon4     
,einstein1 recognizing that in the      
phenomena of gravitation he was not     
dealing with a 'tug' but with a         
curvature of the world1 had to          

reconsider the law of gravitation4 ,he  
could not make any possible law of      
curvature correspond exactly with the   
previously assumed law of tugging4      
,thus he was led to propound a new law  
of gravitation--a law which in  most    
practical cases differs very little from
 that of ,newton1 although  it has an   
essentially different foundation4 ,i    
need not here dwell on the  very        
remarkable way in which ,einstein's     
emendation of the law of  gravitation   
has been confirmed both by the anomalous
 secular change in  the orbit of the    
planet ,mercury1 and by the observed    
displacement of the  stars near the sun 
during the total eclipse of #aiai4 ,i   
might1 however1  remind you that in the 
latter observation the point at issue   
between  ,newton's and ,einstein's      
theory was not the existence of a       
deflexion of  light-rays passing near   
the sun but the amount of the deflexion1
  ,einstein predicting twice the        
deflexion possible on the ,newtonian    

theory4 ,the larger deflexion was       
quantitatively confirmed by the eclipse 
observations4 ,einstein's main          
achievement is a new law1 not a new     
explanation1 of gravitation4 ,he        
attributes the gravitation of massive   
bodies to a curvature of the world in   
the region surrounding them and so      
throws a flood of light on the whole    
problem2 but he is not primarily        
concerned to explain how material bodies
 produce 7or are associated  with7 this 
curvature of the world around them1 nor 
how this curvature is  made subject to a
 law4 ,although it would be an entire   
misunderstanding  of ,einstein's        
attitude in propounding the general     
relativity theory to  regard it as a    
search for an explanation of            
gravitation1 nevertheless ,i  think that
 the further following up of his ideas  
has led to a genuine  explanation as    
complete as could be desired4 ,but ,i am
 not going to give  you the explanation 
in this lecture2 sometimes an           

explanation requires a  great deal of   
explaining4#g    ,i think that we can   
without mathematics form a general idea 
of why  ,einstein found it necessary to 
amend ,newton's law of gravitation4 ,let
 us  return to the illustration of the  
pig1 and imagine that we wish to        
discover the law governing the          
distribution of fat and lean in the     
animal4 ,from the breakfast-table       
standpoint a plausible type of law      
would be that half of each rasher is fat
 and the other half lean2 and if  this  
turned out to be confirmed very         
approximately by observation we  might  
well imagine that we had discovered the 
exact law of porcine  structure4 ,but   
the case is altered if we give up the   
breakfast-table  standpoint and         
contemplate the animal in a more general
 way1 remembering  that he has not been 
designed with any particular reference  
to the  series of rashers into which our
 grocer has chosen to slice him4 ,we    
must  now look for a different type of  

law altogether4 ,two possibilities may  
arise4 ,we may find that our proposed   
law1 although expressed in              
breakfast-table parlance1 is            
nevertheless equivalent to a possible   
biological law2 it may be immediately   
capable of translation into a more      
general statement which makes no        
reference to a particular               
stratification4 ,but on the other hand1 
it may happen that the suggested  law   
cannot be freed from this reference to a
 particular system of  slicing4 ,in that
 case we can only regard it as          
approximate1 perhaps  holding fairly    
well for the slices of which we have    
most experience but  becoming less and  
less accurate in the more tortuous parts
 of the  animal4 ,both these cases are  
illustrated in ,einstein's modifications
 of  classical theory4 ,newton's law of 
gravitation explicitly refers to a      
space-time frame and therefore to a     
world stratified into instantaneous     
states4 ,it proves to be impossible to  

free it from this reference to a        
particular stratification without       
modifying it4 ,in fact if the crucial   
astronomical observations had shown that
 ,newton's law and not ,einstein's  was 
the exact law of gravitation1 this would
 have been evidence of a  real          
stratification of the structure of the  
world--a stratification  revealed by no 
other phenomena4 ,einstein's law is the 
simpler law  because it is consistent   
with what we now know of the general    
plan of  world-structure2 ,newton's law 
could only be made possible by          
introducing  a novel and specialized    
feature--a stratified arrangement of    
structure--which is not revealed in any 
other phenomena4    ,maxwell's laws of  
electromagnetism afford an example of   
the other type4  ,these1 it is true1 are
 stated as relating to the particular   
slices of  the world of events1 which   
are served up to us like rashers instant
 by  instant4 ,but they can be restated1
 without alteration of effect1 in a     

form making no reference to slices4     
,this is a very remarkable property  of 
,maxwell's equations which was quite    
unknown at the time they were  first put
 forward4 ,it was brought to light much 
later by the researches  of ,larmor and 
,lorentz4 ,in consequence of this       
,einstein is able to take  over the     
whole classical theory of               
electromagnetism unaltered2 he  restates
 it so as to show how it applies        
generally and is not bound up  with the 
purely terrestrial point of view1 but he
 does not amend the  laws4 ,he metes out
 different treatment to the             
gravitational laws and  electromagnetic 
laws1 because he finds the latter       
already adapted to his  scheme4    ,if  
,i have succeeded in my object1 you will
 have realized that the  present        
revolution of scientific thought follows
 in natural sequence on  the great      
revolutions at earlier epochs in the    
history of science4  ,einstein's special
 theory of relativity1 which explains   

the  indeterminateness of the frame of  
space and time1 crowns the work of      
,copernicus who first led us to give up 
our insistence on a geocentric  outlook 
on nature2 ,einstein's general theory of
 relativity1 which  reveals the         
curvature or non-,euclidean geometry of 
space and time1  carries forward the    
rudimentary thought of those earlier    
astronomers who  first contemplated the 
possibility that their existence lay on 
something  which was not flat4 ,these   
earlier revolutions are still a source  
of  perplexity in childhood1 which we   
soon outgrow2 and a time will come  when
 ,einstein's amazing revelations have   
likewise sunk into the  commonplaces of 
educated thought4    ,to free our       
thought from the fetters of space and   
time is an aspiration  of the poet and  
the mystic1 viewed somewhat coldly by   
the scientist who  has too good reason  
to fear the confusion of loose ideas    
likely to  ensue4 ,if others have had a 
suspicion of the end to be desired1 it  

has  been left to ,einstein to show the 
way to rid ourselves of these           
'terrestrial adhesions to thought'4 ,and
 in removing our fetters he  leaves us1 
not 7as might have been feared7 vague   
generalities for the  ecstatic          
contemplation of the mystic1 but a      
precise scheme of  world-structure to   
engage the mathematical physicist4      
,footnote #a3 ,the only alternative is  
that 7relatively to a solar  observer7  
the velocity of light differs in        
different directions1 at  least in the  
region where the experiment is          
conducted4 ,this would  presumably be   
due to some influence of the moving     
earth on the  propagation of light      
7convection of the ether74 ,this        
explanation was at  one time favoured1  
but it could not be reconciled with the 
observed  phenomena of the aberration of
 light4    ,footnote #b3 ,the relativity
 theory does not suggest that there is  
such  a thing in nature as a            
four-dimensional space4 ,the whole      

object of the  recognition of the       
four-dimensional world is to eliminate  
the harassing  frame of space4          
,footnote #c3 ,the inclination must not 
exceed a certain limit4 ,this  limiting 
angle may be regarded as a fundamental  
constant of the  world-structure1 and   
owing to its fundamental character it   
appears in  many kinds of phenomena2 for
 example1 it determines the velocity of 
propagation of light4 ,the instant on   
the sun which is simultaneous with  a   
given instant on the earth is           
indeterminate 7varying according to the 
space and time frame employed7 but only 
within a range of #af minutes4  ,any    
event on the sun happening before this  
#af minutes is absolutely in  the past1 
all observers agreeing on this point2 in
 fact it would be  possible for us to   
have already received a wireless message
 announcing  its occurrence4 ,events    
after the #af minutes are in the        
absolute  future4 ,the neutral zone     
which is 7absolutely7 neither past nor  

future  becomes proportionately wider as
 the distance increases2 at the nearest 
fixed star it extends to #h years1 and  
at the most distant stars yet  known it 
reaches #djj1jjj years4    ,footnote #d3
 ,the three events must not be at the   
same place since that  would give a     
time-line not a triangle4 ,the clock    
must move so that the  two events whose 
time-distance is to be determined both  
happen where it  is1 just as the scale  
must be directed so that the two points 
fall on  it4 ,you are not allowed to    
'bend' the clock1 i4 e4 apply force so  
as to  make it move with other than     
uniform velocity1 any more than you are 
allowed to bend the scale by applying   
force4    ,footnote #e3 ,of course1 it  
is not true that any two sides are less 
than the third side4 ,a clock1 unlike a 
scale1 can only measure in one          
direction1 viz4 from past to future1 so 
that the sides ,,ab  ,,bc and  ,,ac can 
be chosen in only one way4    ,footnote 
#f3 ,this involves only a comparatively 

trifling generalization  of ,euclidean  
geometry1 not to be confused with the   
'non-,euclidean'  geometry introduced   
later in the lecture4    ,footnote #g3  
,the following brief outline will give a
 hint of the nature  of the explanation4
 ,einstein's law of gravitation is      
usually expressed  as a set of ten very 
lengthy differential equations2 these   
equations are  exactly equivalent to the
 geometrical statement that 'the radius 
of  spherical curvature of any          
#c-;dimensional section of the          
#d-;dimensional  world is a universal   
constant length1 the same for all points
 of the  world and for all directions of
 the section'4 ,the law therefore       
implies  that the world has a certain   
type of homogeneity and isotropy 7not1  
however1 the complete homogeneity and   
isotropy of a sphere74 ,to  explain the 
law of gravitation and the phenomena    
governed by it1 we have  to explain how 
this isotropy and homogeneity is        
secured4 ,our explanation  is that the  

homogeneity and isotropy is not         
initially in the external  world1 but in
 the measurements which we make of it4  
,it is introduced  in all our operations
 of measurement1 because the appliances 
which we  use for measurement are       
themselves part of the world4 ,in the   
earlier  part of this lecture we saw    
that the contraction of the arm turned  
from  horizontal to vertical is not     
detected by measurements made with a    
yard-measure which shares the           
contraction2 in the same way any        
anisotropy of the world does not appear 
in measurements of it by  appliances    
which1 being part of the world1 share   
the same anisotropy4  ,the law of       
gravitation therefore arises from the   
fact that a certain  type of            
non-homogeneity and non-isotropy of the 
world cannot come into  observational   
experience1 because it is necessarily   
eliminated in all  observations and     
measurements made with material         
appliances4 ,the orderly  phenomena of  

gravitation are due to the absence of   
certain conceivable  effects4 ,we have  
been trying to find a key to the        
mystery2 but the  secret of the lock    
lies not in the key but in the wards4