,,transcribers ,,note3        ,the plain
 text version of this ebook includes    
mathematical      formulas in ,te,x     
notation4 ,these formulas are enclosed  
between dollar signs4 ,the ,,html       
version includes the same      formulas 
rendered in their original form4        
,,einstein',s ,,theories ,,of           
,,relativity ,,and ,,gravitation        
,a ,,selection ,,of ,,material ,,from   
,,the ,,essays                          
,,submitted ,,in ,,the ,,competition    
,,for ,,the ,,eugene ,,higgins ,,prize  
,,of #e1jjj                             
,,compiled ,,and ,,edited1              
,,and ,,introductory ,,matter ,,supplied
                                        
,,by                              ,j4   
,,malcolm ,,bird1                       
,associate ,editor1 ,scientific         
,american                               
,,new ,,york                            
,,scientific ,,american ,,publishing    
,,co41                                  
,,munn  ,,co4                           

#aiba                    ,,preface      
,the obstacles which the layman finds to
 understanding ,einstein's  relativity  
theories lie not so much in the inherent
 difficulty of  these theories          
themselves as in the difficulty of      
preparing the  mind for their reception4
 ,the theory is no more difficult than  
any  scientific development of          
comparable depth2 it is not so difficult
 as  some of these4 ,but it is a fact   
that for a decent understanding of it1  
a large background of scientific        
knowledge and scientific habit of       
thought is essential4 ,the bulk of the  
writers who have attempted to  explain  
,einstein to the general reader have not
 realized the great gulf  which lies    
between the mental processes of the     
trained mathematician  and those of the 
man in the street4 ,they have not       
perceived that  the lay reader must be  
personally conducted for a long distance
 from  the vestibule of the temple of   
science before he comes to ,einstein1   

and that he cannot by any possibility   
make this journey unaided4 ,the  result 
has been to pitchfork the reader into   
the intricacies of the  subject without 
adequate preparation4    ,the present   
volume avoids this mistake with the     
utmost care4 ,it avoids  it1 in fact1   
with such deliberation as to make it in 
order to say a  word in explanation of  
what will at first glance seem an       
extraordinary  arrangement of material4 
,it was to be expected1 doubtless1 that 
this  book would open with a brief      
statement of the genesis and the outcome
  of the ,einstein ,prize ,essay        
,contest for the #e1jjj prize offered by
  ,mr4 ,eugene ,higgins4 ,it was        
doubtless to be expected that1 after    
this  had been dismissed1 the winning   
essay would be given the post of honor  
in advance of all other material bearing
 actually on the ,einstein  theories4   
,when the reader observes that this has 
not been done1 he  will by all means    
expect a word of explanation2 and it is 

mainly for  the purpose of giving this  
that we make these introductory remarks4
    ,the essays submitted in the        
contest1 and in particular the comments 
of  a few disappointed readers upon ,mr4
 ,bolton's prize essay1 make quite      
plain what might have been              
anticipated--that in the small compass  
of #c1jjj words it is not possible both 
to prepare the reader's mind  for a     
discussion of ,relativity and to give a 
discussion that shall be  adequate4 ,mr4
 ,bolton himself1 in replying to a      
protest that he had not  done all this1 
has used the word miracle--we think it a
 well-advised  one4 ,no miracle was     
expected as a result of the contest1 and
 none has  been achieved4 ,but in       
awarding the prize1 the ,judges had to  
decide  whether it was the best         
preliminary exposition or the best      
discussion  that was wanted4 ,they      
decided1 and rightly we believe1 that   
the award  should go to an actual       
statement of what the ,einstein theories

 are  and what they do1 rather than to a
 mere introduction1 however well        
conceived and well executed the latter  
might be4 ,nevertheless1  we should be  
closing our eyes to a very obvious fact 
if we did  not recognize that1 without  
something in the way of preparation1    
the general reader is not going to      
pursue ,mr4 ,bolton's essay1 or  any    
other essay on this subject1 with       
profit4 ,it is in order the  more       
forcefully to hold out inducements to   
him to subject himself to  this         
preparation that we place at the head of
 the book the chapters  designed to give
 it to him4    ,chapter ,,ii4 is        
intended so to bring the mind of the    
reader into contact  with certain       
philosophical problems presented to us  
by our experiences  with the external   
world and our efforts to learn the facts
 about it1  that he may approach the    
subject of relativity with an           
appreciation  of the place it occupies  
as a phase of human thought and a pillar

 of  the scientific structure4 ,until   
the reader is aware of the existence  of
 these problems and the directions taken
 by the efforts1 successful  and        
unsuccessful1 to unravel them1 he is not
 equipped to comprehend  the doctrine of
 relativity at all2 he is in much the   
same case as a  child whose education   
had reached only the primer stage1 if   
asked to  read the masterpieces of      
literature4 ,he lacks not alone the     
vocabulary1  but equally the mental     
background on which the vocabulary is   
based4    ,it will be noted that in this
 and the chapters immediately following 
it1 the ,editor has supplied material   
freely4 ,the obvious interpretation  is 
that satisfactory material covering the 
desired ground was not  found in any of 
the essays2 for we are sure the scope   
and number  of the credited excerpts    
will make it clear that all             
contributions  were adequately          
scrutinized in search of available      
passages4 ,this  inadequacy of the      

competing essays has been severely      
commented  upon by several              
correspondents1 and the inference drawn 
that as  a whole the offerings were not 
up to the mark4 ,such a viewpoint  is   
wholly unjust to the contestants4 ,the  
essays which paid serious  attention to 
the business of paving the way to       
relativity necessarily  did so at the   
expense of completeness in the later    
paragraphs where  specific explanation  
of the ,einstein theories was in order4 
,mr4 ,law1  whose essay was by all means
 the best of those that gave much space 
to introductory remarks1 found himself  
left with only #fjj words  in which to  
tell what it was that he had been       
introducing4 ,the  majority of the      
contestants appear to have faced the    
same question  as to subject matter     
which the ,judges faced1 and to have    
reached  the same decision4 ,they       
accordingly devoted their attention     
toward  the prize1 rather than toward   
the production of an essay that would   

best supplement that of the winner4 ,it 
is for this very reason that1  in these 
preliminary chapters1 so large a        
proportion of the material  has had to  
be supplied by the ,editor2 and this    
very circumstance is  a tribute to the  
good judgment of the competitors1 rather
 than ground  for criticism of their    
work4    ,the general introduction of   
,chapter ,,ii4 out of the way1 ,chapters
  ,,iii4 and ,,iv4 take up the business 
of leading the reader into the  actual  
subject of relativity4 ,the subject is  
here developed in what  may be called   
the historical order--the order in which
 it took form in  ,einstein's own mind4 
,both in and outside the contest of     
which this book  is the outcome1 a      
majority of those who have written on   
relativity  have followed this order1   
which is indeed a very natural one and  
one well calculated to give to the      
rather surprising assumptions of        
relativity a reasonableness which they  
might well appear to the lay  mind to   

lack if laid down more arbitrarily4 ,in 
these two chapters  no effort is made to
 carry the argument beyond the          
formulation of  the ,special ,principle 
of the relativity of uniform motion1 but
 this  principle is developed in        
considerably more detail than would be  
the  case if it were left entirely to   
the competing essayists4 ,the reason    
for this is again that we are dealing   
with a phase of the subject  which is of
 subordinate importance so far as a     
complete statement of  the ,general     
,theory of ,relativity is concerned1 but
 which is of the  greatest significance 
in connection with the effort of the    
layman to  acquire the proper           
preliminary orientation toward the      
larger subject4    ,chapter ,v4 goes    
back again to general ground4 ,among the
 ideas which  the competing essayists   
were forced to introduce into their text
 on a  liberal scale is that of         
non-,euclidean geometry4 ,the entire    
formulation  of the ,general ,theory of 

,relativity is in fact an exercise in   
this4 ,the  essayists--good1 bad and    
indifferent alike--were quite unanimous 
in  their decision that this was one    
thing which the reader would have  to   
assume the responsibility of acquiring  
for himself4 ,certainly they  were      
justified in this2 for the ,editor has  
been able to explain what               
non-,euclidean geometry is only by using
 up considerably more space  than the   
contestants had for an entire essay4 ,no
 effort has been made  to set forth any 
of the details of any of the various    
non-,euclidean  geometries2 it has      
simply been the aim to draw the dividing
 line  between ,euclidean and           
non-,euclidean1 and to make the         
existence of  the latter appear         
reasonable1 so that when the essayists  
come to talk  about it the reader will  
not feel hopelessly at sea4 ,in other   
words1  this is another case of         
providing the mental background1 but on 
such  a scale that it has seemed        

necessary to give a separate chapter to 
it4    ,chapter ,,vi4 completes the     
preliminary course in the fundamentals  
of  relativity by tying up together the 
findings of ,chapter ,v4 and those  of  
,chapters ,,iii4 and ,,iv4 ,it          
represents more or less of a last-minute
  change of plan2 for while it had been 
the ,editor's intent from the  beginning
 to place the material of ,chapters     
,,ii4-,v4 in its present  position1 his 
preliminary impression would have been  
that the work  of the present ,chapter  
,,vi4 would be adequately done by the   
essayists  themselves4 ,his reading of  
the essays1 however1 convinced him that 
it  had not so been done--that with the 
possible exception of ,mr4 ,francis1    
the essayists did not make either a     
serious or a successful effort to  show 
the organic connection between the      
,special ,theory of ,relativity  and the
 ,minkowski space-time structure1 or the
 utter futility of  trying to reconcile 
ourselves to the results of the former  

without  employing the ideas of the     
latter4 ,so ,chapter ,,vi4 was supplied 
to make  good this deficiency1 and to   
complete the mental equipment which the 
reader requires for his battle with the 
,general ,theory4    ,in laying down a  
set of general principles to govern the 
award of  the prize1 one of the first   
things considered by the ,judges was the
  relative importance of the ,special   
and the ,general ,theories4 ,it was     
their opinion that no essay could       
possibly qualify for the prize which    
did not very distinctly give to the     
,general ,theory the center of the      
stage2 and that in fact discussion of   
the ,special ,theory was pertinent  only
 so long as it contributed1 in          
proportion to the space assigned  it1 to
 the attack upon the main subject4 ,the 
same principle has been  employed in    
selecting essays for complete or        
substantially complete  reproduction in 
this volume4 ,writers who dealt with the
 ,special ,theory  in any other sense   

than as a preliminary step toward the   
,general ,theory  have been relegated to
 the introductory chapters1 where such  
excerpts  from their work have been used
 as were found usable4 ,the distinction 
of publication under name and title is  
reserved for those who wrote            
consistently and specifically upon the  
larger subject--with the one  exception 
of ,dr4 ,russell1 whose exposition of   
the ,special ,theory is so  far the best
 of those submitted and at the same time
 so distinctive  that we have concluded 
it will appear to better advantage by   
itself  than as a part of ,chapters     
,,iii4 and ,,iv4    ,following after    
,mr4 ,bolton's essay we have tried to   
arrange the various  contributions1 not 
at all in any order of merit1 but in the
 order  that will make connected reading
 of the book most nearly possible  and  
profitable4 ,each essay should be made  
easier of reading by the  examination of
 those preceding it2 at the same time   
each1 by the choice  of ground covered  

and by the emphasis on points not       
brought out sharply  by its             
predecessors1 should throw new light    
upon these predecessors4    ,the reader 
will find that no two of the essays     
given thus in full  duplicate or even   
come close to duplicating one another4  
,they have  of course been selected with
 this in view2 each represents the best 
of several essays of substantially the  
same character4 ,not all of  them       
require comment here1 but concerning    
some of them a word may  well be said4  
,mr4 ,francis1 we believe1 has succeeded
 in packing more substance  into his    
#c1jjj words than any other competitor4 
,mr4 ,elliot has come  closer than      
anybody else to really explaining       
relativity in terms  familiar to        
everybody1 without asking the reader to 
enlarge his  vocabulary and with a      
minimum demand so far as enlarging his  
mental  outlook is concerned4 ,were it  
not for certain conspicuous defects1    
his essay would probably have taken the 

prize4 ,in justice to the  ,judges1 we  
should state that we have taken the     
liberty of eliminating  ,mr4 ,elliot's  
concluding paragraph1 which was the most
 objectionable  feature of his essay4   
,dr4 ,dushman chose for his title the   
one which we adopted for this  book4 ,it
 became necessary1 therefore1 for us to 
find a new title for  his essay2 aside  
from this instance1 the main titles     
appearing at the  heads of the various  
complete essays are those of the        
authors4 ,the  subtitles have in        
practically every instance been supplied
 editorially4    ,dr4 ,pickering        
submitted two essays1 one written from  
the viewpoint  of the physicist1 the    
other from that of the astronomer4 ,to  
make  each complete1 he naturally found 
it necessary to duplicate between  them 
certain introductory and general        
material4 ,we have run the two  essays  
together into a single narrative1 with  
the elimination of this  duplicated     
material2 aside from this blue-penciling

 no alteration  has been made in ,dr4   
,pickering's text4 ,this text however   
served as  the basis of blue-penciling  
that of several other contestants1 as   
indicated in the foot notes4    ,for the
 reader who is qualified or who can     
qualify to understand it1  ,dr4         
,murnaghan's essay is perhaps the most  
illuminating of all4 ,even  the reader  
who does not understand it all will     
realize that its author  brings to the  
subject a freshness of viewpoint and an 
originality of  treatment which are     
rather lacking in some of the published 
essays1  and which it will readily be   
understood were conspicuously lacking   
in a good many of the unpublished ones4 
,dr4 ,murnaghan of all the  competitors 
has come closest to making a            
contribution to science as  well as to  
the semi-popular literature of science4 
,in the composite chapters1 the brackets
 followed by reference numbers  have    
been used as the most practicable means 
of identifying the  various individual  

contributions4 ,we believe that this    
part of the  text can be read without   
allowing the frequent occurrence of     
these  symbols to distract the eye4 ,as 
to the references themselves1 the       
asterisk marks the contributions of the 
,editor4 ,the numbers are those         
attached to the essays in order of and  
at the time of their receipt2  it has   
been more convenient to use these than  
to assign consecutive  numbers to the   
quoted essays4 ,the several numbers     
identify passages  from the essays of   
the following contestants3       #aj3   
,frederick ,w4 ,shurlock1 ,derby1       
,england4   #ah3   ,l4 ,l4 ,whyte1      
,cambridge1 ,england4   #bd3   ,prof4   
,moritz ,schlick1 ,university of        
,rostock1 ,germany4   #cj3   ,c4 ,e4    
,rose1 ,m4,e41 ,little ,rock1 ,ark4     
#cc3   ,h4 ,gartelmann1 ,bremen1        
,germany4   #ce3   ,prof4 ,joseph ,s4   
,ames1 ,johns ,hopkins ,university1     
,baltimore4   #dg3   ,james ,o4 ,g4     
,gibbons1 ,east ,orange4 ,n4 ,j4   #hb3 

,charles ,h4 ,burr1 ,philadelphia4      
#aja3   ,l4 ,f4 ,h4 de ,miffonis4       
,b4,a41 ,c4,e41 ,ottawa1 ,canada4  #ajb3
   ,charles ,a4 ,brunn1 ,kansas ,city4  
#ajf3   ,j4 ,elias ,fries1 ,fellow      
,a4,i4,e4,e41 ,birmingham1 ,ala4  #aad3 
,dean ,w4 ,p4 ,graham1 ,syracuse        
,university1 ,syracuse1 ,n4 ,y4  #aae3  
,rev4 ,george ,thomas ,manley1 ,london4 
#aaf3   ,prof4 ,j4 ,a4 ,schouten1       
,delft1 ,netherlands4  #aba3   ,elwyn   
,f4 ,burrill1 ,berkeley1 ,cal4  #abe3   
,dorothy ,burr1 ,bryn ,mawr1 ,pa4  #acj3
   ,c4 ,w4 ,kanolt1 ,bureau of          
,standards1 ,washington4  #ace3         
,robert ,stevenson1 ,new ,york4  #aci3  
,leopold ,schorsch1 ,new ,york4  #ada3  
,dr4 ,m4 ,c4 ,mott-,smith1 ,los         
,angeles1 ,calif4  #adg3   ,edward ,a4  
,clarke1 ,columbus1 ,o4  #adi3   ,edward
 ,a4 ,partridge1 ,philadelphia4  #aej3  
,col4 ,john ,millis1 ,u4 ,s4 ,a41       
,chicago4  #aeb3   ,george ,f4          
,marsteller1 ,detroit4  #aef3   ,d4 ,b4 
,hall1 ,cincinnati4  #afe3   ,francis   

,farquhar1 ,york1 ,pa4  #agh3   ,dr4    
,george de ,bothezat1 ,dayton1 ,o4      
#agi3   ,professor ,a4 ,e4 ,caswell1    
,university of ,oregon1 ,eugene1 ,ore4  
#ahb3   ,c4 ,e4 ,dimick1 ,new ,london1  
,conn4  #ahf3   ,earl ,r4 ,evans1       
,washington1 ,d4 ,c4  #ahh3   ,norman   
,e4 ,gilbert1 ,dartmouth ,college1      
,hanover1 ,n4 ,h4  #aib3   ,a4 d',abro4 
,new ,york4  #aid3   ,l4 ,m4 ,alexander1
 ,cincinnati4  #aig3   ,kenneth ,w4     
,reed1 ,east ,cleveland1 ,o4  #aih3     
,prof4 ,e4 ,n4 da ,c4 ,andrade1         
,ordnance ,college1 ,woolwich1 ,england4
  #baf3   ,professor ,andrew ,h4        
,patterson1 ,university of ,north       
,carolina1         ,chapel ,hill1 ,n4   
,c4  #bbj3   ,prof4 ,arthur ,gordon     
,webster1 ,clark ,college1 ,worcester1  
,mass4  #bba3   ,walter van ,b4         
,roberts1 ,princeton ,university1 ,n4   
,j4  #bbc3   ,paul ,m4 ,batchelder1     
,austin1 ,tex4  #bbg3   ,prof4 ,r4 ,w4  
,wood1 ,johns ,hopkins ,university1     
,baltimore4  #bbi3   ,e4 ,p4 ,fairbairn1

 ,m4,c41 ,b4,sc41 ,glasgow4  #bca3   ,r4
 ,f4 ,deimel1 ,hoboken1 ,n4 ,j4  #bcb3  
,lieut4 ,w4 ,mark ,angus1 ,u4 ,s4 ,n41  
,philadelphia4  #bce3   ,edward ,adams  
,richardson1 ,kansas ,city4  #bfc3      
,prof4 ,william ,benjamin ,smith1       
,tulane ,university1 ,new ,orleans4     
#bfd3   ,james ,rice1 ,university of    
,london1 ,london4  #bfg3   ,william     
,hemmenway ,pratt1 ,lynn1 ,mass4  #bgb3 
,r4 ,bruce ,lindsay1 ,new ,bedford1     
,mass4  #bhc3   ,frank ,e4 ,law1        
,montclair1 ,n4 ,j4      ,in addition to
 the specific credit given by these     
references for  specifically quoted     
passages1 the ,editor feels that he     
ought to  acknowledge his general       
indebtedness to the competing essayists1
  collectively1 for the many ideas which
 he has taken away from their  text to  
clothe in his own words4 ,this does not 
mean that the ,editor has  undertaken   
generally to improve upon the language  
of the competitors1  but merely that the
 reading of all their essays has given  

him many  ideas of such complex origin  
that he could not assign credit if  he  
would4                    ,,table ,,of  
,,contents      ,i4--,the ,einstein     
#e1jjj ,prize3 ,how the ,contest ,came  
to be ,held1  and ,some of the ,details 
of ,its ,conduct4 ,by the ,editor       
#a    ,,ii4--,the ,world--,and ,us3 ,an 
,introductory ,discussion of the        
,philosophy  of ,relativity1 and of the 
,mechanism of our ,contact with ,time   
and  ,space4 ,by various contributors   
and the ,editor      #ai    ,,iii4--,the
 ,relativity of ,uniform ,motion3       
,classical ,ideas on the  ,subject2 the 
,ether and the ,apparent ,possibility of
 ,absolute ,motion2  the                
,michelson-,morley ,experiment and the  
,final ,negation of this  possibility4  
,by various contributors and the ,editor
      #dg    ,,iv4--,the ,special       
,theory of ,relativity3 ,what           
,einstein's ,study of  ,uniform ,motion 
,tells ,us ,about ,time and ,space and  
the ,nature of the  ,external ,reality4 

,by various contributors and the ,editor
      #gf    ,v4--,that ,parallel       
,postulate3 ,modern ,geometric ,methods2
 the ,dividing  ,line ,between          
,euclidean and ,non-,euclidean2 and the 
,significance of  the ,latter4 ,by the  
,editor      #aaa    ,,vi4--,the        
,space-,time ,continuum3 ,minkowski's   
,world of ,events1 and the  ,way in     
,which ,it ,fits ,into ,einstein's      
,structure4 ,by the ,editor and a  few  
contributors      #ada                  
,,vii4--,relativity3 ,the ,winning      
,essay in the ,contest for the ,eugene  
,higgins #e1jjj ,prize4 ,by ,lyndon     
,bolton1 ,british ,patent ,office1      
,london      #afi    ,,viii4--,the ,new 
,concepts of ,time and ,space3 ,the     
,essay in ,behalf of  ,which the        
,greatest ,number of ,dissenting        
,opinions ,have ,been ,recorded4 ,by    
,montgomery ,francis1 ,new ,york        
#aha    ,,ix4--,the ,principle of       
,relativity3 ,a ,statement of ,what it  
is ,all  ,about1 in ,ideas of ,one      

,syllable4 ,by ,hugh ,elliot1           
,chislehurst1 ,kent1  ,england      #aie
    ,x4--,space1 ,time and ,gravitation3
 ,an ,outline of ,einstein's ,theory of 
,general ,relativity4 ,by ,w4 de        
,sitter1 ,university of ,leyden         
#bjf    ,,xi4--,the ,principle of       
,general ,relativity3 ,how ,einstein1 to
 a ,degree  ,never ,before ,equalled1   
,isolates the ,external ,reality from   
the  ,observer's ,contribution4 ,by ,e4 
,t4 ,bell1 ,university of ,seattle      
#bah    ,,xii4--,force ,vs4 ,geometry3  
,how ,einstein ,has ,substituted the    
,second  for the ,first in ,connection  
with the ,cause of ,gravitation4 ,by    
,saul  ,dushman1 ,schenectady      #bcj 
,,xiii4--,an ,introduction to           
,relativity3 ,a ,treatment in which the 
,mathematical ,connections of           
,einstein's ,work are ,brought ,out     
,more  ,strongly and ,more ,successfully
 than ,usual in a ,popular ,explanation4
 ,by  ,harold ,t4 ,davis1 ,university of
 ,wisconsin      #bdj    ,,xiv4--,new   

,concepts for ,old3 ,what the ,world    
,looks ,like ,after ,einstein  ,has ,had
 ,his ,way with ,it4 ,by ,john ,g4      
,mc,hardy1 ,commander ,r4 ,n41  ,london 
#bea    ,,xv4--,the ,new ,world3 ,a     
,universe in ,which ,geometry ,takes the
 ,place of  ,physics1 and ,curvature    
that of ,force4 ,by ,george ,frederick  
,hemens1  ,m4,c41 ,b4,sc41 ,london      
#bfe    ,,xvi4--,the ,quest of the      
,absolute3 ,modern ,developments in     
,theoretical  ,physics1 and the ,climax 
,supplied by ,einstein4 ,by ,dr4        
,francis  ,d4 ,murnaghan1 ,johns        
,hopkins ,university1 ,baltimore        
#bgf    ,,xvii4--,the ,physical ,side of
 ,relativity3 ,the ,immediate ,contacts 
between  ,einstein's ,theories and      
,current ,physics and ,astronomy4 ,by   
,professor  ,william ,h4 ,pickering1    
,harvard ,college ,observatory1         
,mandeville1  ,jamaica      #bhg        
,,xviii4--,the ,practical ,significance 
of ,relativity3 ,the ,best ,discussion  
of the ,special ,theory ,among ,all the 

,competing ,essays4 ,by ,prof4 ,henry   
,norris ,russell1 ,princeton ,university
      #cjf    ,,xix4--,einstein's       
,theory of ,relativity3 ,a ,simple      
,explanation of ,his  ,postulates and   
,their ,consequences4 ,by ,t4 ,royds1   
,kodaikanal ,observatory1  ,india       
#cah    ,,xx4--,einstein's ,theory of   
,gravitation3 ,the ,discussion of the   
,general ,theory and ,its ,most         
,important ,application1 from the ,essay
 by  ,prof4 ,w4 ,f4 ,g4 ,swann1         
,university of ,minnesota1 ,minneapolis 
#cbg    ,,xxi4--,the ,equivalence       
,hypothesis3 ,the ,discussion of ,this1 
,with ,its  ,difficulties and the       
,manner in ,which ,einstein ,has        
,resolved ,them1 from  the ,essay by    
,prof4 ,e4 ,n4 da ,c4 ,andrade1         
,ordnance ,college1 ,woolwich1  ,england
      #ccd    ,,xxii4--,the ,general    
,theory3 ,fragments of ,particular      
,merit on ,this  ,phase of the ,subject4
 ,by ,various ,contributors      #cch   
,i4    ,,the ,,einstein #e1jjj ,,prize  

,,how ,,the ,,contest ,,came ,,to ,,be  
,,held1 ,,and ,,some ,,of ,,the         
,,details ,,of ,,its ,,conduct    ,,by  
,,the ,,editor      ,in ,january1 #aiji1
 an anonymous donor who was interested  
in the spread  of correct scientific    
ideas offered through the ,scientific   
,american a  prize of #ejj for the best 
essay explaining1 in simple             
non-technical  language1 that paradise  
of mathematicians and bugaboo of plain  
ordinary folk--the fourth dimension4    
,many essays were submitted in  this    
competition1 and in addition to that of 
the winner some twenty  were adjudged   
worthy of ultimate publication4 ,it was 
felt that the  competition had added    
distinctly to the popular understanding 
of  this significant subject2 that it   
had done much to clear up popular       
misconception of just what the          
mathematician means when he talks  of   
four or even more dimensions2 and that  
it had therefore been as  successful as 
it was unusual in character4    ,in     

,november1 #aiai1 the world was startled
 by the announcement from  ,london that 
examination of the photographs taken    
during the total  solar eclipse of ,may 
#bith had been concluded1 and that      
predictions  based upon the ,einstein   
theories of relativity had been         
verified4 ,in  the reaction from the    
long surfeit of war news an item of this
 sort  was a thoroughly journalistic    
one4 ,long cable dispatches were carried
  in the news columns all over the      
world2 ,einstein and his theories  were 
given a prominent place on the front    
pages day after day2 leading  scientists
 in great number were called upon to    
tell the public through  the reportorial
 medium just what the excitement was all
 about1 just  in what way the classical 
scientific structure had been           
overthrown4    ,instead of being a mere 
nine days' wonder1 the ,einstein        
theories held  their place in the public
 mind4 ,the more serious periodicals    
devoted  space to them4 ,first and last1

 a very notable group of scientific men 
attempted to explain to the general     
reader the scope and content of         
,einstein's system4 ,these efforts1 well
 considered as they were1 could  be no  
more than partially successful on       
account of the very radical  character  
of the revisions which the relativity   
doctrine demands in  our fundamental    
concepts4 ,such revisions cannot be made
 in a day2  the average person has not  
the viewpoint of the mathematician which
  permits a sudden and complete exchange
 of one set of fundamentals  for        
another4 ,but the whole subject had     
caught the popular attention  so        
strongly1 that even complete initial    
failure to discover what it  was all    
about did not discourage the general    
reader from pursuing  the matter with   
determination to come to some           
understanding of what  had happened to  
,newton and ,newtonian mechanics4       
,,the ,,donor ,,and ,,the ,,prize    ,in
 ,may1 #aibj1 ,mr4 ,eugene ,higgins1 an 

,american citizen long resident  in     
,paris1 a liberal patron of the arts and
 sciences1 and a lifelong  friend of the
 ,scientific ,american and its          
proprietors1 suggested that  the success
 of the ,fourth ,dimension ,prize       
,contest of #aiaj had been  so great    
that it might be desirable to offer     
another prize in similar  fashion for   
the best popular essay on the ,einstein 
theories4 ,he stated  that if in the    
opinion of the ,scientific ,american    
these theories  were of sufficient      
importance1 and the probability of      
getting a  good number of meritorious   
essays were sufficiently great1 and the 
public need and desire for enlightenment
 were sufficiently present1  he would   
feel inclined to offer such a prize1    
leaving the conduct  of the contest to  
the ,scientific ,american as in the     
former event4 ,it  was the judgment of  
the editors of the ,scientific ,american
 that all  these provisos should be met 
with an affirmative1 and that ,mr4      

,higgins  accordingly could with        
propriety be encouraged to offer the    
prize4    ,in his preliminary letter    
,mr4 ,higgins had suggested that in view
  of the apparent greater importance of 
the subject to be proposed  for         
discussion by the contestants of #aibj1 
the prize offered should  probably be   
more liberal than in the former         
instance4 ,this view met  with the      
approval of the editors as well2 but    
they were totally  unprepared for the   
receipt1 late in ,june1 of a cablegram  
from  ,mr4 ,higgins stating that he had 
decided to go ahead with the matter1    
and that he was forwarding a draft for  
#e1jjj to represent the amount  of the  
prize4 ,such a sum1 exceeding any award 
open to a professional  man with the    
single exception of the ,nobel ,prize1  
for which he cannot  specifically       
compete1 fairly took the breath of the  
,editors1 and  made it immediately clear
 that the contest would attract the     
widest  attention1 and that it should   

score the most conspicuous success4 ,it 
also made it clear that the handling of 
the contest would be a more  serious    
matter than had been anticipated4    ,in
 spite of the fact that it would not for
 some time be possible to  announce the 
identity of the ,judges1 it was felt    
that the prospective  contestants should
 have every opportunity for extensive   
preparation2  so the contest was        
announced1 and the rules governing it   
printed  as far as they could be        
determined on such short shrift1 in the 
,scientific ,american for ,july #aj1    
#aibj4 ,several points of ambiguity  had
 to be cleared up after this initial    
publication4 ,in particular1  it had    
been ,mr4 ,higgins' suggestion that in  
the very probable event  of the ,judges'
 inability to agree upon the winning    
essay1 the prize  might1 at their       
discretion1 be divided between the      
contributors  of the best two essays4   
,this condition was actually printed in 
the  first announcement1 but the ,post  

,office ,department insisted upon its   
withdrawal1 on the ground that with it  
in force the contestant would  not know 
whether he were competing for #e1jjj or 
for #b1ejj1 and that  this would        
introduce the element of chance which   
alone was necessary1  under the ,federal
 statutes1 to make the contest a        
lottery4 ,so this  provision was        
replaced by one to the effect that in   
the event the  ,judges were not able to 
agree1 the ,einstein ,editor should cast
 the  deciding vote between the essays  
respectively favored by them4    ,the   
announcement attracted the widest       
attention1 and was copied in  newspapers
 and magazines all over the world4      
,inquiries poured in from  all quarters1
 and the ,einstein ,editor found it     
almost impossible to  keep himself      
supplied with proofs of the conditions  
and rules to mail  in response to these 
inquiries4 ,it was immediately clear    
that there  was going to be a large     
number of essays submitted1 and that    

many  distinguished names would be      
listed among the competitors4           
,,the ,,judges    ,in the ,scientific   
,american for ,september #ah1           
announcement was carried  in the        
following words3    ,we are assured with
 complete certainty that the competition
 for the  five-thousand-dollar prize    
will be very keen1 and that many essays 
will be submitted which1 if they bore   
the names of their authors1 would  pass 
anywhere as authoritative statements4   
,the judges will confront a  task of    
extraordinary difficulty in the effort  
to determine which of  these efforts is 
the best2 and we believe the            
difficulties are such  that             
multiplication of judges would merely   
multiply the obstacles to  an agreement4
 ,it is altogether likely that the      
initial impressions  of two or three or 
five judges would incline toward two or 
three  or five essays1 and that any     
final decision would be attainable  only
 after much consultation and discussion4

 ,it seems to us that by  making the    
committee as small as possible while    
still preserving the  necessary feature 
that its decision represent a consensus1
 we shall  simplify both the mental and 
the physical problem of coming to an    
agreement4 ,we believe that the award   
should if possible represent  a         
unanimous decision1 without any minority
 report1 and that such a  requirement is
 far more likely to be met among two men
 than among  three or five4 ,at the same
 time1 the bringing together of two men 
and  the details of general             
administration of their work together   
are far  simpler than if there were     
three or five4 ,so we have finally      
decided  to have but two judges1 and in 
this we have the endorsement of all  the
 competent opinion that we have         
consulted4    ,the gentlemen who have   
consented to act as ,judges are         
,professors  ,leigh ,page and ,edwin    
,plimpton ,adams1 of the departments of 
physics  of ,yale and ,princeton        

,universities1 respectively4 ,both are  
of the  younger generation of physicists
 that has paid special attention to     
those phases of mathematics and physics 
involved in the ,einstein  theories1 and
 both have paid special attention to    
these theories  themselves4 ,we are     
gratified to be able to put forward as  
,judges  two men so eminently qualified 
to act4 ,we feel that we may here       
appropriately quote ,professor ,page1   
who says in his acceptance3  ',as the   
large prize offers a great inducement1  
,i had thought of  entering the contest4
 ,however ,i realize that not many      
people in  this country have made a     
considerable study of ,einstein's       
theory1  and if all who have should     
enter the contest1 it would be difficult
  to secure suitable ,judges4' ,without 
any desire to put the gentleman  in the 
position of pleading for himself1 we    
think this suggests very  well the      
extent to which the ,scientific         
,american1 the contestants1  and the    

public at large1 are indebted to        
,professors ,page and ,adams  for their 
willingness to serve in the difficult   
capacity of ,judges4    ,it might       
appropriately have been added to this   
announcement that it  was altogether to 
the credit of science and the scientific
 spirit  that the first two gentlemen   
approached with the invitation to act   
as ,judges were willing to forego their 
prospects as contestants in  order thus 
to contribute to the success of the     
contest4          ,,three ,,thousand    
,,words    ,of the conditions1 the one  
which evoked most comment was that      
stating  the word limit4 ,this limit was
 decided upon after the most careful    
discussion of the possibilities of the  
situation4 ,it was not imagined  for a  
moment that any contestant would succeed
 in getting within  #c1jjj words a      
complete discussion of all aspects of   
the ,special  and the ,general ,theories
 of ,relativity4 ,it was however felt   
that  for popular reading a single essay

 should not be much if any longer  than 
this4 ,moreover1 ,i will say quite      
frankly that we should never  have      
encouraged ,mr4 ,higgins to offer such a
 prize if we had supposed  that the     
winning essay was the only thing of     
value that would come  from the contest1
 or if we had not expected to find in   
many of the  other essays material which
 would be altogether deserving of the   
light4 ,from the beginning we had in    
view the present volume1 and the  severe
 restriction in length was deliberately 
imposed for the purpose  of forcing     
every contestant to stick to what he    
considered the most  significant        
viewpoints1 and to give his best skill  
to displaying the  theories of ,einstein
 to the utmost advantage from these     
viewpoints4 ,we  felt that divergent    
viewpoints would be more advantageously 
treated in  this manner than if we gave 
each contestant enough space to discuss 
the subject from all sides2 and that the
 award of the prize to the  essay which1

 among other requirements1 seemed to the
 ,judges to embody  the best choice of  
material1 would greatly simplify the    
working of the  contest without         
effecting any injustice against those   
contestants who  displayed with equal   
skill less happily chosen material4     
,perhaps on  this point ,i may again    
quote with profit the editorial page of 
the  ,scientific ,american3    ,an essay
 of three thousand words is not long    
enough to lose a  reader more than once2
 if it does lose him it is a failure1   
and if  it doesn't it is a competitor   
that will go into the final elimination 
trials for the prize4 ,if we can        
present1 as a result of the contest1    
six or a dozen essays of this length    
that will not lose the lay reader  at   
all1 we shall have produced something   
amply worth the expenditure  of ,mr4    
,higgins' money and our time4 ,for such 
a number of essays of  such character   
will of necessity present many different
 aspects of  the ,einstein theories1 and

 in many different ways1 and in doing so
  will contribute greatly to the popular
 enlightenment4    ,really the          
significant part of what has already    
appeared is  not the part that is       
intelligible1 but rather the part that1 
being unintelligible1 casts the shadow  
of doubt and suspicion  on the whole4   
,the successful competitor for the prize
 and  his close contestants will have   
written essays that1 without  any claim 
to completeness1 will emphasize what    
seems to each  author the big           
outstanding feature2 and every one of   
them will be  intelligible4 ,together   
they will in all probability be         
reasonably  complete1 and will retain   
the individual characteristic of        
intelligibility4 ,they will approach the
 various parts of the field  from       
various directions--we could fill this  
page with suggestions as  to how the one
 item of the four-dimensional character 
of ,einstein's  time-space might be set 
forth for the general reader4 ,and when 

a man  must say in three thousand words 
as much as he can of what eminent       
scientists have said in whole           
volumes--well1 the result in some cases 
will be sheer failure1 and in others a  
product of the first water4 ,the  best  
of the essays will shine through        
intelligent selection of what  is to be 
said1 and brilliant success in saying   
it4 ,it is to get a  group of essays of 
this character1 not to get the single   
essay which  will earn the palm1 that   
the prize is offered4          ,,the    
,,competing ,,essays    ,at all times   
after the first announcement the        
,einstein ,editor had a  heavy          
correspondence2 but the first real      
evidence that the contest was  under way
 came with the arrival of the first     
essay1 which wandered  into our office  
in the middle of ,september4 ,about a   
week later they  began to filter in at  
the rate of one or two per day--mostly  
from  foreign contestants who were      
taking no chances on the mails4 ,heavy  

returns did not commence until about ten
 days before the closing  date4 ,the    
great avalanche1 however1 was reserved  
for the morning of  ,monday1 ,november  
#ast4 ,here we had the benefit of three 
days' mail2  there were about #abj      
essays4 ,among those which were thrown  
out on  the ground of lateness the      
honors should no doubt go to the man who
  mailed his offering in ,the ,hague on 
,october #cast4    ,essays were received
 in greater quantity from ,germany than 
from any  other foreign country1        
doubtless because of the staggering     
value of  #e1jjj when converted into    
marks at late #aibj rates4 ,england     
stood  next on the list2 and one or more
 essays were received from ,austria1    
,czechoslovakia1 ,jugoslavia1 ,france1  
,switzerland1 the ,netherlands1         
,denmark1 ,italy1 ,chile1 ,cuba1        
,mexico1 ,india1 ,jamaica1 ,south       
,africa  and the ,fiji ,islands4        
,canada1 of course1 contributed her fair
 share2  and few of our own states were 

missing on the roll-call4    ,the       
general level of ,english composition   
among the essays from                   
non-,english-speaking sources was about 
what might have been expected4 ,a  man  
may have a thorough utilitarian         
knowledge of a foreign tongue1  but when
 he attempts intensive literary         
competition with a man who was  brought 
up in that tongue he is at a            
disadvantage4 ,we read ,french and      
,german with ease and ,spanish and      
,italian without too much difficulty1   
ourselves2 we should never undertake    
serious writing in any of these         
languages4 ,not many of the foreign     
contributions1 of course1 were  as      
ludicrous as the one we quote to some   
extent in our concluding  chapter1 but  
most of them were distinctly below par  
as literary  compositions4 ,drs4 ,de    
,sitter and ,schlick were the notable   
exceptions  to this2 both showed the    
ability to compete on a footing of      
absolute  equality with the best of the 

native product4    ,we dare say it was a
 foregone conclusion that many essays   
should have  been over the limit1 and   
that a few should have been over it to  
the  point of absurdity4 ,the winning   
essay contains #b1iai words1 plus or    
minus a reasonable allowance for error  
in counting2 that it should  come so far
 from being on the ragged edge should be
 sufficient answer  to those who        
protested against the severity of the   
limitation4 ,one  inquirer1 by the way1 
wanted to know if #c1jjj words was not a
 misprint  for #cj1jjj4 ,another        
contestant suggested that instead of    
disqualifying  any essay that was over  
the line1 we amputate the superfluous   
words  at the end4 ,this was a plausible
 enough suggestion1 since any essay     
able to compete after such amputation   
must necessarily have been one  of      
extreme worth2 but fortunately we did   
not have to decide whether we  should   
follow the scheme4 ,perhaps twenty of   
the essays submitted were  so seriously 

in excess of the limit that it was not  
even necessary  to count their words in 
detail2 most of these offenders ran to  
#c1ejj  words or thereabouts1 and one--a
 good one1 too1 from which we use  a    
good deal of material in this           
volume--actually had #d1gjj4 ,on the    
other extreme were a few competitors who
 seemed to think that the  shortest     
essay was necessarily the best1 and who 
tried to dismiss  the subject with #ejj 
or #a1jjj words4    ,by a curious trick 
of chance there were submitted in       
competition  for the prize exactly #cjj 
essays4 ,of course a few of these did   
not  require serious consideration--this
 is inevitable in a contest of such     
magnitude4 ,but after excluding all the 
essays that were admittedly  not about  
the ,einstein theories at all1 and all  
those whose ,english  was so execrable  
as to make them quite out of the        
question1 and all  those which took the 
subject so lightly as not to write      
reasonably  close to the limit of #c1jjj

 words1 and all those which were given  
over to explanation of the manner in    
which ,einstein's theories verify  those
 of the writer1 and all those in which  
the writer attempted  to substitute his 
own cosmic scheme for ,einstein's--after
 all this1  there remained some #bge    
essays which were serious efforts to    
explain  in simple terms the nature and 
content and consequences of ,special    
and ,general ,relativity4               
,,looking ,,for ,,the ,,winner    ,the  
,einstein ,editor was in sufficiently   
close touch with the details  of the    
adjudication of the essays to have every
 realization of  the difficulty of this 
work4 ,the caliber of the essays        
submitted  was on the whole high4 ,there
 were many which would have been well   
worthy of the prize in the absence of   
others that were distinctly             
better--many which it was not possible  
to eliminate on the ground of  specific 
faults1 and which could only be adjudged
 not the best by  detailed comparison   

with specific other essays4 ,it was this
 detailed  comparison which took time1  
and which so delayed the award that we  
were not able to publish the winning    
essay any sooner than ,february  #eth4  
,especially difficult was this process  
of elimination after the  number of     
surviving essays had been reduced to    
twenty or less4 ,the  advantages of plan
 possessed by one essay had to be       
weighed against  those of execution     
exhibited in another4 ,a certain essay  
had to be  critically compared with     
another so like it in plan that the two 
might  have been written from a common  
outline1 and at the same time with  a   
third as unlike it in scope and content 
as day and night4 ,and all  the time    
there was present in the background the 
consciousness that  a prize of #e1jjj   
hung upon the decision to be reached4   
,for anyone  who regards this as an easy
 task we have no worse wish than that he
  may some day have to attack a similar 
one4    ,we had anticipated that the    

bulk of the superior essays would be    
among those received during the last day
 or two of the contest2  for we felt    
that the men best equipped to attack the
 subject would  be the most impressed   
with its seriousness4 ,here we were     
quite off  the track4 ,the seventeen    
essays which withstood most stubbornly  
the  ,judges' efforts at elimination    
were1 in order of receipt1 numbers #h1  
#ah1 #bh1 #dj1 #da1 #dc1 #ib1 #ie1 #ig1 
#acj1 #aha1 #aid1 #aih1 #bbc1 #bfg1     
#bgj1 #bge3  a fairly even distribution4
 ,the winner was the #ibnd essay        
received4    ,the ,judges held their    
final meeting in the editorial office on
  ,january #ah1 #aiba4 ,the four essays 
which were before the committee  at the 
start of the session were speedily cut  
to three1 and then  to two2 and after an
 all-day session the ,judges found      
themselves  conscientiously able to     
agree on one of these as the best4 ,this
  unanimity was especially gratifying1  
the more so since it by no  means was to

 be confidently expected1 on a priori   
grounds1 that it  would be possible of  
attainment4 ,even the ,einstein ,editor1
 who might  have been called upon for a 
final decision but wasn't1 can hardly be
  classed as a dissenter2 for with some 
slight mental reservations in  favor of 
the essay by ,mr4 ,francis which did not
 enter the ,judges'  final discussion at
 all1 and which he rather suspects      
appeals more  to his personal taste than
 to his soundest judgment1 he is        
entirely  in accord with the verdict    
rendered4    ,the fact that the prize   
went to ,england was no surprise to     
those  acquainted with the history of   
,einstein's theories4 ,the ,special     
,theory1  promulgated fifteen years ago1
 received its fair share of attention   
from mathematicians all over the world1 
and is doubtless as well  known and as  
fully appreciated here as elsewhere4    
,but it has never  been elevated to a   
position of any great importance in     
mathematical  theory1 simply because of 

itself1 in the absence of its extension 
to the general case1 it deserves little 
importance4 ,it is merely an            
interesting bit of abstract speculation4
    ,the ,general ,theory was put out by
 ,einstein in finished form during  the 
war4 ,owing to the scientific           
moratorium1 his paper1 and hence a      
clear understanding of the new methods  
and results and of the sweeping         
consequences if the ,general ,theory    
should prevail1 did not attain  general 
circulation outside ,germany until some 
time in #aiah or even  later4 ,had it   
not been for ,eddington it is doubtful  
that the ,british  astronomers would    
have realized that the eclipse          
expeditions were of  particular         
consequence4 ,therefore at the time of  
these expeditions1  and even as late as 
the ,november announcement of the       
findings1 the  general body of          
scientific men in ,america had not      
adequately realized  the immense        
distinction between the ,special and the

 ,general ,theories1  had not adequately
 appreciated that the latter led to     
distinctive  consequences of any import1
 and we fear in many cases had not even 
realized explicitly that the deflection 
of light and the behavior of  ,mercury  
were matters strictly of the ,general   
and in no sense of the  ,special        
,theory4 ,certainly when the ,american  
newspapers were searching  frantically  
for somebody to interpret to their      
public the great stir  made by the      
,british announcement that ,einstein's  
predictions had  been verified1 they    
found no one to do this decently2 nor   
were our  magazines much more successful
 in spite of the greater time they had  
to devote to the search4 ,in a word1    
there is not the slightest room  for    
doubt that ,american science was in     
large measure caught asleep  at the     
switch--perhaps for no reason within its
 control2 and that  ,american writers   
were in no such favorable case to write 
convincingly  on the subject as were    

their ,british and continental          
contemporaries4    ,so it was quite in  
accord with what might have been        
expected to find1  on opening the       
identifying envelopes1 that not alone   
the winning essay1  but its two most    
immediate rivals1 come from members of  
that school of  ,british thought which  
had been in contact with the ,einstein  
theories  in their entirety for two     
years longer than the average ,american 
of  equal competence4 ,this riper       
familiarity with the subject was bound  
to yield riper fruit4 ,indeed1 had it   
not been for the handicap of  writing in
 a strange language1 it is reasonable to
 assume that the  scientists of ,germany
 would have made a showing superior to  
that of  either ,americans or           
,british--and for the same reason that  
,britain  showed to better advantage    
than ,america4          ,,the ,,winner  
,,of ,,the ,,prize    ,mr4 ,bolton1 the 
winner of the big prize1 we suppose may 
fairly be  referred to as unknown in a  

strict scientific sense4 ,indeed1 at    
the time of the publication of his essay
 in the ,scientific ,american  nothing  
could be learned about him on the       
,american side of the water  beyond the 
bare facts that he was not a young man1 
and that he had  for a good many years  
occupied a position of rank in the      
,british  ,patent ,office4 7,it will be 
recalled that ,einstein himself was in  
the  ,swiss ,patent ,office for some    
time47 ,in response to the request of   
the  ,scientific ,american for a brief  
biographical sketch that would serve  to
 introduce him better to our readers1   
,mr4 ,bolton supplied such a  concise   
and apparently such a characteristic    
statement that we can  do no better than
 quote it verbatim4    ,i was born in   
,dublin in #ahfj1 but ,i have lived in  
,england since  #ahfi4 ,my family       
belonged to the landed gentry class1 but
 ,i owe nothing  to wealth or position4 
,i was in fact put through school and   
college  on an income which a workman   

would despise nowadays4 ,after attending
  sundry small schools1 ,i entered      
,clifton ,college in #ahgc4 ,my career  
there was checkered1 but it ended well4 
,i was always fairly good at  natural   
science and very fond of all sorts of   
mechanical things4 ,i  was an honest    
worker but no use at classics1 and as ,i
 did practically  nothing else for the  
first four years at ,clifton1 ,i came to
 consider  myself something of a dunce4 
,but a big public school is a little    
world4 ,everyone gets an opportunity1   
often seemingly by accident1  and it is 
up to him to take it4 ,mine did not come
 till ,i was nearly  #ag4 ,as ,i was    
intended for the engineering profession1
 ,i was sent to  the military side of   
the school in order to learn some       
mathematics1  at which subject ,i was   
then considered very weak4 ,this was    
certainly  true1 as at that time ,i     
barely knew how to solve a quadratic1 ,i
 was  only about halfway through the    
third book of ,euclid1 and ,i knew no   

trigonometry4 ,but the teaching was     
inspiring1 and ,i took readily  to      
mathematics4 ,one day it came out that  
,i had been making quite a  good start  
with the differential calculus on my own
 without telling  anybody4 ,after that  
all was well4 ,i left ,clifton in #ahhj 
with a ,school  ,exhibition and a       
mathematical scholarship at ,clare      
,college1 ,cambridge4    ,after taking  
my degree in #ahhc as a ,wrangler1 ,i   
taught science and  mathematics at      
,wellington ,college1 but ,i was        
attracted by what ,i had  heard of the  
,patent ,office and ,i entered it       
through open competition in  #ahhe4     
,during my official career ,i have been 
one of the ,comptroller's  private      
secretaries and ,i am now a ,senior     
,examiner4 ,during the  war ,i was      
attached to the ,inventions ,department 
of the ,ministry of  ,munitions1 where  
my work related mainly to anti-aircraft 
gunnery4 ,i  have contributed1 and am   
still contributing to official          

publications  on the subject4    ,i have
 written a fair number of essays on     
various subjects1 even  on literature1  
but my only extra-official publications 
relate to  stereoscopic photography4 ,i 
read a paper on this subject before the 
,royal ,photographic ,society in #aijc  
which was favorably noticed by  ,dr4 von
 ,rohr of ,messrs4 ,zeiss of ,jena4 ,i  
have also written in the  ,amateur      
,photographer4    ,i have been fairly   
successful at athletics1 and ,i am a    
member of  the ,leander ,club4    ,that 
,mr4 ,bolton did not take the prize     
through default of serious  competition 
should be plain to any reader who       
examines the text from  competing essays
 which is to be found in this volume4   
,the reference  list of these           
competitors1 too1 supplemented by the   
names that appear  at the heads of      
complete essays1 shows a notable array  
of distinguished  personalities1 and ,i 
could mention perhaps a dozen more very 
well  known men of science whose        

excellent essays have seemed a trifle   
too  advanced for our immediate use1 but
 to whom ,i am under a good deal  of    
obligation for some of the ideas which  
,i have attempted to clothe  in my own  
language4    ,before leaving the        
subject1 we wish to say here a word of  
appreciation  for the manner in which   
the ,judges have discharged their       
duties4 ,the  reader will have          
difficulty in realizing what it means to
 read such  a number of essays on such a
 subject4 ,we were fortunate beyond all 
expectation in finding ,judges who      
combined a thorough scientific  grasp of
 the mathematical and physical and      
philosophical aspects  of the matter    
with an extremely human viewpoint which 
precluded any  possibility of an award  
to an essay that was not properly a     
popular  discussion1 and with a         
willingness to go to meet each other's  
opinions  that is rare1 even among those
 with less ground for confidence in     
their own views than is possessed by    

,drs4 ,page and ,adams4                 
,,ii4    ,,the ,,world--,,and ,,us      
,an ,introductory ,discussion of the    
,philosophy of ,relativity1 and of  the 
,mechanism of ,our ,contact with ,time  
and ,space    ,,by ,,various            
,,contributors ,,and ,,the ,,editor     
,from a time beyond the dawn of history1
 mankind has been seeking  to explain   
the universe4 ,at first the effort did  
not concern itself  further probably    
than to make a supposition as to what   
were the causes  of the various         
phenomena presented to the senses4 ,as  
knowledge  increased1 first by          
observation and later by experiment     
also1  the ideas as to these causes     
passed progressively through three      
stages--the theological 7the causes were
 thought to be spirits or  gods72 the   
metaphysical 7the causes were thought in
 this secondary  or intermediate stage  
to be some inherent1 animating1         
energizing  principles72 and the        
scientific 7the causes were finally     

thought of  as simply mechanical1       
chemical1 and magneto-electrical        
attractions  and repulsions1 qualities  
or characteristics of matter itself1 or 
of  the thing of which matter is itself 
composed47    ,with increase of         
knowledge1 and along with the inquiry as
 to the nature  of causes1 there arose  
an inquiry also as to what reality was4 
,what was  the essential nature of the  
stuff of which the universe was made1   
what  was matter1 what were things in   
themselves1 what were the noumena 7the  
realities71 lying back of the phenomena 
7the appearances78 ,gradually  ideas    
explaining motion1 force1 and energy    
were developed4 ,at the  same time      
inquiry was made as to the nature of    
man1 the working  of his mind1 the      
nature of thought1 the relation of his  
concepts  7ideas7 to his perceptions    
7knowledge gained through the sense7    
and the relations of both to the noumena
 7realities74#bhc    ,the general       
direction taken by this inquiry has been

 that of a  conflict between two schools
 of thought which we may characterize   
as those of absolutism and of           
relativism499 ,the ancient ,greek       
philosophers believed that they could   
tap a source of knowledge  pure and     
absolute by sitting down in a chair and 
reasoning about the  nature of time and 
space1 and the mechanism of the physical
 world4#bba  ,they maintained that the  
mind holds in its own right certain     
concepts  than which nothing is more    
fundamental4 ,they considered it proper 
to  conceive of time and space and      
matter and the other things presented   
to their senses by the world as having a
 real existence in the mind1  regardless
 of whether any external reality could  
be identified  with the concept as      
ultimately put forth4 ,they could even  
dispute  with significance the qualities
 which were to be ascribed to this      
abstract conceptual time and space and  
matter4 ,all this was done  without     
reference to the external reality1 often

 in defiance of that  reality4 ,the mind
 could picture the world as it ought to 
be2 if the  recalcitrant facts refused  
to fit into the picture1 so much the    
worse  for them4 ,we all have heard the 
tale of how generation after generation 
of ,greek philosophers disputed         
learnedly why and how it was that a live
  fish could be added to a brimming pail
 of water without raising the  level of 
the fluid or increasing the weight2     
until one day some common  person       
conceived the troublesome idea of trying
 it out experimentally  to learn whether
 it were so--and found that it was not4 
,true or false1  the anecdote admirably 
illustrates the subordinate place which 
the  externals held in the absolutist   
system of ,greek thought499    ,under   
this system a single observer is        
competent to examine a  single          
phenomenon1 and to write down the       
absolute law of nature  by referring the
 results to his innate ideas of absolute
 qualities  and states4 ,the root of the

 word absolute signifies taking away1   
and in its philosophical sense the word 
implies the ability of the  mind to     
subtract away the properties or         
qualities from things1 and  to consider 
these abstract qualities detached from  
the things2 for  example1 to take away  
the coldness from ice1 and to consider  
pure or  abstract coldness apart from   
anything that is cold2 or to take away  
motion from a moving body1 and to       
consider pure motion apart from         
anything that moves4 ,this assumed power
 is based upon the ,socratic  theory of 
innate ideas4 ,according to this theory 
the mind is endowed  by nature with the 
absolute ideas of hardness1 coldness1   
roundness1  equality1 motion1 and all   
other absolute qualities and states1 and
  so does not have to learn them4 ,thus 
a ,socratic philosopher could  discuss  
pure or absolute being1 absolute space  
and absolute time4#aba                  
,,getting ,,away ,,from ,,the ,,greek   
,,ideas    ,this ,greek mode of thought 

persisted into the late ,middle ,ages1  
at which time it was still altogether in
 order to dispose of a  troublesome fact
 of the external world by quoting       
,aristotle against  it4 ,during the     
,renaissance1 which intellectually at   
least marks the  transition from ancient
 to modern1 there came into being       
another  type of absolutism1 equally    
extreme1 equally arbitrary1 equally     
unjustified4 ,the revolt against the    
mental slavery to ,greek ideas  carried 
the pendulum too far to the other side1 
and early modern  science as a          
consequence is disfigured by what we    
must now recognize  as gross            
materialism4 ,the human mind was        
relegated to the position  of a mere    
innocent bystander4 ,the external       
reality was everything1  and aside from 
his function as a recorder the observer 
did not in the  least matter4 ,the whole
 aim of science was to isolate and      
classify the  elusive external fact4    
,the rle of the observer was in every   

possible  way minimized4 ,it was of     
course his duty to get the facts right1 
but so far as any contribution to these 
was concerned he did not  count--he was 
definitely disqualified4 ,he really     
played the part of  an intruder2 from   
his position outside the phenomena he   
was searching  for the absolute truth   
about these phenomena4 ,the only        
difference  between his viewpoint and   
that of ,aristotle was that the latter  
looked  entirely inside himself for the 
elusive truth1 while the classical      
scientist1 as we call him now1 looked   
for it entirely outside himself4    ,let
 me illustrate the difference between   
the two viewpoints which  ,i have       
discussed1 and the third one which ,i am
 about to outline1  by another concrete 
instance4 ,the ,greeks1 and the         
medievals as well1  were fond of        
discussing a question which embodies the
 whole of what  ,i have been saying4    
,this question involved1 on the part of 
one who  attempted to answer it1 a      

choice between the observer and the     
external  world as the seat of reality4 
,it was put in many forms2 a familiar   
one is the following3 ,if the wind blew 
down a great tree at a time  and place  
where there was no conscious being to   
hear1 would there  be any noise8 ,the   
,greek had to answer this question in   
the negative  because to him the noise  
was entirely a phenomenon of the        
listener4 ,the  classical scientist had 
to answer it in the affirmative because 
to him  the noise was entirely a        
phenomenon of the tree and the air and  
the  ground4 ,today we answer it in the 
negative1 but for a very different      
reason from that which swayed the       
,greek4 ,we believe that the noise is  a
 joint phenomenon of the observer and   
the externals1 so that in the  absence  
of either it must fail to take          
existence4 ,we believe there are  sound 
waves produced1 and all that2 but what  
of it8 ,there is no noise  in the       
presence of the falling tree and the    

absence of the observer1  any more than 
there would be in the presence of the   
observer and the  absence of the tree   
and the wind2 the noise1 a joint        
phenomenon of  the observer and the     
externals1 exists only in their joint   
presence4          ,,relativism ,,and   
,,reality    ,this is the viewpoint of  
relativism4 ,the statue is golden for   
one observer and silver to the other4   
,the sun is rising here and  setting in 
another part of the world4 ,it is       
raining here and clear in  ,chicago4    
,the observer in ,delft hears the       
bombardment of ,antwerp and  the        
observer in ,london does not4 ,if they  
were to be consistent1 both  the ,greek 
and the medieval-modern absolutist would
 have to dispute  whether the statue    
were really golden or silver1 whether   
the sun  were really rising or setting1 
whether the weather were really  fair or
 foul1 whether the bombardment were     
really accompanied by  loud noises or   
not2 and on each of these questions they

 would have to  come to an agreement or 
confess their methods inadequate4 ,but  
to the  relativist the answer is        
simple--whether this or that be true    
depends  upon the observer4 ,in simple  
cases we understand this full well1 as  
we have always realized it4 ,in less    
simple cases we recognize it less       
easily or not at all1 so that some of   
our thought is absolutist in  its       
tendencies while the rest is            
relativistic4 ,einstein is the first    
ever to realize this fully--or if not   
this1 then the first ever to  realize it
 so fully as to be moved toward a       
studied effort to free  human thought   
from the mixture of relativism and      
absolutism and make  it consistently the
 one or the other4    ,this brings it   
about that the observed fact occupies a 
position of  unexpected significance4   
,for when we discuss matters of physical
  science under a strictly relativistic 
philosophy1 we must put away as         
metaphysical everything that smacks of a

 reality partly concealed  behind our   
observations4 ,we must focus attention  
upon the reports  of our senses and of  
the instruments that supplement them4   
,these  observations1 which join our    
perceptions to their external objects1  
afford us our only objective            
manifestations2 them we must accept as  
final--subject always to such correction
 as more refined observations  may      
suggest4 ,the question whether a true   
length or area or mass  or velocity or  
duration or temperature exists back of  
the numerical  determination1 or in the 
presence of a determination that is     
subject  to correction1 or in the       
absence of any determination at all1 is 
a  metaphysical one and one that the    
physicist must not ask4 ,length1  area1 
mass1 velocity1 duration1               
temperature--none of these has any      
meaning other than the number obtained  
by measurement499 ,if several  different
 determinations are checked over and no 
error can be found  in any of them1 the 

fault must lie not with the observers   
but with  the object1 which we must     
conclude presents different values to   
different observers4#cc    ,we are after
 all accustomed to this viewpoint2 we do
 not demand that  ,pittsburgh shall     
present the same distance from ,new     
,york and from  ,philadelphia1 or that  
the ,new ,yorker and the ,philadelphian 
come to  any agreement as to the real   
distance of ,pittsburgh4 ,the distance  
of ,pittsburgh depends upon the position
 of the observer4 ,nor do we  demand    
that the man who locates the magnetic   
pole in one spot in #aijj  and in       
another in #aiba come to a decision as  
to where it really  is2 we accept his   
statement that its position depends upon
 the time  of the observation4    ,what 
this really means is that the distance  
to ,pittsburgh and the  position of the 
magnetic pole are joint properties of   
the observer  and the                   
observed--relations between them1 as we 
might put it4 ,this  is obvious enough  

in the case of the distance of          
,pittsburgh2 it is  hardly so obvious in
 the case of the position of the        
magnetic pole1  varying with the lapse  
of time4 ,but if we reflect that the    
observation  of #aijj and that of #aiba 
were both valid1 and both represented   
the  true position of the pole for the  
observer of the date in question1 we    
must see that this is the only          
explanation that shows us the way out4  
,i do not wish to speak too definitely  
of the ,einstein theories in  these     
introductory remarks1 and so shall      
refrain from mentioning  explicitly in  
this place the situation which they     
bring up and upon  which what ,i have   
just said has direct bearing4 ,it will  
be recognized  when it arises4 ,what    
must be pointed out here1 however1 is   
that we  are putting the thing which the
 scientist calls the observed value  on 
a footing of vastly greater consequence 
than we should have been  willing       
offhand to concede to it4 ,so far as any

 single observer is  concerned1 his own 
best observed values are themselves the 
external  world2 he cannot properly go  
behind the conditions surrounding  his  
observations and speak of a real        
external world beyond these             
observations4 ,any world which he may   
think of as so existing is purely  a    
conceptual world1 one which for some    
reason he infers to exist  behind the   
deceptive observations4 ,provided he    
makes this reservation  he is quite     
privileged to speculate about this      
concealed world1 to  bestow upon it any 
characteristics that he pleases2 but it 
can have  no real existence for him     
until he becomes able to observe it4    
,the  only reality he knows is the one  
he can directly observe4          ,,laws
 ,,of ,,nature    ,the observations     
which we have been discussing1 and which
 we have  been trying to endow with     
characteristics of reality which they   
are  frequently not realized to possess1
 are the raw material of physical       

science4 ,the finished product is the   
result of bringing together a  large    
number of these observations499 ,the    
whole underlying thought  behind the    
making of observations1 in fact1 is to  
correlate as many  as possible of them1 
to obtain some generalization1 and      
finally to  express this in some simple 
mathematical form4 ,this formulation is 
then called a law of nature4#ce    ,much
 confusion exists because of a          
misunderstanding in the lay mind  of    
what is meant by a law of nature4 ,it is
 perhaps not a well chosen  term4 ,one  
is accustomed to associate the word law 
with the idea of  necessity or          
compulsion4 ,in the realm of nature the 
term carries  no such meaning4 ,the laws
 of nature are man's imperfect attempts 
to  explain natural phenomena2 they are 
not inherent in matter and the          
universe1 not an iron bar of necessity  
running through worlds1 systems  and    
suns4 ,laws of nature are little more   
than working hypotheses1  subject to    

change or alteration or enlargement or  
even abandonment1  as man's vision      
widens and deepens4 ,no sanctity        
attaches to them1  and if any one1 or   
all1 of them fail to account for any    
part1 or all1  of the phenomena of the  
universe1 then it or they must be       
supplemented  or abandoned4#ajb    ,the 
test of one of these laws is that it can
 be shown to include  all the related   
phenomena hitherto known and that it    
enables us to  predict new phenomena    
which can then be verified4 ,if new     
facts are  discovered that are not in   
agreement with one of these generalized 
statements1 the assumptions on which the
 latter is based are examined1  those   
which are not in accordance with the new
 facts are given up1  and the statement 
is modified so as to include the new    
facts4#aj  ,and if one remembers that   
the laws of physics were formerly based 
on a range of observations much narrower
 than at present available1  it seems   
natural that in the light of this       

widening knowledge one law  or another  
may be seen to be narrow and            
insufficient4 ,new theories  and laws do
 not necessarily disprove old ones1 but 
explain certain  discrepancies in them  
and penetrate more deeply into their    
underlying  principles1 thereby         
broadening our ideas of the universe4   
,to follow the  new reasoning we must   
rid ourselves of the prejudice behind   
the old1  not because it is wrong but   
because it is insufficient4 ,the        
universe  will not be distorted to fit  
our rules1 but will teach us the rules  
of existence4#abe    ,always1 however1  
we must guard against the too easy error
 of  attributing to these rules anything
 like absolute truth499 ,the  modern    
scientist has attained a very           
business-like point of view  toward his 
laws of nature4 ,to him a law is        
fundamentally nothing  but a short-hand 
way of expressing the results of a large
 number of  experiments in a single     
statement4 ,and it is important to      

remember that  this mere shortening of  
the description of a lot of diverse     
occurrences  is by no means any real    
explanation of how and why they         
happened4 ,in  other words1 the aim of  
science is not ultimately to explain but
 only  to discover the relations that   
hold good among physical quantities and 
to embody all these relations in as few 
and as simple physical laws  as         
possible4#bba ,this is inherently the   
method of relativism499  ,under it a set
 of phenomena is observed4 ,there are   
two or many  observers1 and they write  
down their several findings4 ,these are 
reviewed by a final observer or judge1  
who strains out the bias due  to the    
different viewpoints of the original    
observers4 ,he then writes  down1 not   
any absolute law of nature governing the
 observed phenomena1  but a law as      
general as possible expressing their    
interrelations4#aba  ,and through this  
procedure modern science and philosophy 
reveal  with increasing emphasis that we

 superimpose our human qualities  on    
external nature to such an extent       
that#ajf we have at once the  strongest 
practical justification1 in addition to 
the arguments  of reason1 for our       
insistence that the contact between     
objective  and subjective represented by
 the observation is the only thing      
which we shall ever be able to recognize
 as real4 ,we may indulge in  abstract  
metaphysical speculation to our heart's 
content1 if we be  metaphysically       
inclined2 we may not attempt to impose  
the dicta of  metaphysics upon the      
physical scientist499                   
,,concepts ,,and ,,realities    ,from   
the inquiry and criticism which have    
gone on for centuries has  emerged the  
following present-day attitude of mind  
toward the sum  total of our knowledge4 
,the conceptual universe in our minds in
 some  mysterious way parallels the real
 universe1 but is totally unlike  it4   
,our conceptions 7ideas7 of matter1     
molecules1 atoms1 corpuscles1           

electrons1 the ether1 motion1 force1    
energy1 space1 and time stand  in the   
same or similar relation to reality as  
the x's and y's of the  mathematician do
 to the entities of his problem4        
,matter1 molecules1  atoms1 corpuscles1 
electrons1 the ether1 motion1 force1    
energy1 space1  and time do not exist   
actually and really as we conceive them1
 nor do  they have actually and really  
the qualities and characteristics with  
which we endow them4 ,the concepts are  
simply representations of things        
outside ourselves2 things which1 while  
real1 have an essential nature  not     
known to us4 ,matter1 molecules1 atoms1 
corpuscles1 electrons1  the ether1      
motion1 force1 energy1 space and time   
are merely devices1  symbols1 which     
enable us to reason about reality4 ,they
 are parts of  a conceptual mechanism in
 our minds which operates1 or enables   
our  minds to operate1 in the same      
sequence of events as the sequence of   
phenomena in the external universe1 so  

that when we perceive by our  senses a  
group of phenomena in the external      
universe1 we can reason  out what result
 will flow from the interaction of the  
realities  involved1 and thus predict   
what the situation will be at a given   
stage in the sequence4    ,but while our
 conceptual universe has thus a         
mechanical aspect1  we do not regard the
 real universe as mechanical in its     
nature4#bhc  ,this may be illustrated by
 a little story4 ,entering his friend's 
house1 a gentleman is seized unawares   
from behind4 ,he turns his  head but    
sees nothing4 ,his hat and coat are     
removed and deposited  in their proper  
places by some invisible agent1 seats   
and tables and  refreshments appear in  
due time where they are required1 all   
without  any apparent cause4 ,the       
visitor shivers with horror and asks his
  host for an explanation4 ,he is then  
told that the ideas order and           
regularity are at work1 and that it is  
they who acquit themselves so  well of  

their tasks4 ,these ideas cannot be seen
 nor felt nor seized  nor weighed2 they 
reveal their existence only by their    
thoughtful  care for the welfare of     
mankind4 ,i think the guest1 coming     
home1  will relate that his friend's    
house is haunted4 ,the ghosts may be    
kind1 benevolent1 even useful2 yet      
ghosts they are4 ,now in ,newtonian     
mechanics1 absolute space and absolute  
time and force and inertia and  all the 
other apparatus1 altogether             
imperceptible1 appearing only at  the   
proper time to make possible a proper   
building up of the theory1  play the    
same mysterious part as the ideas order 
and regularity  in my story4 ,classical 
mechanics is haunted4#aaf    ,as a      
matter of fact1 we realize this and do  
not allow ourselves to  be imposed upon 
with regard to the true nature of these 
agencies499  ,we use a mechanistic      
terminology and a mechanistic mode of   
reasoning  only because we have found by
 experience that they facilitate our    

reasoning4 ,they are the tools which we 
find produce results4 ,they  are adapted
 to our minds1 but perhaps it would be  
better to say that  our minds are so    
constructed as to render our conceptual 
universe  necessarily mechanical in its 
aspect in order that our minds may      
reason  at all4 ,two things antithetic  
are involved--subject 7our perceiving   
mind which builds up concepts7 and      
object 7the external reality72 and      
having neither complete nor absolute    
knowledge of either1 we cannot  affirm  
which is more truly to be said to be    
mechanistic in its nature1  though we   
may suspect that really neither is4 ,we 
no longer think  of cause and effect as 
dictated by inherent necessity1 we      
simply  regard them as sequences in the 
routine of our sense-impressions of     
phenomena4 ,in a word1 we have at length
 grasped the idea that our  notions of  
reality1 at present at least1 whatever  
they may become  ultimately1 are not    
absolute1 but simply relative4 ,we see1 

too1 that  we do not explain the        
universe1 but only describe our         
perceptions of  its contents4    ,the   
so-called laws of nature are simply     
statements of formul which  resume or   
sum up the relationships and sequences  
of phenomena4 ,our  effort is constantly
 to find formul which will describe the 
widest  possible range of phenomena4 ,as
 our knowledge increases1 that is1  as  
we perceive new phenomena1 our laws or  
formul break down1  that is1 they fail  
to afford a description in brief terms  
of all  of our perceptions4 ,it is not  
that the old laws are untrue1 but       
simply that they are not comprehensive  
enough to include all of our            
perceptions4 ,the old laws are often    
particular or limiting instances  of the
 new laws4#bhc    ,from what we have    
said of the reality of observations it  
follows that  we must support that      
school of psychology1 and the parallel  
school  of philosophy1 which hold that  
concepts originate in perceptions4 ,but 

this does not impose so strong a        
restriction upon conceptions as might   
appear4 ,the elements of all our        
concepts do come to us from outside2  we
 manufacture nothing out of whole cloth4
 ,but when perception  has supplied a   
sufficient volume of raw material1 we   
may group its  elements in ways foreign 
to actual occurrence in the perceptual  
world1  and in so doing get conceptual  
results so entirely different from  what
 we have consciously perceived that we  
are strongly tempted to  look upon them 
as having certainly been manufactured in
 our minds  without reference to the    
externals4 ,of even more significance is
 our  ability to abstract from concrete 
objects and concrete incidents the      
essential features which make them alike
 and different4 ,but unlike  the        
,greeks1 we see that our concept of     
coldness is not something with  which we
 were endowed from the beginning1 but   
merely an abstraction  from concrete    
experiences with concrete objects that  

have been cold4          ,,the          
,,concepts ,,of ,,space ,,and ,,time    
,when we have formed the abstract ideas 
of coldness and warmth1 and  have had   
experience indicating that the          
occurrence of these properties  varies  
in degree1 we are in a position to form 
the secondary abstract  notion covered  
by the word temperature4 ,when we have  
formed the  abstract ideas of size and  
position and separation1 we are         
similarly  in a position to form a      
secondary abstraction to which we give  
the name  space4 ,not quite so easy to  
trace to its definite source but none   
the  less clearly an abstraction based  
on experience1 is our idea of what  we  
call time4 ,none of us are deceived as  
to the reality of these  abstractions499
 ,we do not regard space as real in the 
sense that  we regard a chair as real2  
it is merely an abstract idea convenient
  for the location of material objects  
like the chair4#aih ,nor do we  regard  
time as real in this sense4 ,things     

occupy space1 events occupy  time2 space
 and time themselves we realize are     
immaterial and unreal2  space does not  
exist and time does not happen in the   
same sense that  material objects exist 
and events occur4 ,but we find it       
absolutely  necessary to have1 among the
 mental machinery mentioned above as    
the apparatus by aid of which we keep   
track of the external world1  these     
vessels for that world to exist in and  
move in4    ,space and time1 then1 are  
concepts499 ,it is not strange1 however1
  that when confronted with the vast and
 bewildering complexity of  the universe
 and the difficulty of keeping separate 
and distinct in  our minds our          
perceptions and conceptions1 we should  
at times and as  respects certain things
 project our conceptions illegitimately 
into the  perpetual universe and mistake
 them for perceptions4 ,the most notable
  example perhaps of this projection has
 occurred in the very case of  space and
 time1 most fundamental of all of our   

concepts4 ,we got to  think of these as 
absolute1 as independent of each other  
and of all  other things1 and as always 
existing and continuing to exist whether
  or not we or anything else            
existed--space as a three-dimensional1  
uniform continuum1 having the same      
properties in all directions2  time as a
 one-dimensional1 irreversible          
continuum1 flowing in one  direction4   
,it is difficult to get back to the idea
 that space and  time so described and  
defined are concepts merely1 for the    
idea of  their absolute existence is    
ingrained in us as the result probably  
of long ancestral experience4#bhc       
,newton's definitions of course         
represent the classical idea of time    
and space4 ,he tells us that absolute1  
true and mathematical time  flows in    
virtue of its own nature1 uniformly and 
without reference to  any external      
object2 and that absolute space1 by     
virtue of its own  nature and without   
reference to any external object1 always

 remains  the same and is immovable4 ,of
 course from modern standpoints it is   
absurd to call either of these          
pronouncements a definition2 but they   
represent about as well as any words can
 the ideas which ,newton had  about time
 and space1 and they make it clear      
enough that he regarded  both as having 
real existence in the external world4   
,if space and time are to be the vessels
 of our universe1 and if the  only thing
 that really matters is measured        
results1 it is plain enough  that we    
must have1 from the very beginning1     
means of measuring space  and time4     
,whether we believe space and time to   
have real existence or  not1 it is      
obvious that we can measure neither     
directly4 ,we shall have  to measure    
space by measuring from one material    
object to another2  we shall have to    
measure time by some similar convention 
based  on events4 ,we shall later have  
something further to say about the      
measurement of time2 for the present we 

need only point out that99  ,newtonian  
time is measured independently of space2
 and the existence  is presupposed of a 
suitable timekeeper4#aj    ,the space of
 ,galileo and ,newton was conceived of  
as empty1 except in  so far as certain  
parts of it were occupied by matter4    
,positions of  bodies in this space were
 in general determined by reference     
to#bhc  a coordinate system of some     
kind4 ,this is again something that     
demands a certain amount of discussion4 
,,the ,,reference ,,frame ,,for ,,space 
,the mathematician1 following the lead  
of the great ,french all-around  genius1
 ,descartes1 shows us very clearly how  
to set up1 for the  measurement of      
space1 the framework known as the       
,cartesian coordinate  system4 ,the     
person of most ordinary mathematical    
attainments  will realize that to locate
 a point in a plane we must have two    
measurements2 and we could probably show
 this person1 without too  serious      
difficulty1 that we can locate a point  

in any surface by  two measurements4 ,an
 example of this is the location of     
points on the  earth's surface by means 
of their latitude and longitude4 ,it is 
equally  clear that if we add a third   
dimension and attempt to locate points  
in space1 we must add a third           
measurement4 ,in the case of points on  
the earth's surface1 this might be the  
elevation above sea level1  which would 
define the point not as part of the     
spherical surface  of the earth but as  
part of the solid sphere4 ,or we may    
fall back  on ,dr4 ,slosson's suggestion
 that in order to define completely the 
position of his laboratory1 we must make
 a statement about ,broadway1  and one  
about #aafth ,street1 and one telling   
how many flights of stairs  there are to
 climb4 ,in any event1 it should be     
clear enough that the  complete         
definition of a point in space calls for
 three measurements4    ,the            
mathematician formulates all this with  
the utmost precision4 ,he  asks us to99 

pick out any point whatever in space and
 call it ,o4 ,we  then draw or conceive 
to be drawn through this point three    
mutually  perpendicular lines called    
coordinate axes1 which we may designate 
,,ox1 ,,oy and ,,oz1 respectively4      
,finally1 we consider the three  planes 
also mutually perpendicular like the two
 walls and the floor of  a room that    
meet in one common corner1 which are    
formed by the lines  ,,ox and ,,oy1 ,,oy
 and ,,oz1 and ,,oz and ,,ox1           
respectively4 ,these  three planes are  
called coordinate planes4 ,and then any 
other point ,p  in space can be         
represented with respect to ,o by its   
perpendicular  distances from each of   
the three coordinate planes--the        
distances x1  y1 z in the figure4 ,these
 quantities are called the coordinates  
of the point4#bgb    ,to the layman     
there seems something altogether naive  
in this notion  of the scientist's      
setting up the three sides of a box in  
space and  using them as the basis of   

all his work4 ,the layman somehow feels 
that while it is perfectly all right for
 him to tell us that he  lives at #ajfe 
7one coordinate7 #aefth ,street 7two    
coordinates7 on  the third floor 7three 
coordinates71 it is rather trivial      
business  for the serious-minded        
scientist to consider the up-and-down1  
the  forward-and-back1 the              
right-and-left of every point with which
 he has  occasion to deal4 ,there seems 
to the layman something particularly    
inane and foolish and altogether puerile
 about a set of coordinate  axes1 and   
you simply can't make him believe that  
the serious-minded  scientist has to    
monkey with any such funny business4 ,he
 can't be  induced to take this         
coordinate-axis business seriously4     
,nevertheless1  the fact is that the    
scientist takes it with the utmost      
seriousness4 ,it  is necessary for him  
to define the positions of points2 and  
he does  do it by means of a set of     
coordinate axes4    ,the scientist1     

however1 is not interested in points of 
empty space4 ,the  point is to him      
merely part again of the conceptual     
machinery which  he uses in his effort  
to run along with the external world4   
,he knows  there are no real points1 but
 it suits his convenience to keep track 
of certain things that are real by      
representing them as points4 ,but  these
 things are in practically every        
instance material bodies2 and  in       
practically every instance1 instead of  
staying put in one spot1  they insist   
upon moving about through space4 ,the   
scientist has to use  his coordinate    
system1 not merely to define a single   
position of such  a point1 but to keep  
track of the path over which it moves   
and to  define its position in that path
 at given moments4          ,,time ,,and
 ,,the ,,coordinate ,,system    ,this   
introduces the concept of time into     
intimate relationship with  the spatial 
coordinate system4 ,and at once we feel 
the lack of a  concrete1 visualized     

fourth dimension499 ,if we want to fix  
objects  in the floor alone1 the edge of
 the room running toward the ceiling    
would become unnecessary and could be   
dropped from our coordinate  system4    
,that is1 we need only two coordinates  
to fix the position of a  point in a    
plane4 ,suppose instead of discarding   
the third coordinate1  we use it to     
represent units of time4 ,it then       
enables us to record  the time it took a
 moving point in the floor to pass from 
position  to position4 ,certain points  
in the room would be vertically above   
the corresponding points occupied by the
 moving point in its path  across the   
floor2 and the vertical height above the
 floor of such  points corresponds to a 
value of the time-coordinate which      
indicates  the time it took the point to
 move from position to position4#aeb    
,just as the path of the point across   
the floor is a continuous curve  7for   
the mathematician1 it should be         
understood1 this term curve  includes   

the straight line1 as a special case in 
which the curvature  happens to be      
zero72 so the series of points above    
these in the room  forms a continuous   
curve which records for us1 not merely  
the path of  the point across the floor1
 but in addition the time of its arrival
  at each of its successive positions4  
,in the algebraic work connected  with  
such a problem1 the third coordinate    
behaves exactly the same1  regardless of
 whether we consider it to represent    
time or a third  spatial dimension2 we  
cannot even tell from the algebra what  
it  does represent4    ,when we come to 
the more general case of a point moving 
freely through  space1 we have but three
 coordinates at our disposal2 there is  
not  a fourth one by aid of which we can
 actually diagram its time-space        
record4 ,nevertheless1 we can write down
 the numerical and algebraic  relations 
between its three space-coordinates and 
the time which it  takes to pass from   
one position to another2 and by this    

means we can  make all necessary        
calculations4 ,its motion is completely 
defined  with regard both to space and  
to time4 ,we are very apt to call       
attention to the fact that if we did    
have at our disposal a fourth1          
space-coordinate1 we could use it to    
represent the time graphically1  as     
before1 and actually construct a        
geometric picture of the path of  our   
moving point with regard to space and   
time4 ,and on this account  we are very 
apt to speak as though the time         
measurements constituted  a fourth      
coordinate1 regardless of any question  
of our ability to  construct a picture  
of this coordinate4 ,the arrival of a   
point in  a given position constitutes  
an event2 and this event is completely  
defined by means of four                
coordinates--three in space1 which we   
can  picture on our coordinate axes1 and
 one in time which we cannot4    ,the   
set of coordinate axes in space1        
together with the zero point from  which

 we measure time1 constitute what we    
call a frame of reference4 ,if  we are  
not going to pay any attention to time1 
we can think of the  space coordinate   
system alone as constituting our        
reference frame4 ,this  expression      
appears freely throughout the subsequent
 text1 and always  with one or the other
 of these interpretations4    ,we see1  
then1 how we can keep track of a moving 
point by keeping track  of the          
successive positions which it occupies  
in our reference frame499  ,now we have 
implied that these coordinate axes are  
fixed in space2  but there is nothing to
 prevent us from supposing that they    
move4#bgb  ,if they do1 they carry with 
them all their points2 and any motion   
of these points which we may speak about
 will be merely motion with  reference  
to the coordinate system4 ,if we find   
something outside  our coordinate system
 that is not moving1 the motion of      
points in  our system with regard to    
those outside it will be a combination  

of  their motion with regard to our     
coordinate axes and that of these axes  
with regard to the external points4     
,this will be a great nuisance2  and it 
represents a state of affairs which we  
shall try to avoid4 ,we  shall avoid it1
 if at all1 by selecting a coordinate   
system with  reference to which we1     
ourselves1 are not moving2 one which    
partakes  of any motion which we may    
have4 ,or perhaps we shall sometimes    
wish to  reverse the process1 in        
studying the behavior of some group of  
bodies1  and seek a set of axes which is
 at rest with respect to these bodies2  
one which partakes of any motion they   
may have4          ,,the ,,choice ,,of  
,a ,,coordinate ,,frame    ,all this    
emphasizes the fact that our coordinate 
axes are not  picked out for us in      
advance by nature1 and set down in some 
one  particular spot4 ,we select them   
for ourselves1 and we select them in    
the most convenient way4 ,but different 
observers1 or perhaps the same  observer

 studying different problems1 will find 
it advantageous to  utilize different   
coordinate systems499 ,the astronomer   
has found it  possible1 and highly      
convenient1 to select a coordinate frame
 such  that the great majority of the   
stars have1 on the whole1 no motion     
with respect to it4#bhc ,such a system  
would be most unsuited for              
investigations confined to the earth2   
for these we naturally select  a        
framework attached to the earth1 with   
its origin ,o at the earth's  center if 
our investigation covers the entire     
globe and at some more  convenient point
 if it does not1 and in either event    
accompanying the  earth in its rotation 
and revolution4 ,but such a framework1  
as well  as the one attached to the     
fixed stars1 would be highly            
inconvenient  for an investigator of the
 motions of the planets2 he would       
doubtless  attach his reference frame to
 the sun4#aja    ,in this connection a  
vital question suggests itself4 ,is the 

expression of natural law independent of
 or dependent upon the choice  of a     
system of coordinates8 ,and to what     
extent shall we be able to  reconcile   
the results of one observer using one   
reference frame1 and  a second observer 
using a different one8 ,the answer to   
the second  question is obvious499      
,true1 if any series of events is       
described  using two different sets of  
axes1 the descriptions will be          
different1  depending upon the time     
system adopted and the relative motion  
of the  axes4 ,but if the connection    
between the reference systems is known1 
it is possible by mathematical processes
 to deduce the quantities  observed in  
one system if those observed in the     
other are known4#ce  ,this process of   
translating the results of one observer 
into those of  another is known as a    
transformation2 and the mathematical    
statement  of the rule governing the    
transformation is called the equation or
 the  equations 7there are usually      

several of them7 of the                 
transformation499  ,transformations of  
this character constitute a             
well-developed branch  of               
mathematics4#ce    ,when we inquire     
about the invariance of natural law it  
is necessary to  be rather sure of just 
what we mean by this expression4 ,the   
statement  that a given body is moving  
with a velocity of #ge miles per hour is
 of  course not a natural law2 it is a  
mere numerical observation4 ,but aside  
from such numerical results1 we have a  
large number of mathematical  relations 
which give us a more or less general    
statement of the  relations that exist  
between velocities1 accelerations1      
masses1  forces1 times1 lengths1        
temperatures1 pressures1 etc41 etc4     
,there  are some of these which we would
 be prepared to state at once as        
universally valid--distance travelled   
equals velocity multiplied by  time1 for
 instance4 ,we do not believe that any  
conceivable change of  reference systems

 could bring about a condition in which 
the product  of velocity and time1 as   
measured from a certain framework1 would
  fail to equal distance as measured    
from this same framework4 ,there  are   
other relations more or less of the same
 sort which we probably  believe to be  
in the same invariant category2 there   
are others1  perhaps1 of which we might 
be doubtful2 and presumably there are   
still others which we should suspect of 
restricted validity1 holding  in certain
 reference systems only and not in      
others4    ,the question of invariance  
of natural law1 then1 may turn out to   
be one which may be answered in the     
large by a single statement2  it may    
equally turn out to be one that has to  
be answered in the  small1 by           
considering particular laws in          
connection with particular              
transformations between particular      
reference systems4 ,or1 perhaps1  we may
 find ourselves justified in taking the 
stand that an alleged  law of nature is 

truly such a law only in the event that 
it is  independent of the change from   
one reference system to another4 ,in    
any event1 the question may be          
formulated as follows3    ,observer ,a1 
using the reference system ,r1 measures 
certain quantities  t1 w1 x1 y1 z4      
,observer ,b1 using the reference system
 ,s1  measures the same items and gets  
the values t'1 w'1 x'1 y'1  z'4 ,the    
appropriate transformation equations for
 calculating the  one set of values from
 the other is found4 ,if a mathematical 
relation  of any sort is found to exist 
between the values t1 w1 x1 y1  z1 will 
the same relation exist between the     
values t'1 w'1  x'1 y'1 z'8 ,if it does 
not1 are we justified in still calling  
it a law of nature8 ,and if it does not1
 and we refrain from calling  it such a 
law1 may we expect in every case to find
 some relation that  will be invariant  
under the transformation1 and that may  
therefore  be recognized as the natural 
law connecting t1 w1 x1 y and z8    ,i  

have found it advisable to discuss this 
point in such detail because  here more 
than in any other single place the      
competing essayists  betray uncertainty 
of thought and sloppiness of expression4
 ,it doesn't  amount to much to talk    
about the invariance of natural laws and
  their persistence as we pass from one 
coordinate system to another1  unless we
 are fairly well fortified with respect 
to just what we mean  by invariance and 
by natural law4 ,we don't expect the    
velocity of a  train to be #fj miles per
 hour alike when we measure it with     
respect  to a signal tower along the    
line and with respect to a moving train 
on the other track4 ,we don't expect the
 angular displacement of  ,mars to      
change as rapidly when he is on the     
other side of the sun  as when he is on 
our side4 ,but we do1 ,i think1 rather  
expect that  in any phenomenon which we 
may observe1 we shall find a natural law
  of some sort which is dependent for   
its validity neither upon the  units we 

employ1 nor the place from which we make
 our measurements1  nor anything else   
external to the phenomenon itself4 ,we  
shall see1  later1 whether this         
expectation is justified1 or whether it 
will have  to be discarded in the final 
unravelling of the absolutist from the  
relativistic philosophy which1 with     
,einstein1 we are to undertake499       
,,iii    ,,the ,,relativity ,,of        
,,uniform ,,motion    ,classical ,ideas 
on the ,subject2 the ,ether and the     
,apparent ,possibility  of ,absolute    
,motion2 the ,michelson-,morley         
,experiment and the ,final  ,negation of
 ,this ,possibility    ,,by ,,various   
,,contributors ,,and ,,the ,,editor     
,when we speak of a body as being in    
motion1 we mean that this  body is      
changing its position in space4 ,now it 
is clear that the  position of an object
 can only be determined with reference  
to other  objects3 in order to describe 
the place of a material thing we must1  
for example1 state its distances from   

other things4 ,if there were no  such   
bodies of reference1 the words position 
in space would have  no definite meaning
 for us4#bd ,the number of such external
 bodies of  reference which it is       
necessary to cite in order to define    
completely  the position of a given body
 in space depends upon the character  of
 the space dealt with4 ,we have seen    
that when we visualize the  space of our
 experience as a surface of any         
character1 two citations  are           
sufficient2 and that when we conceive of
 it as surrounding us  in three         
dimensions we require three4 ,it will be
 realized that the  mathematician is    
merely meeting this requirement when he 
sets up his  system of coordinate axes  
to serve as a reference frame499        
,what is true of place must be true also
 of motion1 since the  latter is nothing
 but change of place4 ,in fact1 it would
 be impossible  to ascribe a state of   
motion or of rest to a body poised all  
alone  in empty space4 ,whether a body  

is to be regarded as resting or as      
moving1 and if the latter at what speed1
 depends entirely upon the  objects to  
which we refer its positions in         
space4#bd ,as ,einstein  sits at his    
desk he appears to us to be at rest2 but
 we know that  he is moving with the    
rotation of the earth on its axis1 with 
the  earth in its orbit about the sun1  
and with the solar system in its  path  
through space--a complex motion of which
 the parts or the whole  can be detected
 only by reference to appropriately     
chosen ones of the  heavenly bodies4 ,no
 mechanical test has ever been devised  
which will  detect this motion1#ahb if  
we reserve for discussion in its proper 
place the ,foucault pendulum experiment 
which will reveal the axial  rotation of
 our globe499 ,no savage1 if he were to 
stand still1  could be convinced that he
 was moving with a very high velocity   
or in fact that he was moving at all4#cj
 ,you drop a coin straight  down a      
ship's side3 from the land its path     

appears parabolic2 to a  polar onlooker 
it whirls circle-wise2 to dwellers on   
,mars it darts  spirally about the sun2 
to a stellar observer it gyrates through
  the sky#bfc in a path of many         
complications4 ,to you it drops in a    
straight line from the deck to the      
sea499 ,yet its various tracks  in      
ship-space1 sea-space1 earth-space1     
sun-space1 star-space1 are  all equally 
real1#bfc and the one which will be     
singled out for  attention depends      
entirely upon the observer1 and the     
objects to  which he refers the         
motion499 ,the earth moves in the solar 
system1  which is itself approaching a  
distant star-cluster4 ,but we cannot say
  whether we are moving toward the      
cluster1 or the cluster toward us1#ah   
or both1 or whether we are conducting a 
successful stern chase of it1  or it of 
us199 unless we have in mind some third 
body with reference  to which the       
motions of earth and star-cluster are   
measured4#ah ,and  if we have this1 the 

measurements made with reference to it  
are of  significance with regard to it1 
rather than with regard to the earth    
and the star-cluster alone499    ,we can
 express all this by saying ,all motions
 are relative2 there  is no such thing  
as absolute motion4 ,this line of       
argument has in  fact been followed by  
many natural philosophers4 ,but is its  
result  in agreement with actual        
experience8 ,is it really impossible to 
distinguish between rest and motion of a
 body if we do not take into            
consideration its relations to other    
objects8 ,in fact it can easily  be seen
 that1 at least in many cases1 no such  
distinction is possible4          ,,who 
,,is ,,moving8    ,imagine yourself     
sitting in a railroad car with veiled   
windows and  running on a perfectly     
straight track with unchanging velocity3
 you  would find it absolutely          
impossible to ascertain by any          
mechanical  means whether the car were  
moving or not4 ,all mechanical          

instruments  behave exactly the same1   
whether the car be standing still or in 
motion4#bd ,if you drop a ball you will 
see it fall to the floor in  a straight 
line1 just as though you had dropped it 
while standing on  the station platform4
 ,furthermore1 if you drop the ball from
 the same  height in the two cases1 and 
measure the velocities with which it    
strikes the car floor and the station   
platform1 or the times which  it        
requires for the descent1 you will find 
these identical in the  two cases4#ahb  
,any changes of speed or of direction   
7as when the car speeds up  or slows    
down or rounds a curve7 can be detected 
by observing the  behavior of bodies in 
the car1 without apparent reference to  
any  outside objects4 ,this becomes     
particularly obvious with sudden        
irregularities of motion1 which manifest
 themselves by shaking  everything in   
the car4 ,but a uniform motion in a     
straight line does  not reveal itself by
 any phenomenon within the vehicle4#bd  

,moreover1 if we remove the veil from   
our window to the extent that  we may   
observe the train on the adjoining      
track1 we shall be able to  make no     
decision as to whether we or it be      
moving4 ,this is indeed an  experience  
which we have all had499 ,often when    
seated in a train  about to leave the   
station1 we have thought ourselves under
 way1  only to perceive as the motion   
becomes no longer uniform that another  
train has been backing into the station 
on the adjoining track4 ,again1  as we  
were hurried on our journey1 we have1   
raising suddenly our eyes1  been puzzled
 to say whether the passing train were  
moving with us or  against us or indeed 
standing still2 or more rarely we have  
had the  impression that both it and we 
seemed to be at rest1 when in truth     
both were moving rapidly with the same  
speed4#hb ,even this phrase in  truth is
 a relative one1 for it arises through  
using the earth as an  absolute         
reference body4 ,we are indeed naive if 

we cannot appreciate  that there is no  
reason for doing this beyond            
convenience1 and that  to an observer   
detached from the earth it were just as 
reasonable  to say that the rails are   
sliding under the train as that the     
train  is advancing along the rails4    
,one of my own most vivid childhood     
recollections is of the terror with     
which1 riding on a train that  passed   
through a narrow cut1 ,i hid my head in 
the maternal lap to shut  out the horrid
 sight of the earth rushing past my     
window4 ,the absence  of a background in
 relatively slow retrograde motion was  
sufficient  to prevent my consciousness 
from drawing the accustomed conclusion  
that after all it was really the train  
that was moving499          ,,mechanical
 ,,relativity    ,so we can enunciate   
the following principle3 ,when a body is
 in  uniform rectilinear motion         
relatively to a second body1 then all   
phenomena take place on the first in    
exactly the same manner as on  the      

second2 the physical laws for the       
happenings on both bodies are           
identical4#bd ,and between a system of  
bodies1 nothing but relative  motion may
 be detected by any mechanical means    
whatever2 any attempt  to discuss       
absolute motion presupposes a           
super-observer on some body  external to
 the system4 ,even then1 the absolute   
motion is nothing  but motion relative  
to this super-observer4 ,by no          
mechanical means  is uniform            
straight-line motion of any other than  
relative character  to be detected4     
,this is the ,principle of ,mechanical  
,relativity4    ,there is nothing new in
 this4 ,it was known to ,galileo1 it was
 known  to ,newton1 it has been known   
ever since4 ,but the curious persistence
  of the human mind in habits of thought
 which confuse relativity with          
absolutism brought about a state of     
affairs where we attempted to  know this
 and to ignore it at the same time4 ,we 
shall have to return  to the            

mathematical mode of reasoning to see   
how this happened4 ,the  mathematician  
has a way all his own of putting the    
statement of  relativity which we have  
made4 ,he recalls1 what we have already 
seen1  that the observer on the earth   
who is measuring his absolute motion    
with respect to the earth has merely    
attached his reference framework  to the
 earth2 that the passenger in the train 
who measures all motion  naively with   
respect to his train is merely carrying 
his coordinate  axes along with his     
baggage1 instead of leaving them on the 
solid  ground2 that the astronomer who  
deals with the motion of the earth      
about the sun1 or with that of the fixed
 stars against one another1  does so    
simply by the artifice of hitching his  
frame of reference to  the sun or to one
 of the fixed stars4 ,so the            
mathematician points out  that dispute  
as to which of two bodies is in motion  
comes right down  to dispute as to which
 of two sets of coordinate axes is the  

better  one1 the more nearly natural or 
absolute4 ,he therefore phrases  the    
mechanical principle of relativity as   
follows3    ,among all coordinate       
systems that are merely in uniform      
straight-line  motion to one another1 no
 one occupies any position of unique    
natural  advantage2 all such systems are
 equivalent for the investigation of    
natural laws2 all systems lead to the   
same laws and the same results4    ,the 
mathematician has thus removed the      
statement of relativity  from its       
intimate association with the external  
observed phenomena1  and transferred it 
to the observer and his reference frame4
 ,we must  either accept the principle  
of relativity1 or seek a set of         
coordinate  axes that have been singled 
out by nature as an absolute reference  
frame4 ,these axes must be in some way  
unique1 so that when we refer  phenomena
 to them1 the laws of nature take a form
 of exceptional  simplicity not attained
 through reference to ordinary axes4    

,where  shall we look for such a        
preferred coordinate system899          
,,the ,,search ,,for ,,the ,,absolute   
,older theory clung to the belief that  
there was such a thing as  absolute     
motion in space4#aig ,as the body of    
scientific law developed  from the      
sixteenth century onward1 the not       
unnatural hypothesis crept  in1 that    
these laws 7that is to say1 their       
mathematical formulations  rather than  
their verbal statements7 would reveal   
themselves in  especially simple forms1 
were it possible for experimenters to   
make  their observations from some      
absolute standpoint2 from an absolutely 
fixed position in space rather than from
 the moving earth4#bfd  ,somewhere a set
 of coordinate axes incapable of motion 
was to be  found1#aig a fixed set of    
axes for measuring absolute motion2 and 
for two hundred years the world of      
science strove to find it1#adg in  spite
 of what should have been assurance that
 it did not exist4 ,but  the search     

failed1 and gradually the universal     
applicability of the  principle of      
relativity1 so far as it concerned      
mechanical phenomena1  grew into general
 acceptance499 ,and after the           
development1 by the  great              
mathematicians of the eighteenth        
century1 of ,newton's laws of  motion   
into their most complete mathematical   
form1 it was seen that so  far as these 
laws are concerned the absolutist       
hypothesis mentioned is  quite          
unsupported4 ,no complication is        
introduced into ,newton's laws  if the  
observer has to make his measurements in
 a frame of reference  moving uniformly 
through space2 and for measurements in a
 frame like  the earth1 which moves with
 changing speed and direction about the 
sun and rotates on its axis at the same 
time1 the complication is  not of so    
decisive a nature as to give us any clue
 to the earth's  absolute motion in     
space4    ,but mechanics1 albeit the    
oldest1 is yet only one of the physical 

sciences4 ,the great advance made in the
 mathematical formulation  of optical   
and electromagnetic theory during the   
nineteenth century  revived the hope of 
discovering absolute motion in space by 
means of  the laws derived from this    
theory4#bfd ,newton had supposed light  
to be a material emanation1 and if it   
were so1 its passage across  empty space
 from sun and stars to the earth raised 
no problem4 ,but  against ,newton's     
theory ,huyghens1 the ,dutch astronomer1
 advanced the  idea that light was a    
wave motion of some sort4 ,during the   
,newtonian  period and for many years   
after1 the corpuscular theory prevailed2
  but eventually the tables were        
turned499 ,men made rays of light       
interfere1 producing darkness 7see page 
#fa74 ,from this1 and from  other       
phenomena like polarization1 they had   
deduced that light was a  form of wave  
motion similar to water ripples2 for    
these interfere1  producing level       
surfaces1 or reinforce each other1      

producing waves  of abnormal height4    
,but if light were to be regarded as a  
form of  wave motion--and the phenomena 
could apparently be explained on no     
other basis--then there must be some    
medium capable of undergoing  this form 
of motion4#ace ,transmission of waves   
across empty space  without the aid of  
an intermediary material medium would be
 action  at a distance1 an idea         
repugnant to us4 ,trammeled by our      
tactual1  wire-pulling conceptions of a 
material universe1 we could not accustom
  ourselves to the idea of              
something--even so immaterial a         
something  as a wave--being transmitted 
by nothing4 ,we needed a word--ether--to
  carry light if not to shed it2 just as
 we need a word--inertia--to  carry a   
projectile in its flight4#bca ,it was   
necessary to invest this  medium with   
properties to account for the observed  
facts4 ,on the whole  it was regarded as
 the perfect fluid4#bce ,the ether was  
imagined as  an all-pervading1          

imponderable substance filling the vast 
emptiness  through which light reaches  
us1 and as well the intermolecular      
spaces  of all matter4 ,nothing more was
 known definitely1 yet this much served 
as a good working hypothesis on the     
basis of which ,maxwell was enabled  to 
predict the possibility of radio        
communication4 ,by its fruits the  ether
 hypothesis justified itself2 but does  
the ether exist8#bca          ,,the     
,,ether ,,and ,,absolute ,,motion    ,if
 it does exist1 it seems quite          
necessary1 on mere philosophical        
grounds1 that it shall be eligible to   
serve as the long-sought  reference     
frame for absolute motion4 ,surely it   
does not make sense to  speak of a      
homogeneous medium filling all space1   
sufficiently material  to serve as a    
means of communication between remote   
worlds1 and in  the next breath to deny 
that motion with respect to this medium 
is a  concept of significance499 ,such a
 system of reference as was offered  by 

the ether1 coextensive with the entire  
known region of the universe1  must     
necessarily serve for all motions within
 our perceptions4#ahf  ,the conclusion  
seems inescapable that motion with      
respect to the  ether ought to be of a  
sufficiently unique character to stand  
out  above all other motion4 ,in        
particular1 we ought to be able to use  
the  ether to define1 somewhere1 a      
system of axes fixed with respect to    
the ether1 the use of which would lead  
to natural laws of a uniquely  simply   
description4    ,maxwell's work added   
fuel to this hope499 ,during the last   
century1  after the units of electricity
 had been defined1 one set for static   
electrical calculations and one for     
electromagnetic calculations1 it  was   
found that the ratio of the metric units
 of capacity for the two  systems was   
numerically equal to what had already   
been found as the  velocity with which  
light is transmitted through the        
hypothetical  ether4 ,one definition    

refers to electricity at rest1 the other
  to electricity in motion4 ,maxwell1   
with little more working basis  than    
this1 undertook to prove that electrical
 and optical phenomena  were merely two 
aspects of a common cause1#bce to which 
the general  designation of             
electromagnetic waves was applied4      
,maxwell treated  this topic in great   
fullness and with complete success4 ,in 
particular1  he derived certain         
equations giving the relations between  
the various  electrical quantities      
involved in a given phenomenon4 ,but it 
was  found1 extraordinarily enough1 that
 these relations were of such  character
 that1 when we subject the quantities   
involved to a change  of coordinate     
axes1 the transformed quantities did not
 preserve these  relations if the new   
axes happened to be in motion with      
respect to  the original ones4 ,this1 of
 course1 was taken to indicate that     
motion  really is absolute when we come 
to deal with electromagnetic phenomena1 

and that the ether which carries the    
electromagnetic waves really may  be    
looked to to display the properties of  
an absolute reference frame4            
,reference to the phenomenon of         
aberration1 which ,dr4 ,pickering has   
discussed adequately in his essay and   
which ,i need therefore mention  here   
only by name1 indicated that the ether  
was not dragged along  by material      
bodies over and through which it might  
pass4 ,it seemed  that it must filter   
through such bodies1 presumably via the 
molecular  interstices1 without         
appreciable opposition4 ,were this not  
the case1  we should be in some doubt as
 to the possibility of observing the    
velocity through the ether of material  
bodies2 if the ether adjacent  to such  
bodies is not dragged along or thrown   
into eddies1 but stands  still while the
 bodies pass1 there seems no imaginable 
reason for  anything other than the     
complete success of such observations4  
,and  of course these are of the utmost 

importance1 the moment we assign  to the
 ether the rle of absolute reference    
frame4          ,,the ,,earth ,,and     
,,the ,,ether    ,one body in motion    
with respect to the ether is our earth  
itself4 ,we  do not know in advance in  
what direction to expect this motion or 
what  magnitude to anticipate that it   
will have4 ,but one thing is clear499   
,in its motion around the sun1 the earth
 has1 at opposite points on its  orbit1 
a difference in velocity with respect to
 the surrounding medium  which is double
 its orbital velocity with respect to   
the sun4 ,this  difference comes to #cg 
miles per second4 ,the earth should     
therefore1  at some time in the year1   
show a velocity equal to or greater than
  #ah #a_/b miles per second1 with      
reference to the universal medium4 ,the 
famous ,michelson-,morley experiment of 
#ahhg was carried out with the          
expectation of observing this           
velocity4#bfg    ,the ether1 of course1 
and hence velocities through it1 cannot 

be  observed directly4 ,but it acts as  
the medium for the transmission of      
light499 ,if the velocity of light      
through the ether is ,c and that  of the
 earth through the ether is v1 then the 
velocity of light  past the earth1 so   
the argument runs1 must vary from ,c-v  
to ,cv1  according as the light is      
moving exactly in the same direction as 
the  earth1 or in the opposite          
direction1#ahb or diagonally across the 
earth's path so as to get the influence 
only of a part of the earth's  motion4  
,this of course assumes that ,c has     
always the same value2  an assumption   
that impresses one as inherently        
probable1 and one that  is at the same  
time in accord with ordinary            
astronomical observation4    ,it is not 
possible to measure directly the        
velocity of light 7#ahf1ccj  miles per  
second1 more or less7 with sufficient   
accuracy to give any  meaning to the    
variation in this velocity which might  
be effected by  adding or subtracting   

that of the earth in its orbit 7a mere  
#ah #a_/b  miles per second74 ,it is1   
however1 possible to play a trick on the
  light by sending it back and forth    
over several paths1 and comparing  7not 
measuring absolutely1 but merely        
comparing7 with great minuteness  the   
times consumed in these several round   
trips4          ,a ,,journey ,,upstream 
,,and ,,back    ,the number of letters  
the ,scientific ,american has received  
questioning  the ,michelson-,morley     
experiment indicates that many people   
are not  acquainted with the fundamental
 principle on which it is based4 ,so    
let us look at a simple analogous case4 
,suppose a swimmer or a rower  make a   
return trip upstream and down1          
contending with the current  as he goes 
up and getting its benefit when he comes
 down4 ,obviously1  says snap judgment1 
since the two legs of the journey are   
equal1 he  derives exactly as much      
benefit from the current when he goes   
with it  as he suffers handicap from it 

when he goes against it2 so the round   
trip must take exactly the same time as 
a journey of the same length  in still  
water1 the argument applying equally in 
the case where the  swimmer is a wave of
 light in the ether stream4    ,but let 
us look now at a numerical case4 ,a man 
can row in still water  at four miles   
per hour4 ,he rows twelve miles upstream
 and back1 in a  current of two miles   
per hour4 ,at a net speed of two miles  
per hour  he arrives at his turning     
point in six hours4 ,at a net speed of  
six  miles per hour he makes the        
down-stream leg in two hours4 ,the      
elapsed  time for the journey is eight  
hours2 in still water he would row the  
twenty-four miles in six hours4    ,if  
we were to attempt an explanation of    
this result in words we should  say that
 by virtue of the very fact that it does
 delay him1 the adverse  current        
prolongs the time during which it       
operates2 while by virtue of  the very  
fact that it accelerates his progress1  

the favoring current  shortens its      
venue4 ,the careless observer realizes  
that distances are  equal between the   
two legs of the journey1 and            
unconsciously assumes  that times are   
equal4    ,if the journey be made       
directly with and directly against the  
stream  of water or ether or what not1  
retardation is effected to its fullest  
extent4 ,if the course be a diagonal    
one1 retardation is felt to an  extent  
measurable as a component1 and depending
 for its exact value  upon the exact    
angle of the path4 ,felt1 however1 it   
must always be4    ,here is where we    
begin to get a grip on the problem of   
the earth  and the ether4 ,in any       
problem involving the return-trip       
principle1  there will enter two        
velocities--that of the swimmer and that
 of the  medium2 and the time of        
retardation4 ,if we know any two of     
these items  we can calculate the third4
 ,when the swimmer is a ray of light and
 the  velocity of the medium is that of 

the ether as it flows past the earth1   
we know the first of these two2 we hope 
to observe the retardation  so that we  
may calculate the second velocity4 ,the 
apparatus for the  experiment is        
ingenious and demands description4      
,,the ,,michelson-,,morley ,,experiment 
,the machine is of structural steel1    
weighing #a1ijj pounds4 ,it has two     
arms which form a ,greek cross4 ,each   
arm is #ad feet in length4 ,the whole   
apparatus is floated in a trough        
containing #hjj pounds of mercury4      
,four mirrors are arranged on the end of
 each arm1 sixteen in all1  with a      
seventeenth mirror1 ,m1 set at one of   
the inside corners of the  cross1 as    
diagrammed4 ,a source of light 7in this 
case a calcium flame7  is provided1 and 
its rays directed by a lens toward the  
mirror ,m4 ,part  of the light is       
allowed to pass straight through ,m to  
the opposite  arm of the cross1 where it
 strikes mirror #a4 ,it is reflected    
back  across the arm to mirror #b1      

thence to #c1 and so on until it reaches
  mirror #h4 ,thence it is reflected    
back to mirror #g1 to #f1 and so on1    
retracing its former path1 and finally  
is caught by the reverse side  of the   
mirror ,m and is sent to an observer at 
,o4 ,in retracing its  path the light   
sets up an interference phenomenon 7see 
below7 and the  interference bands are  
visible to the observer1 who is provided
 with  a telescope to magnify the       
results4    ,a second part of the       
original light-beam is reflected off at 
right  angles by the mirror ,m1 and is  
passed to and fro on the adjacent arms  
of the machine1 in exactly the same     
manner and over a similar path1  by     
means of the mirrors ,i1 ,,ii1 ,,iii1   
444 ,,viii4 ,this light finally  reaches
 the observer at the telescope1 setting 
up a second set of  interference bands1 
parallel to the first4    ,a word now   
about this business of light            
interference4 ,light is a wave  motion4 
,the length of a wave is but a few      

millionths of an inch1  and the         
amplitude is correspondingly minute2 but
 none the less1  these waves behave in a
 thoroughly wave-like manner4 ,in       
particular1  if the crests of two waves 
are superposed1 there is a double       
effect2  while if a crest of one wave   
falls with a trough of another1 there   
is a killing-off or interference4       
,under ordinary circumstances           
interference of light waves does not    
occur4 ,this is simply because under    
ordinary circumstances light waves  are 
not piled up on one another4 ,but       
sometimes this piling up occurs2  and   
then1 just so sure as the piled-up waves
 are in the same phase  they reinforce  
one another1 while if they are in       
opposite phase they  interfere4 ,and the
 conditions which we have outlined      
above1 with the  telescope and the      
mirrors and the ray of light retracing  
the path  over which it went out1 are   
conditions under which interference does
  occur4 ,if the returning wave is in   

exact phase with the outgoing one1  the 
effect is that of uniform double        
illumination2 if it is in exactly       
opposite phase the effect is that of    
complete extinguishing of the  light1   
the reversed wave exactly cancelling out
 the original one4 ,if  the two rays are
 partly in phase1 there is partial      
reinforcement or  partial cancelling    
out1 according to whether they are      
nearly in phase  or nearly out of phase4
 ,finally1 if the mirrors are not set   
absolutely  parallel--as must in        
practice be the case when we attempt to 
measure  their parallelism in terms of  
the wave-length of light--adjacent      
parts of the light ray will vary in the 
extent to which they are out  of phase1 
since they will have travelled a        
fraction of a wave-length  further to   
get to and from this1 that or the other 
mirror4 ,there  will then appear in the 
telescope alternate bands of            
illumination  and darkness1 whose width 
and spacing depend upon all the factors 

entering into the problem4    ,if it    
were possible for us to make the        
apparatus with such a degree  of        
refinement that the path from mirror ,m 
via mirrors #a1 #b1 #c1 etc41  back     
through ,m and into the telescope1 were 
exactly the same length  as that from   
flame to telescope by way of the mirrors
 ,i1 ,,ii1 ,,iii1  etc4--exactly the    
same to a margin of error materially    
less than a  single wave-length of      
light--why1 then1 the two sets of       
interference  fringes would come out    
exactly superposed provided the motion  
of  the earth through the ether turn out
 to have no influence upon the  velocity
 of light2 or1 if such influence exist1 
these fringes would  be displaced from  
one another to an extent measuring the  
influence in  question4 ,but our ability
 to set up this complicated pattern of  
mirrors  at predetermined distances     
falls far short of the wave-length as a 
measure of error4 ,so in practice all   
that we can say is that having  once set

 the instrument up1 and passed a beam of
 light through it1  there will be       
produced two sets of parallel           
interference fringes4 ,these  sets will 
fail of superposition--each fringe of   
one set will be removed  from the       
corresponding fringe of the other       
set--by some definite  distance4 ,then1 
any subsequent variation in the speed of
 light along  the two arms will at once 
be detected by a shifting of the        
interference  bands through a distance  
which we shall be able to measure4      
,,the ,,verdict    ,under the theories  
and assumptions governing at the time of
 the  original performance of this      
experiment1 it will be readily seen that
  if this machine be set up in an ether 
stream with one arm parallel  to the    
direction of the stream and the other at
 right angles thereto1  there will be a 
difference in the speed of the light    
along the two  arms4 ,then if the       
apparatus be shifted to a position      
oblique to the  ether stream1 the excess

 velocity of the light in the one arm   
would  be diminished1 and gradually come
 to zero at the #de-;degree angle1      
after which the light traveling along   
the other arm would assume the  greater 
speed4 ,in making observations1         
therefore1 the entire apparatus  was    
slowly rotated1 the observers walking   
with it1 so that changes  of the sort   
anticipated would be observed4    ,the  
investigators were1 however1 ignorant of
 the position in which the  apparatus   
ought to be set to insure that one of   
the arms lie across  the ether drift2   
and they were ignorant of the time of   
year at which  the earth's maximum      
velocity through the ether was to be    
looked for4 ,in  particular1 it is plain
 that if the solar system as a whole is 
moving  through the ether at a rate less
 than the earth's orbital velocity1     
there is a point in our orbit where our 
velocity through the ether  and that    
around the sun just cancel out and leave
 us temporarily in  a state of absolute 

rest4 ,so it was anticipated that the   
experiment  might have to be repeated in
 many orientations of the machine and at
  many seasons of the year in order to  
give a series of readings from  which   
the true motion of the earth through the
 ether might be deduced4    ,for those  
who have a little algebra the           
demonstration which ,dr4 ,russell  gives
 on a subsequent page will be           
interesting as showing the  situation in
 perfectly general terms4 ,it will be   
realized that the  more complicated     
arrangement of mirrors in the experiment
 as just  described is simply an        
eightfold repetition of the simple      
experiment  as outlined by ,dr4         
,russell1 and that it was done so for   
the mere  sake of multiplying by eight  
the distances travelled and hence the   
difference in time and in phase4    ,and
 now for the grand climax4 ,the         
experiment was repeated many times1     
with the original and with other        
apparatus1 indoors and outdoors1 at  all

 seasons of the year1 with variation of 
every condition that could  imaginably  
affect the result4 ,the apparatus was   
ordinarily such that a  shift in the    
fringes of anywhere from one-tenth to   
one one-hundredth  of that which would  
have followed from any reasonable value 
for  the earth's motion through the     
ether would have been systematically    
apparent4 ,the result was uniformly     
negative4 ,at all times and in all      
directions the velocity of light past   
the earth-bound observer was  the same4 
,the earth has no motion with reference 
to the ether6    ,the amazing character 
of this result is not by any possibility
 to  be exaggerated499 ,according to one
 experiment the ether was carried  along
 by a rapidly moving body and according 
to another equally  well-planned and    
well-executed experiment a rapidly      
moving body did  not disturb the ether  
at all4 ,this was the blind alley into  
which  science had been led4#bcb        
,,the ,,contraction ,,hypothesis        

,numerous efforts were made to explain  
the contradiction499 ,it  is indeed a   
very puzzling one1 and it gave          
physicists no end of  trouble4 ,however 
,lorentz and ,fitzgerald finally put    
forward an  ingenious explanation1 to   
the effect that the actual motion of    
the earth through the ether is balanced1
 as far as the ability of  our measuring
 instruments is concerned1 by a         
contraction of these  same instruments  
in the direction of their motion4 ,this 
contraction  obviously cannot be        
observed directly because all bodies1   
including  the measuring instruments    
themselves 7which after all are only    
arbitrary guides71 will suffer the      
contraction equally4 ,according to  this
 theory1 called the ,lorentz-,fitzgerald
 contraction theory1#bgb  all bodies in 
motion suffer such contraction of their 
length in the  direction of their       
motion2#bhc the contraction being made  
evident  by our inability to observe the
 absolute motion of the earth1 which  it

 is assumed must exist4#bgb ,this would 
suffice to show why the                 
,michelson-,morley experiment gave a    
negative result1 and would preserve  the
 concept of absolute motion with        
reference to the ether4#bhc    ,this    
proposal of ,lorentz and ,fitzgerald    
loses its startling aspect  when we     
consider that all matter appears to be  
an electrical structure1  and that the  
dimensions of the electric and magnetic 
fields which  accompany the electrons of
 which it is constituted change with the
  velocity of motion4#bfg ,the forces of
 cohesion which determine the  form of a
 rigid body are held to be              
electromagnetic in nature2 the          
contraction may be regarded as due to a 
change in the electromagnetic  forces   
between the molecules4#aj ,as one writer
 has put it1 the  orientation1 in the   
electromagnetic medium1 of a body       
depending for  its very existence upon  
electromagnetic forces is not           
necessarily a  matter of indifference499

    ,granting the plausibility of all   
this1 on the basis of an                
electromagnetic theory of matter1 it    
leaves us in an unsatisfactory          
position4 ,we are left with a fixed     
ether with reference to which  absolute 
motion has a meaning1 but that motion   
remains undetected and  apparently      
undetectable4 ,further1 if we on shore  
measure the length  of a moving ship1   
using a yard-stick which is stationary  
on shore1  we shall obtain one result4  
,if we take our stick aboard it         
contracts1  and so we obtain a greater  
length for the ship4 ,not knowing our   
real motion through the ether1 we cannot
 say which is the true  length4 ,is it  
not1 then1 more satisfactory to discard 
all notion  of true length as an        
inherent quality of bodies1 and1 by     
regarding  length as the measure of a   
relation between a particular object and
  a particular observer1 to make one    
length as true as the other8#ahb  ,the  
opponents of such a viewpoint contend   

that ,michelson's result  was due to a  
fluke2 some mysterious counterbalancing 
influence was  for some reason at work1 
concealing the result which should      
normally  have been expected4 ,einstein 
refuses to accept this explanation2#aib 
he refuses to believe that all nature is
 in a contemptible conspiracy  to delude
 us499    ,the ,fitzgerald suggestion is
 further unsatisfactory because it      
assumes all substances1 of whatever     
density1 to undergo the same            
contraction2 and above all for the      
reason that it sheds no light upon      
other phenomena4#aid ,it is indeed a    
very special explanation2 that  is1 it  
applies only to the particular          
experiment in question4 ,and  indeed it 
is only one of many possible            
explanations4 ,einstein  conceived the  
notion that it might be infinitely more 
valuable to  take the most general      
explanation possible1 and then try to   
find  from this its logical             
consequences4 ,this most general        

explanation  is1 of course1 simply that 
it is impossible in any way whatever to 
measure the absolute motion of a body in
 space4#bgb ,accordingly  ,einstein     
enunciated1 first the ,special ,theory  
of ,relativity1 and  later the ,general 
,theory of ,relativity4 ,the special    
theory was so  called because it was1   
limited to uniform rectilinear and      
non-rotary  motions4 ,the general       
theory1 on the other hand1 dealt not    
only with  uniform rectilinear motions1 
but with any arbitrary motion whatever4 
,,taking ,,the ,,bull ,,by ,,the ,,horns
    ,the hypothesis of relativity       
asserts that there can be no such       
concept  as absolute position1 absolute 
motion1 absolute time2 that space and   
time are inter-dependent1 not           
independent2 that everything is relative
  to something else4 ,it thus accords   
with the philosophical notion of  the   
relativity of all knowledge4#bhc        
,knowledge is based1 ultimately1  upon  
measurement2 and clearly all measurement

 is relative1 consisting  merely in the 
application of a standard to the        
magnitude measured4 ,all  metric numbers
 are relative2 dividing the unit        
multiplies the metric  number4          
,moreover1 if measure and measured      
change proportionately1  the measuring  
number is unchanged4 ,should space with 
all its contents  swell in fixed ratio  
throughout1 no measurement could detect 
this2  nor even should it pulse         
uniformly throughout4 ,furthermore1 were
  space and space-contents in any way   
systematically transformed 7as by       
reflection in curved mirrors7 point for 
point1 continuously1 without  rending1  
no measurement could reveal this        
distortion2 experience  would proceed   
undisturbed4#bfc    ,mark ,twain said   
that the street in ,damascus which is   
called  straight1 is so called because  
while it is not as straight as a        
rainbow it is straighter than a         
corkscrew4 ,this expresses the basic    
idea of relativity--the idea of         

comparison4 ,all our knowledge is       
relative1 not absolute4 ,things are big 
or little1 long or short1  light or     
heavy1 fast or slow1 only by comparison4
 ,an atom may be  as large1 compared to 
an electron1 as is a cathedral compared 
to  a fly4 ,the relativity theory of    
,einstein emphasizes two cases of       
relative knowledge2 our knowledge of    
time and space1 and our knowledge  of   
motion4#baf ,and in each case1 instead  
of allowing the notions of  relativity  
to guide us only so far as it pleases us
 to follow them1  there abandoning them 
for ideas more in accord with what we   
find  it easy to take for granted1      
,einstein builds his structure on the   
thesis that relativity must be admitted1
 must be followed out to the  bitter    
end1 in spite of anything that it may do
 to our preconceived  notions4 ,if      
relativity is to be admitted at all1 it 
must be admitted  in toto2 no matter    
what else it contradicts1 we have no    
appeal from  its conclusions so long as 

it refrains from contradicting itself499
    ,the hypothesis of relativity was   
developed by ,einstein through  a priori
 methods1 not the more usual a          
posteriori ones4 ,that is1  certain     
principles were enunciated as probably  
true1 the consequences  of these were   
developed1 and these deductions tested  
by comparison of  the predicted and the 
observed phenomena4 ,it was in no sense 
attained  by the more usual procedure of
 observing groups of phenomena and      
formulating a law or formula which would
 embrace them and correctly  describe   
the routine or sequence of phenomena4   
,the first principle thus enunciated is 
that it is impossible to  measure or    
detect absolute translatory motion      
through space1 under  any circumstances 
or by any means4 ,the second is that the
 velocity of  light in free space       
appears the same to all observers       
regardless of  the relative motion of   
the source of light and the observer4   
,this  velocity is not affected by      

motion of the source toward or away     
from the observer1#bhc if we may for the
 moment use this expression  with its   
implication of absolute motion499 ,but  
universal relativity  insists that      
motion of the source toward the observer
 is identical  with motion of the       
observer toward the source4#bhc    ,it  
will be seen that we are at once on the 
horns of a dilemma4 ,either  we must    
give up relativity before we get fairly 
started on it1 or we  must overturn the 
foundations of common sense by admitting
 that time  and space are so constituted
 that when we go to meet an advancing   
light-impulse1 or when we retreat from  
it1 it still reaches us with  the same  
velocity as though we stood still       
waiting for it4 ,we shall  find when we 
are through with our investigation that 
common sense is  at fault2 that our     
fixed impression of the absurdity of the
 state of  affairs just outlined springs
 from a confusion between relativism    
and absolutism which has heretofore     

dominated our thought and gone          
unquestioned4 ,the impression of        
absurdity will vanish when we have      
resolved this confusion499              
,,questions ,,of ,,common ,,sense       
,but it is obvious from what has just   
been said that if we are to  adopt      
,einstein's theory1 we must make very   
radical changes in some of  our         
fundamental notions1 changes that seem  
in violent conflict with  common sense4 
,it is unfortunate that many            
popularizers of relativity  have been   
more concerned to astonish their readers
 with incredible  paradoxes than to give
 an account such as would appeal to     
sound  judgment4 ,many of these         
paradoxes do not belong essentially to  
the  theory at all4 ,there is nothing in
 the latter that an enlarged and        
enlightened common sense would not      
readily endorse4 ,but common sense  must
 be educated up to the necessary        
level4#ada    ,there was a time when it 
was believed1 as a result of centuries  

of  experience1 that the world was flat4
 ,this belief checked up with the  known
 facts1 and it could be used as the     
basis for a system of science  which    
would account for things that had       
happened and that were to  happen4 ,it  
was entirely sufficient for the time in 
which it prevailed4    ,then one day a  
man arose to point out that all the     
known facts were  equally accounted for 
on the theory that the earth was a      
sphere4 ,it  was in order for his       
contemporaries to admit this1 to say    
that so far  as the facts in hand were  
concerned they could not tell whether   
the  earth was flat or round--that new  
facts would have to be sought that      
would contradict one or the other       
hypothesis4 ,instead of this the  world 
laughed and insisted that the earth     
could not be round because  it was flat2
 that it could not be round because then
 the people would  fall off the other   
side4    ,but the field of              
experimentation widened1 and men were   

able to observe  facts that had been    
hidden from them4 ,presently a man      
sailed west and  arrived east2 and it   
became clear that in spite of previously
 accepted  facts to the contrary1 the   
earth was really round4 ,the previously 
accepted facts were then revised to fit 
the newly discovered truth2  and finally
 a new system of science came into      
being1 which accounted  for all the old 
facts and all the new ones4    ,at      
intervals this sort of thing has been   
repeated4 ,a ,galileo shows  that       
preconceived ideas with regard to the   
heavens are wrong1 and  must be revised 
to accord with his newly promulgated    
principles4 ,a  ,newton does the same   
for physics--and people unlearn the fact
  that motion has to be supported by    
continued application of force1         
substituting the new idea that it       
actually requires force to stop a       
moving body4 ,a ,harvey shows that the  
things which have been known  for       
generations about the human body are not

 so4 ,a ,lyell and a ,darwin  force men 
to throw overboard the things they have 
always believed about  the way in which 
the earth and its creatures came into   
being4 ,every  science we possess has   
passed through one or more of these     
periods  of readjustment to new facts4  
,,shifting ,,the ,,mental ,,gears       
,now we are apt to lose sight of the    
true significance of this4 ,it  is not  
alone our opinions that are altered2 it 
is our fundamental  concepts4 ,we get   
concepts wholly from our perceptions1   
making them  to fit those perceptions4  
,whenever a new vista is opened to our  
perceptions1 we find facts that we never
 could have suspected from the          
restricted viewpoint4 ,we must then     
actually alter our concepts to make  the
 new facts fit in with the greatest     
degree of harmony4 ,and we must  not    
hesitate to undertake this alteration1  
through any feeling that  fundamental   
concepts are more sacred and less freely
 to be tampered  with than derived      

facts499 ,we do1 to be sure1 want       
fundamental  concepts that are easy for 
a human mind to conceive2 but we also   
want our laws of nature to be simple4   
,if the laws begin to become1           
intricate1 why not reshape1 somewhat1   
the fundamental concepts1 in  order to  
simplify the scientific laws8           
,ultimately it is the simplicity  of the
 scientific system as a whole that is   
our principal aim4#agh    ,as a fair    
example1 see what the acceptance of the 
earth's sphericity  did to the idea     
represented by the word down4 ,with a   
flat earth1  down is a single direction1
 the same throughout the universe2 with 
a round earth1 down becomes merely the  
direction leading toward  the center of 
the particular heavenly body on which we
 happen to be  located4 ,it is so with  
every concept we have4 ,no matter how   
intrinsic  a part of nature and of our  
being a certain notion may seem1 we can 
never know that new facts will not      
develop which will show it to be a      

mistaken one4 ,today we are merely      
confronted by a gigantic example of     
this sort of thing4 ,einstein tells us  
that when velocities are attained  which
 have just now come within the range of 
our close investigation1  extraordinary 
things happen--things quite             
irreconcilable with our  present        
concepts of time and space and mass and 
dimension4 ,we are  tempted to laugh at 
him1 to tell him that the phenomena he  
suggests  are absurd because they       
contradict these concepts4 ,nothing     
could be  more rash than this4    ,when 
we consider the results which follow    
from physical velocities  comparable    
with that of light1 we must confess that
 here are  conditions which have never  
before been carefully investigated4 ,we 
must be quite as well prepared to have  
these conditions reveal some            
epoch-making fact as was ,galileo when  
he turned the first telescope  upon the 
skies4 ,and if this fact requires that  
we discard present  ideas of time and   

space and mass and dimension1 we must be
 prepared  to do so quite as thoroughly 
as our medieval fathers had to discard  
their notions of celestial perfection   
which demanded that there  be but seven 
major heavenly bodies and that          
everything center about  the earth as a 
common universal hub4 ,we must be       
prepared to revise  our concepts of     
these or any other fundamentals quite as
 severely as  did the first philosopher 
who realized that down in ,london was   
not  parallel to down in ,bagdad or on  
,mars499    ,in all ordinary terrestrial
 matters we take the earth as a fixed   
body1 light as instantaneous4 ,this is  
perfectly proper1 for such  matters4    
,but we carry our earth-acquired habits 
with us into the  celestial regions4    
,though we have no longer the earth to  
stand on1  yet we assume1 as on the     
earth1 that all measurements and        
movements  must be referred to some     
fixed body1 and are only then valid4 ,we
 cling  to our earth-bound notion that  

there is an absolute up-and-down1       
back-and-forth1 right-and-left1 in      
space4 ,we may admit that we can  never 
find it1 but we still think it is there1
 and seek to approach  it as nearly as  
possible4 ,and similarly from our earth 
experiences1  which are sufficiently in 
a single place to make possible this    
simplifying assumption1 we get the idea 
that there is one universal  time1      
applicable at once to the entire        
universe4#ada ,the difficulty  in       
accepting ,einstein is entirely the     
difficulty in getting away from  these  
earth-bound habits of thought499        
,,iv    ,,the ,,special ,,theory ,,of   
,,relativity    ,what ,einstein's ,study
 of ,uniform ,motion ,tells ,us ,about  
,time and ,space  and the ,nature of the
 ,external ,reality    ,,by ,,various   
,,contributors ,,and ,,the ,,editor     
,whatever the explanation adopted for   
the negative result of the              
,michelson-,morley experiment1 one thing
 stands out clearly3 the attempt  to    

isolate absolute motion has again       
failed499 ,einstein generalizes  this   
with all the other and older negative   
results of similar sort into  a negative
 deduction to the effect that no        
experiment is possible upon  two systems
 which will determine that one of them  
is in motion and  the other at rest4#aba
 ,he elevates the repeated failure to   
detect  absolute motion through space   
into the principle that experiment  will
 never reveal anything in the nature of 
absolute velocities4 ,he  postulates    
that all laws of nature can and should  
be enunciated in  such forms that they  
are as true in these forms for one      
observer  as for another1 even though   
these observers with their frames of    
reference be in motion relative to one  
another4#bfd    ,there are various ways 
of stating the principle of the         
relativity  of uniform motion which has 
been thus arrived at1 and which forms   
the basis of the ,special ,theory of    
,einstein4 ,if we care to emphasize  the

 rle of mathematics and the reference   
frame we may say that99  any coordinate 
system having a uniform rectilinear     
motion with  respect to the bodies under
 observation may be interchangeably used
  with any other such system in         
describing their motions2#bcb or that   
the unaccelerated motion of a system of 
reference cannot be detected  by        
observations made on this system        
alone4#aid ,or we can let this  aspect  
of the matter go1 and state the         
relativity postulate in a form  more    
intelligible to the non-mathematician by
 simply insisting that  it is impossible
 by any means whatever to distinguish   
any other than  the relative motion     
between two systems that are moving     
uniformly4 ,as  ,dr4 ,russell puts it on
 a later page1 we can assume boldly that
 the  universe is so constituted that   
uniform straight-ahead motion of an     
observer and all his apparatus will not 
produce any difference whatever  in the 
result of any physical process or       

experiment of any kind4    ,as we have  
seen1 this is entirely reasonable1 on   
philosophical grounds1  until we come to
 consider the assumptions of the past   
century with  regard to light and its   
propagation4 ,on the basis of these     
assumptions  we had expected the        
,michelson-,morley experiment to produce
 a result  negativing the notion of     
universal relativity4 ,it refused to do 
this1  and we agree with ,einstein that 
the best explanation is to return  to   
the notion of relativity1 rather than to
 invent a forced and  special hypothesis
 to account for the experiment's        
failure4 ,but we  must now investigate  
the assumptions underlying the theory of
 light1  and remove the one that        
requires the ether to serve as a        
universal  standard of absolute motion4 
,,light ,,and ,,the ,,ether    ,it is   
among the possibilities that the wave   
theory of light itself  will in the end 
be more or less seriously modified4 ,it 
is even more  definitely among the      

possibilities that the ether will be    
discarded499  ,certainly when ,lord     
,kelvin estimates that its mass per     
cubic  centimeter is                    
4#jjj1jjj1jjj1jjj1jjj1jja gram1 while   
,sir ,oliver ,lodge  insists that the   
correct figure is #a1jjj1jjj1jjj1jjj1jjj
 grams1 it is  quite evident that we    
know so little about it that it is      
better to  get along without it if we   
can4#baf ,but to avoid confusion we must
  emphasize that ,einstein makes no     
mention whatsoever of the ether2  his   
theory is absolutely independent of any 
theory of the ether4#aci  ,save as he   
forbids us to employ the ether as a     
standard of absolute  motion1 ,einstein 
does not in the least care what         
qualities we assign  to it1 or whether  
we retain it at all4 ,his demands are   
going to be made  upon light itself1 not
 upon the alleged medium of light       
transmission4    ,when two observers in 
relative motion to one another measure  
their  velocities with respect to a     

third material object1 they expect to   
get  different results4 ,their          
velocities with regard to this object   
properly  differ1 for it is no more to  
be taken as a universal super-observer  
than either of them4 ,but if they get   
different results when they come  to    
measure the velocity with which light   
passes their respective  systems1       
relativity is challenged4 ,light is with
 some propriety  to be regarded as a    
universal observer2 and if it will      
measure our  velocities against each    
other we cannot deny it rank as an      
absolute  standard4 ,if we are not      
prepared to abandon universal           
relativity1  and adopt one of the fluke 
explanations for the ,michelson-,morley 
result1 we must boldly postulate that in
 free space light presents the  same    
velocity ,c to all observers--whatever  
the source of the light1  whatever the  
relative motion between source and      
observer1 whatever the  relative motion 
between the several observers4 ,the     

departure here from  the old assumption 
lies in the circumstance that the old   
physics with  its ether assigned to     
light a velocity universally constant in
 this  ether2 we have stopped talking   
about the medium and have made the      
constant ,c refer to the observer's     
measured value of the velocity  of light
 with regard to himself4    ,we are     
fortified in this assumption by the     
,michelson-,morley  result and by all   
other observations bearing directly upon
 the  matter4 ,nevertheless1 as ,mr4    
,francis says in his essay1 we feel     
instinctively that space and time are   
not so constituted as to  make it       
possible1 if ,i pass you at #ajj miles  
per hour1 for the  same light-impulse to
 pass us both at the same speed ,c499   
,the  implicit assumptions underlying   
this feeling1 be they true or false1    
are now so interwoven with the commonly 
received notions of space and  time that
 any theory which questions them has all
 the appearance of  a fantastic and     

unthinkable thing4#aae ,we cannot1      
however1 go back  on our relativity2 so 
when99 ,einstein shows us that an       
entirely new  set of time and space     
concepts is necessary to reconcile      
universe  relativity with this          
fundamental fact of the absolute        
constancy of  the observed velocity of  
light in vacuo1#ah all that is left for 
us  to do is to inquire what revisions  
are necessary1 and submit to them499    
,the conceptual difficulties of the     
theory arise principally from           
attributing to space and time the       
properties of things4 ,no portion of    
space can be compared with another1 save
 by convention2 it is things  which we  
compare4 ,no interval of time can be    
compared with another1  save by         
convention4 ,the first has gone when the
 second becomes  now4#adi ,it is events 
that we compare1 through the            
intervention  of things4 ,our           
measurements are never of space or of   
time1 but only  of the things and the   

events that occupy space and time4 ,and 
since  the measurements which we deal   
with as though they were of space and   
of time lie at the foundation of all    
physical science1 while at the  same    
time themselves constituting1 as we have
 seen1 the only reality  of which we are
 entitled to speak1 it is in order to   
examine with  the utmost care the       
assumptions underlying them4 ,that there
 are such  assumptions is clear--the    
very possibility of making measurements 
is  itself an assumption1 and every     
technique for carrying them out rests   
on an assumption4 ,let us inquire which 
of these it is that relativity  asks us 
to revise499          ,,the             
,,measurement ,,of ,,time ,,and ,,space 
,time is generally conceived as         
perfectly uniform4 ,how do we judge     
about it8 ,what tells us that the second
 just elapsed is equal to the  one      
following8 ,by the very nature of time  
the superposition of its  successive    
intervals is impossible4 ,how then can  

we talk about the  relative duration of 
these intervals8 ,it is clear that any  
relationship  between them can only be  
conventional4#agh ,as a matter of fact1 
we  habitually measure time in terms of 
moving bodies4 ,the simplest method  is 
to agree that some entity moves with    
uniform velocity4 ,it will be           
considered as travelling equal distances
 in equal intervals of time1  the       
distances to be measured as may be      
specified by our assumptions  governing 
this department of investigation4#agi   
,the motions of the  earth through which
 we ultimately define the length of day 
and year1  the division of the former   
into #hf1djj equal intervals as defined 
by  the motions of pendulum or balance  
wheel through equal distances1 are      
examples of this convention of time     
measurement4 ,even when we correct  the 
motions of the earth1 on the basis of   
what our clocks tell us of  these       
motions1 we are following this lead2 the
 earth and the clocks  fall out1 it is  

plain that one of them does not satisfy 
our assumption  of equal lengths in     
equal times1 and we decide to believe   
the clock499    ,the foregoing          
concerning time may be accepted as      
inherent in  time itself4 ,but          
concerning lengths it may be thought    
that we  are able to verify absolutely  
their equality and especially their     
invariability4 ,let us have the audacity
 to verify this statement4 ,we  have two
 lengths1 in the shape of two rods1     
which coincide perfectly  when brought  
together4 ,what may we conclude from    
this coincidence8 ,only  that the two   
rods so considered have equal lengths at
 the same  place in space and at the    
same moment4 ,it may very well be that  
each  rod has a different length at     
different locations in space and at     
different times2 that their equality is 
purely a local matter4 ,such  changes   
could never be detected if they affected
 all objects in the  universe4 ,we      
cannot even ascertain that both rods    

remain straight  when we transport them 
to another location1 for both can very  
well  take the same curvature and we    
shall have no means of detecting it4    
,euclidean geometry assumes that        
geometrical objects have sizes and      
shapes independent of position and of   
orientation in space1 and  equally      
invariable in time4 ,but the properties 
thus presupposed are  only conventional 
and in no way subject to direct         
verification4 ,we  cannot even ascertain
 space to be independent of time1       
because when  comparing geometrical     
objects we have to conceive them as     
brought to  the same place in space and 
in time4#agh ,even the statement that   
when they are made to coincide their    
lengths are equal is1 after  all1 itself
 an assumption inherent in our ideas of 
what constitutes  length4 ,and certainly
 the notion that we can shift them from 
place  to place and from moment to      
moment1 for purposes of comparison1 is  
an assumption2 even ,euclid1 loose as he

 was from modern standards  in this     
business of axioms1 knew this and       
included a superposition  axiom among   
his assumptions4    ,as a matter of     
fact1 this procedure for determining    
equality of lengths  is not always      
available4 ,it assumes1 it will be      
noted1 that we have  free access to the 
object which is to be measured--which is
 to say1  it assumes that this object is
 at rest with respect to us4 ,if it  is 
not so at rest1 we must employ at least 
a modification of this  method2 a       
modification that will in some manner   
involve the sending  of signals4 ,even  
when we employ the ,euclidean method of 
superposition  directly1 we must be     
assured that the respective ends of the 
lengths  under comparison coincide at   
the same time4 ,the observer cannot be  
present at both ends simultaneously2 at 
best he can only be present  at one end 
and receive a signal from the other end4
          ,,the ,,problem ,,of          
,,communication    ,accordingly1 in     

making the necessary assumptions to     
cover the matter  of measuring lengths1 
we must make one with regard to the     
character  of the signals which are to  
be employed for this purpose4 ,if we    
could assume a system of signalling that
 would consume no time in  transmission 
all would be simple enough4 ,but we have
 no experience with  such a system4     
,even if we believe that it ought to be 
possible thus  to transmit signals at   
infinite velocity1 we may not1 in the   
absence  of our present ability to do   
this1 assume that it is possible4 ,so we
  may only assume1 with ,einstein1 that 
for our signals we shall employ  the    
speediest messenger with which we are at
 present acquainted4 ,this  of course is
 light1 the term including any of the   
electromagnetic  impulses that travel at
 the speed ,c4    ,of course in the vast
 majority of cases the distance that any
 light  signal in which we are          
interested must go to reach us is so    
small that  the time taken by its       

transmission can by no means be         
measured4 ,we  are then1 to all intents 
and purposes1 at both places--the point 
of  origin of the signal and the point  
of receipt--simultaneously4 ,but  this  
is not the question at all4 ,waiving the
 fact that in astronomical              
investigations this approximation no    
longer holds1 the fact remains  that it 
is1 in every case1 merely an            
approximation4 ,approximations are  all 
right in observations1 where we know    
that they are approximations  and act   
accordingly4 ,but in the conceptual     
universe that parallels the  external   
reality1 computation is as good an agent
 of observation as  visual or auditory  
or tactile sensation2 if we can compute 
the error  involved in a wrong procedure
 the error is there1 regardless of      
whether  we can see it or not4 ,we must 
have methods which are conceptually     
free from error2 and if we attempt to   
ignore the velocity of our  light       
signals we do not meet this condition4  

,the measurement of lengths demands that
 we have a criterion of  simultaneity   
between two remote points--remote in    
inches or remote  in light-years1 it    
does not matter which4 ,there is no     
difficulty in  defining simultaneity of 
two events that fall in the same        
point--or  rather1 in agreeing that we  
know what we mean by such simultaneity4 
,but  with regard to two events that    
occur in remote places there may be a   
question4 ,a scientific definition      
differs from a mere description in  that
 it must afford us a means of testing   
whether a given item comes  under the   
definition or not4 ,there is some       
difficulty in setting up  a definition  
of simultaneity between distant events  
that satisfies  this requirement4 ,if we
 try simply to fall back upon our       
inherent  ideas of what we mean by the  
same instant we see that this is not    
adequate4 ,we must lay down a procedure 
for determining whether two  events at  
remote points occur at the same instant1

 and check up  alleged simultaneity by  
means of this procedure4    ,einstein   
says1 and we must agree with him1 that  
he can find but one  reasonable         
definition to cover this ground4 ,an    
observer can tell  whether he is located
 half way between two points of his     
observation2  he can have mirrors set up
 at these points1 send out              
light-signals1  and note the time at    
which he gets back the reflection4 ,he  
knows  that the velocity of both        
signals1 going and coming1 is the same2 
if he observes that they return to him  
together so that their time of  transit 
for the round trip is the same1 he must 
accept the distances as  equal4 ,he is  
then at the mid-point of the line       
joining the two points  under           
observation2 and he may define          
simultaneity as follows1 without        
introducing anything new or             
indeterminate3 ,two events are          
simultaneous  if an observer midway     
between them sees them at the same      

instant1 by  means1 of course1 of light 
originating at the points of            
occurrence499    ,it is this definition 
of simultaneity1 coupled with the       
assumption  that all observers1 on      
whatever uniformly moving systems1 would
 obtain  the same experimental value for
 the velocity of light1 that leads to   
the apparent paradoxes of the ,special  
,theory of ,relativity4 ,if it be  asked
 why we adopt it1 we must in turn ask   
the inquirer to propose  a better system
 for defining simultaneous events on    
different moving  bodies4#aih    ,there 
is nothing in this definition to        
indicate1 directly1 whether             
simultaneity persists for all observers1
 or whether it is relative1  so that    
events simultaneous to one observer are 
not so to another4 ,the  question must  
then be investigated2 and the answer1 of
 course1 will  hinge upon the           
possibility of making proper allowances 
for the time of  transit of the light   
signals that may be involved4 ,it seems 

as though  this ought to be possible2   
but a simple experiment will indicate   
that  it is not1 unless the observers   
involved are at rest with respect to    
one another4          ,,an ,,einsteinian
 ,,experiment    ,let us imagine an     
indefinitely long1 straight railroad    
track1 with  an observer located        
somewhere along it at the point ,m4     
,according to  the convention suggested 
above1 he has determined points ,a and  
,b  in opposite directions from him     
along the track1 and equally distant    
from him4 ,we shall imagine1 further1   
than a beneficent ,providence  supplies 
two lightning flashes1 one striking at  
,a and one at ,b1  in such a way that   
observer ,m finds them to be            
simultaneous4    ,while all this is     
going on1 a train is passing--a very    
long train1  amply long enough to       
overlap the section ,,amb of the track4 
,among  the passengers there is one1    
whom we may call ,m'1 who is directly   
opposite ,m at the instant when1        

according to ,m1 the lightning  strikes4
 ,observe he is not opposite ,m when ,m 
sees the flashes1  but a brief time     
earlier--at the instant when1 according 
to ,m's  computation1 the simultaneous  
flashes occurred4 ,at this instant there
  are definitely determined the points  
,a' and ,b'1 on the train2  and since we
 may quite well think of the two        
systems--train-system  and              
track-system--as in coincidence at this 
instant1 ,m' is midway  between ,a' and 
,b'1 and likewise is midway between ,a  
and ,b4    ,now if we think of the train
 as moving over the track in the        
direction  of the arrow1 we see very    
easily that ,m' is running away from the
  light from ,a and toward that from ,b1
 and that1 despite--or if  you prefer   
because of--the uniform velocity of     
these light signals1  the one from ,b   
reaches him1 over a slightly shorter    
course1 sooner  than the one from ,a1   
over the slightly longer course4 ,when  
the light  signals reach ,m1 ,m' is no  

longer abreast of him but has moved     
along  a wee bit1 so that at this       
instant when ,m has the two signals1 one
  of these has passed ,m' and the other 
has yet to reach him4 ,the upshot  is   
that the events which were simultaneous 
to ,m are not so to ,m'4    ,it will    
probably be felt that this result is due
 to our having1  somewhat unjustifiably 
and inconsistently1 localized on the    
train  the relative motion between train
 and track4 ,but if we think of  the    
track as sliding back under the train in
 the direction opposite  to the arrow1  
and carrying with it the points ,a and  
,b2 and if  we remember that this in no 
way affects ,m's observed velocity of   
light or the distances ,,am and ,,bm as 
he observes them3 we can  still accept  
his claim that the flashes were         
simultaneous4 ,then we  have again the  
same situation3 when the flashes from ,a
 and from  ,b reach ,m at the same      
moment1 in his new position a trifle to 
the left of his initial position of the 

diagram1 the flash from ,a  has not yet 
reached ,m' in his original position    
while that from  ,b has passed him4     
,regardless of what assumption we make  
concerning  the motion between          
train-system and track-system1 or more  
elegantly  regardless of what coordinate
 system we use to define that motion1   
the event at ,b precedes that at ,a in  
the observation of ,m'4 ,if  we         
introduce a second train moving on the  
other track in the opposite  direction1 
the observer on it will of course find  
that the flash at ,a  precedes that at  
,b--a disagreement not merely as to     
simultaneity but  actually as to the    
order of two events6 ,if we conceive the
 lightning  as striking at the points   
,a' and ,b' on the train1 these points  
travel with ,m' instead of with ,m2 they
 are fixed to his coordinate  system    
instead of to the other4 ,if you carry  
out the argument now1  you will find    
that when the flashes are simultaneous  
to ,m'1 the  one at ,a precedes that at 

,b in ,m's observation4    ,a large     
number of experiments more or less      
similar in outline to  this one can be  
set up to demonstrate the consequences1 
with regard  to measured values of time 
and space1 of relative motion between   
two  observers4 ,i do not believe that a
 multiplicity of such demonstrations    
contributes to the intelligibility of   
the subject1 and it is for  this reason 
that ,i have cut loose from immediate   
dependence upon the  essayists in this  
part of the discussion1 concentrating   
upon the single  experiment to which    
,einstein himself gives the place of    
importance4          ,,who ,,is ,,right8
    ,we may permit ,mr4 ,francis to     
remind us here that neither ,m nor  ,m' 
may correct his observation to make it  
accord with the other  fellow's4 ,the   
one who does this is admitting that the 
other is at  absolute rest and that he  
is himself in absolute motion2 and this 
cannot be4 ,they are simply in          
disagreement as to the simultaneity of  

two events1 just as two observers might 
be in disagreement about the  distance  
or the direction of a single event4     
,this can mean nothing  else than that1 
under the assumptions we have made1     
simultaneity is  not an absolute        
characteristic as we had supposed it to 
be1 but1 like  distance and direction1  
is in fact merely a relation between    
observer  and objective1 and therefore  
depends upon the particular observer    
who happens to be operating and upon the
 reference frame he is using4    ,but   
this is serious4 ,my time measurements  
depend ultimately upon  my space        
measurements2 the latter1 and hence     
both1 depend closely  upon my ideas of  
simultaneity4 ,yours depend upon your   
reading of  simultaneity in precisely   
the same way499 ,suppose the observer on
  the track1 in the above experiment1   
wants to measure the length of          
something on the car1 or the observer on
 the car something on the  track4 ,the  
observer1 or his assistant1 must be at  

both ends of the  length to be measured 
at the same time1 or get simultaneous   
reports  in some way from these ends2   
else they will obtain false results4 ,it
  is plain1 then1 that with different   
criteria of what the same time  is1 the 
observers in the two systems may get    
different values for  the measured      
lengths in question4#bbj    ,who is     
right8 ,according to the principle of   
relativity a decision on  this question 
is absolutely impossible4 ,both parties 
are right from  their own points of     
view2 and we must admit that two events 
in two  different places may be         
simultaneous for certain observers1 and 
yet  not simultaneous for other         
observers who move with respect to the  
first  ones4 ,there is no contradiction 
in this statement1 although it is not   
in accordance with common opinion1 which
 believes simultaneousness to  be       
something absolute4 ,but this common    
opinion lacks foundation4 ,it  cannot be
 proved by direct perception1 for       

simultaneity of events  can be perceived
 directly1#bd and in a manner involving 
none of our  arbitrary assumptions199   
only if they happen at the same place2  
if the events are distant from each     
other1 their simultaneity or  succession
 can be stated only through some method 
of communicating by  signals4 ,there is 
no logical reason why such a method     
should not lead  to different results   
for observers who move with regard to   
one another4    ,from what we have said1
 it follows immediately that in the new 
theory  not only the concept of         
simultaneousness but also that of       
duration is  revealed as dependent on   
the motion of the observer4#bd          
,demonstration  of this should be       
superfluous2 it ought to be plain       
without argument  that if two observers 
cannot agree whether two instants are   
the same  instant or not1 they cannot   
agree on the interval of time between   
instants4 ,in the very example which we 
have already examined1 one  observer    

says that a certain time-interval is    
zero1 and another gives  it a value     
different from zero4 ,the same thing    
happens whenever the  observers are in  
relative motion499 ,two physicists who  
measure the  duration of a physical     
process will not obtain the same result 
if  they are in relative motion with    
regard to one another4    ,they will    
also find different results for the     
length of a body4 ,an  observer who     
wants to measure the length of a body   
which is moving  past him must in one   
way or another hold a measuring rod     
parallel to  its motion and mark those  
points on his rod with which the ends of
  the body come into simultaneous       
coincidence4 ,the distance between the  
two marks will then indicate the length 
of the body4 ,but if the two  markings  
are simultaneous for one observer1 they 
will not be so for  another one who     
moves with a different velocity1 or who 
is at rest1  with regard to the body    
under observation4 ,he will have to     

ascribe  a different length to it4 ,and 
there will be no sense in asking which  
of them is right3 length is a purely    
relative concept1 just as well  as      
duration4#bd          ,,the ,,relativity
 ,,of ,,time ,,and ,,space    ,the      
degree to which distance and time become
 relative instead of  absolute          
quantities under the ,special ,theory of
 ,relativity can be  stated very        
definitely4 ,in the first place1 we must
 point out that the  relativity of      
lengths applies with full force only to 
lengths that lie  parallel to the       
direction of relative motion4 ,those    
that lie exactly  perpendicular to that 
direction come out the same for both    
observers2  those that lie obliquely to 
it show an effect1 depending upon the   
angle1 which of course becomes greater  
and greater as the direction  of        
parallelism is approached4    ,the      
magnitude of the effect is easily       
demonstrated1 but with this             
demonstration we do not need to be      

concerned here4 ,it turns out  that if  
an observer moving with a system finds  
that a certain time  interval in the    
system is ,t seconds and that a certain 
length in  the system is ,l inches1 then
 an observer moving parallel with ,l    
and with a velocity v relative to the   
system will find for these  the         
respective values ,t div ,k and ,l times
 ,k1 where        ,k  sqrt#a -          
v#b_/;,c#b4    ,c in this expression of 
course represents the velocity of light4
 ,it  will be noted that the fraction   
v#b_/;,c#b is ordinarily very small2    
that the expression under the radical is
 therefore less than #a but  by a very  
slight margin2 and that the entire      
expression ,k is itself  therefore less 
than #a but by an even slighter margin4 
,this means1 then1  that the observer   
outside the system finds the lengths in 
the system  to be a wee bit shorter and 
the time intervals a wee bit longer than
  does the observer in the system4      
,another way of putting the matter  is  

based1 ultimately1 upon the fact that in
 order for the observer  in the system  
to get the larger value for distance and
 the smaller  value for time1 his       
measuring rod must go into the distance 
under  measurement more times than that 
of the moving observer1 while his  clock
 must beat a longer second in order that
 less of them shall be  recorded in a   
given interval between two events4 ,so  
it is often said  that the measuring rod
 as observed from without is contracted 
and  the clock runs slow4 ,this does not
 impress me as a happy statement1       
either in form or in content499    ,the 
argument that these formulae are        
contradicted by human experience  can be
 refuted by examining a concrete        
instance4 ,if a train is  #a1jjj feet   
long at rest1 how long will it be when  
running a mile a  minute8#bcb ,i have   
quoted this question exactly as it      
appears in the  essay from which it is  
taken1 because it is such a capital     
example of  the objectionable way in    

which this business is customarily put4 
,for  the statement that lengths        
decrease and time-intervals increase    
with  velocity is not true in just this 
form4 ,the velocity1 to have meaning1   
must be relative to some external       
system2 and it is the observations  from
 that external system that are affected4
 ,so long as we confine  ourselves to   
the system in which the alleged         
modifications of size  are stated as    
having taken place1 there is nothing to 
observe that  is any different from what
 is usual2 there is no way to establish 
that we are enjoying a velocity1 and in 
fact within the intent of  the          
relativity theory we are not enjoying a 
velocity1 for we are  moving with the   
objects which we are observing4 ,it is  
inter-systemic  observations1 and these 
alone1 that show the effect4 ,when we   
travel  with the system under           
observation1 we get the same results as 
any  other observer on this system2 when
 we do not so travel1 we must  conduct  

our observations from our own system1 in
 relative motion to  the other1 and     
refer our results to our system4    ,now
 when no particular observer is         
specified1 we must of course  assume an 
observer connected with the train1 or   
with whatever the body  mentioned4 ,to  
that observer it doesn't make the       
slightest difference  what the train    
does2 it may stand at rest with respect 
to some external  system or it may move 
at any velocity whatsoever2 its length  
remains  always #a1jjj feet4 ,in order  
for this question to have the           
significance  which its propounder means
 it to have1 ,i must restate it as      
follows3  ,a train is #a1jjj feet long  
as measured by an observer travelling   
with  it4 ,if it passes a second        
observer at #fj miles per hour1 what is 
its  length as observed by him8 ,the    
answer is now easy499 ,according to the 
formula the length of the moving train  
as seen from the ground will be         
#a1jjj times sqrt#a - 7#hh7b _/         

7#ahf1jjj times #e1bhj7b                
#iii4iii1iii1iii1iif    feet1 a change  
entirely too small for detection by the 
most delicate  instruments4 ,examination
 of the expression ,k shows that in so  
far  as terrestrial movements of        
material objects are concerned it is    
equal to #abcb within a far smaller     
margin than we can ever hope to  make   
our observations4 ,even the diameter of 
the earth1 as many of the  essayists    
point out1 will be shortened only #b    
#a_/b inches for an outside  observer   
past whom it rushes with its orbital    
speed of #ah4e miles per  second4 ,but  
slight as the difference may be in these
 familiar cases1  its scientific        
importance remains the same499          
,,relativity ,,and ,,reality    ,a      
simple computation shows that this      
effect is exactly the amount  suggested 
by ,lorentz and ,fitzgerald to explain  
the ,michelson-,morley  experiment4#ahh 
,this ought not to surprise us1 since   
both that  explanation and the present  

one are got up with the same purpose4   
,if  they both achieve that purpose they
 must1 numerically1 come to the  same   
thing in any numerical case4 ,it is1    
however1 most emphatically  to be       
insisted that the present shortening of 
lengths99 no longer  appears as a       
physical shortening caused by absolute  
motion through  the ether but is simply 
a result of our methods of measuring    
space  and time4#ahh ,where ,fitzgerald 
and ,lorentz had assumed that a body in 
motion has its dimensions shortened in  
the direction of its motion1#bbj  this  
very form of statement ceases to possess
 significance under  the relativity     
assumption499 ,for if we cannot tell    
which of two  bodies is moving1 which   
one is shortened8 ,the answer is1       
both--for  the other fellow4 ,for each  
frame of reference there is a scale of  
length and a scale of time1 and these   
scales for different frames  are related
 in a manner involving both the length  
and the time4#bbj  ,but we must not     

yield to the temptation to say that all 
this is not  real2 the confinement of a 
certain scale of length and of time to  
a single observation system does not in 
the least make it unreal499  ,the       
situation is real--as real as any other 
physical event4#afe    ,the word        
physical is used in two senses in the   
above paragraph4 ,it  is denied that the
 observed variability in lengths        
indicates any  physical contraction or  
shrinkage2 and on the heels of this it  
is asserted that this observed          
variability is of itself an actual      
physical event4 ,it is difficult to     
express in words the distinction        
between the two senses in which the term
 physical is employed in these  two     
statements1 but ,i think this           
distinction ought to be clear once  its 
existence is emphasized4 ,there is no   
material contraction2 it is  not right  
to say that objects in motion contract  
or are shorter2 they  are not shorter to
 an observer in motion with them4 ,the  

whole thing is  a phenomenon of         
observation4 ,the definitions which we  
are obliged to  lay down and the        
assumptions which we are obliged to make
 in order1  first1 that we shall be able
 to measure at all1 and second1 that we 
shall be able to escape the inadmissible
 concept of absolute motion1  are such  
that certain realities which we had     
supposed ought to be the  same for all  
observers turn out not to be the same   
for observers who  are in relative      
motion with respect to one another4 ,we 
have found this  out1 and we have found 
out the numerical relation which holds  
between  the reality of the one observer
 and that of the other4 ,we have found  
that this relation depends upon nothing 
save the relative velocity  of the two  
observers4 ,as good a way of emphasizing
 this as any is to  point out that two  
observers who have the same velocity    
with respect  to the system under       
examination 7and whose mutual relative  
velocity  is therefore zero7 will always

 get the same results when measuring    
lengths and times on that system4 ,the  
object does not go through any  process 
of contraction2 it is simply shorter    
because it is observed  from a station  
with respect to which it is moving4     
,similar remarks  might be made about   
the time effect2 but the time-interval  
is not so  easily visualized as a       
concrete thing and hence does not offer 
such  temptation for loose statement4   
,the purely relative aspect of the      
matter is further brought out if  we    
consider a single example both backwards
 and forwards4 ,systems ,s  and ,s' are 
in relative motion4 ,an object in ,s    
which to an observer  in ,s is ,l units 
long1 is shorter for an observer in     
,s'--shorter  by an amount indicated    
through the correction factor ,k4 ,now  
if  we have1 in the first instance1 made
 the objectionable statement  that      
objects are shorter in system ,s' than  
they are in ,s1 it  will be quite       
natural for us to infer from this that  

objects in ,s  must be longer than those
 in ,s'2 and from this to assert that   
when  the observer in ,s measures       
objects lying in ,s'1 he gets for them  
greater lengths than does the home      
observer in ,s'4 ,but if we have1  in   
the first instance1 avoided the         
objectionable statement referred  to1 we
 shall be much better able to realize   
that the whole business is  quite       
reciprocal2 that the phenomena are      
symmetric with respect to the  two      
systems1 to the extent that we can      
interchange the systems in any  of our  
statements without modifying the        
statements in any other way4    ,objects
 in ,s appear shorter and times in ,s   
appear longer to the  external moving   
observer in ,s' than they do to the     
domestic  observer in ,s4 ,exactly in   
the same way1 objects in ,s' appear     
shorter to observers in the foreign     
system ,s than to the home  observer in 
,s'1 who remains at rest with respect to
 them4 ,i think  that when we get the   

right angle upon this situation1 it     
loses the  alleged startling character  
which has been imposed upon it by many  
writers4 ,the apparent size of the      
astronomer is an analogy in  point4     
,objects on the moon1 by virtue of their
 great distance1 look  smaller to       
observers on the earth than to observers
 on the moon4 ,do  objects on the earth1
 on this account1 look larger to a moon 
observer  than they do to us8 ,they do  
not2 any suggestion that they do we     
should  receive with appropriate scorn4 
,the variation in size introduced by    
distance is reciprocal1 and this        
reciprocity does not in the least       
puzzle us4 ,why1 then1 should that      
introduced by relative motion  puzzle   
us8          ,,time ,,and ,,space ,,in  
,a ,,single ,,package    ,our old1      
accustomed concepts of time and space1  
which have grown  up through countless  
generations of our ancestors1 and been  
handed  down to us in the form in which 
we are familiar with them1 leave no     

room for a condition where time         
intervals and space intervals are  not  
universally fixed and invariant4 ,they  
leave no room for us to  say that99 one 
cannot know the time until he knows     
where he is1  nor where he is until he  
knows the time1#bbj nor either time or  
place until he knows something about    
velocity4 ,but in this concise          
formulation of the difference between   
what we have always believed  and what  
we have seen to be among the            
consequences of ,einstein's  postulates 
of the universal relativity of uniform  
motion1 we may  at once locate the      
assumption which1 underlying all the old
 ideas1  is the root of all the trouble4
 ,the fact is we have always supposed   
time and space to be absolutely distinct
 and independent entities499    ,the    
concept of time has ever been one of the
 most absolute of all  the categories4  
,it is true that there is much of the   
mysterious about  time2 and philosophers
 have spent much effort trying to clear 

up  the mystery--with unsatisfactory    
results4 ,however1 to most persons  it  
has seemed possible to adopt an         
arbitrary measure or unit of  duration  
and to say that this is absolute1       
independent of the state  of the body or
 bodies on which it is used for         
practical purposes4#bgb  ,time has thus 
been regarded as something which of     
itself flows on  regularly and          
continuously1 regardless of physical    
events concerning  matter4#aej ,in other
 words1 according to this view1 time is 
not  affected by conditions or motions  
in space4#bgb ,we have deliberately     
chosen to ignore the obvious fact that  
time can never appear to us1  be        
measured by us1 or have the least       
significance for us1 save as  a measure 
of something that is closely tied up    
with space and with  material           
space-dimensions4 ,not merely have we   
supposed that time and  space are       
separated in nature as in our easiest   
perceptions1 but we  have supposed that 

they are of such fundamentally distinct 
character  that they can never be tied  
up together4 ,in no way whatever1       
assumes  the ,euclidean and ,newtonian  
intellect1 may space ever depend upon   
time or time upon space4 ,this is the   
assumption which we must remove  in     
order to attain universal relativity2   
and while it may come hard1  it will not
 come so hard as the alternative4 ,for  
this alternative is  nothing other than 
to abandon universal relativity4 ,this  
course would  leave us with logical     
contradictions and discrepancies that   
could not  be resolved by any revision  
of fundamental concepts or by any       
cleaning  out of the ,augean stables of 
old assumptions2 whereas the relativity 
doctrine as built up by ,einstein       
requires only such a cleaning out in    
order to leave us with a strictly       
logical and consistent whole4 ,the  rle 
of ,hercules is a very difficult one for
 us to play4 ,einstein  has played it   
for the race at large1 but each of us   

must follow him  in playing it for      
himself4          ,,some ,,further      
,,consequences    ,i need not trespass  
upon the subject matter of those essays 
which  appear in full by going here into
 any details with regard to the  manner 
in which time and space are finally     
found to depend upon one  another and to
 form the parts of a single universal   
whole4 ,but ,i  may appropriately point 
out that if time and space are found to 
be  relative1 we may surely expect some 
of the less fundamental concepts  that  
depend upon them to be relative also4   
,in this expectation we are  not        
disappointed4 ,for one thing199 mass has
 always been assumed to  be a constant1 
independent of any motion or energy     
which it might  possess4 ,just as       
lengths and times depend upon relative  
motion1  however1 it is found that mass1
 which is the remaining factor in  the  
expression for energy due to motion1    
also depends upon relative  velocities4 
,the dependence is such that if a body  

takes up an amount  of energy ,e with   
respect to a certain system1 the body   
behaves1  to measurements made from that
 system1 as though its mass had been    
increased by an amount ,e_/,c#b1 where  
,c is as usual the velocity  of         
light4#aid    ,this should not startle  
us4 ,the key to the situation lies in   
the  italicized words above1 which      
indicate that the answer to the query   
whether a body has taken up energy or   
not depends upon the seat of            
observation4 ,if ,i take up my location 
on the system ,s1 and you  on the system
 ,s'1 and if we find that we are in     
relative motion1  we must make some     
assumption about the energy which was   
necessary1  initially1 to get us into   
this condition4 ,suppose we are on two  
passing trains499 ,the chances are that 
either of us will assume that  he is at 
rest and that it is the other train     
which moves1 although  if sufficiently  
sophisticated one of us may assume that 
he is moving  and that the other train  

is at rest4#bgb ,whatever our           
assumption1  whatever the system1 the   
localization of the energy that is      
carried  in latent form by our systems  
depends upon this assumption4 ,indeed1  
if our systems are of differing mass1   
our assumptions will even govern  our   
ideas of the amount of energy which is  
represented by our relative  motion2 if 
your system be the more massive1 more   
energy would have to  be localized in it
 than in mine to produce our relative   
motion4 ,if  we did not have the        
universal principle of relativity to    
forbid1  we might make an arbitrary     
assumption about our motions and hence  
about our respective latent energies2 in
 the presence of this veto1  the only   
chance of adjustment lies in our masses1
 which must differ  according to whether
 you or ,i observe them499    ,for most 
of the velocities with which we are     
familiar ,e_/,c#b is1  like the         
difference between ,k and unity1 such an
 extremely small  quantity that the most

 delicate measurements fail to detect   
it4 ,but  the electrons in a highly     
evacuated tube and the particles shot   
out from radioactive materials attain in
 some cases velocities as  high as      
eight-tenths that of light4 ,when we    
measure the mass of such  particles at  
different velocities we find that it    
actually increases  with the velocity1  
and in accordance with the foregoing    
law4#aid  ,this observation1 in fact1   
antedates ,einstein's explanation1 which
  is far more satisfactory than the     
earlier differentiation between  normal 
mass and electrical mass which was      
called upon to account  for the         
increase499    ,but if the quantity     
,e_/,c#b is to be considered as an      
actual increase  in mass1 may it not be 
possible that all mass is energy8 ,this 
would  lead to the conclusion that the  
energy stored up in any mass is  m,c#b4 
,the value is very great1 since ,c is so
 large2 but it is in  good agreement    
with the internal energy of the atom as 

calculated  from other considerations4  
,it is obvious that conservation of mass
  and of momentum cannot both hold good 
under a theory that translates  the one 
into the other4 ,mass is then not       
considered by ,einstein as  conservative
 in the ordinary sense1 but it is the   
total quantity of  mass plus energy in  
any closed system that remains constant4
 ,small  amounts of energy may be       
transformed into mass1 and vice         
versa4#aid    ,other features of the    
theory which are often displayed as     
consequences  are really more in the    
nature of assumptions4 ,it will be      
recalled that  when we had agreed upon  
the necessity of employing signals of   
some  sort1 we selected as the means of 
signalling the speediest messenger  with
 which we happened to be acquainted4    
,our subsequent difficulties  were      
largely due to the impossibility of     
making a proper allowance  for this     
messenger's speed1 even though we knew  
its numerical value2  and as a          

consequence1 this speed enters into our 
formulae4 ,now we have  not said in so  
many words that ,c is the greatest speed
 attainable1  but we have tacitly       
assumed that it is4 ,we need not1       
therefore1 be  surprised if our formulae
 give us absurd results for speeds      
higher than  ,c1 and indicate the       
impossibility of ever attaining these4  
,whatever  we put into a problem the    
algebra is bound to give us back4 ,if we
 look  at our formula for ,k1 we see    
that in the event of v equalling ,c1    
lengths become zero and times infinite4 
,the light messenger itself1  then1 has 
no dimension2 and for it time stands    
still4    ,if we suppose v to be greater
 than ,c1 we get even more bizarre      
results1 for then the factor ,k is the  
square root of a negative  number1 or as
 the mathematician calls it an imaginary
 quantity2  and with it1 lengths and    
times become imaginary too4    ,the fact
 that time stops for it1 and the fact   
that it is the limiting  velocity1 give 

to ,c certain of the attributes of the  
mathematician's  infinity4 ,certainly if
 it can never be exceeded1 we must have 
a new  formula for the composition of   
velocities4 ,otherwise when my system   
passes yours at a speed of #ajj1jjj     
miles per second1 while yours  passes a 
third in the same direction at the same 
velocity1 ,i shall  be passing this     
third framework at the forbidden        
velocity of #bjj1jjj  miles per         
second--greater than ,c4 ,in fact       
,einstein is able to show  that an old  
formula1 which had already been found to
 connect the speed  of light in a       
material medium with the speed of that  
medium1 will now  serve universally for 
the composition of velocities4 ,when we 
combine  the velocities v and u1 instead
 of getting the resultant vu  as we     
would have supposed1 we get the         
resultant        7v  u7 _/ 7#a  7uv _/  
,c#b7711 or ,c#b7v  u7 _/ 7,c#b  uv7    
,this need not surprise us either1 if we
 will but reflect that  the second      

velocity effects a second revision of   
length and time  measurements between   
the systems involved4 ,and now1 if we   
let either  v1 or u1 or even both of    
them1 take the value ,c1 the resultant  
still is ,c4 ,in another way we have    
found ,c to behave like the             
mathematician's infinity1 to which1 in  
the words of the blind poet1  if we add 
untold thousands1 we effect no real     
increment4          ,,assumption ,,and  
,,consequence    ,a good many           
correspondents who have given the       
subject sufficient  thought to realize  
that the limiting character of the      
velocity ,c is  really read into        
,einstein's system by assumption have   
written1 in more  or less perturbed     
inquiry1 to know whether this does not  
invalidate  the whole structure4 ,the   
answer1 of course1 is yes--provided you 
can  show this assumption to be invalid4
 ,the same answer may be made of  any   
scientific doctrine whatever1 and in    
reference to any one of the             

multitudinous assumptions underlying it4
 ,if we were to discover1  tomorrow1 a  
way of sending signals absolutely       
instantaneously1  ,einstein's whole     
structure would collapse as soon as we  
had agreed to  use this new method4 ,if 
we were to discover a signalling agent  
with  finite velocity greater than that 
of light1 relativity would persist  with
 this velocity written in its formulae  
in the place of ,c4    ,it is a mistake 
to quote ,einstein's theory in support  
of the  statement that such a velocity  
can never be4 ,an assumption proves its 
consequences1 but never can prove       
itself2 it must remain always an        
assumption4 ,but in the presence of long
 human experience supporting            
,einstein's assumption that no velocity 
in excess of ,c can be found1  it is    
fair to demand that it be disputed not  
with argument but with  demonstration4  
,the one line of argument that would    
hold out a priori  hope of reducing the 
assumption to an absurdity would be one 

based  on the familiar idea of adding   
velocities2 but ,einstein has spiked    
this argument before it is started by   
replacing the direct addition  of       
velocities with another method of       
combining them that fits in  with his   
assumption and as well with the observed
 facts4 ,the burden  of proof is then on
 the prosecution2 anyone who would      
contradict our  assertion that ,c is the
 greatest velocity attainable may do so 
only  by showing us a greater one4      
,until this has been done1 the admission
  that it may properly be attempted can 
in no way be construed as a  confession 
of weakness on the part of ,einstein4   
,it may be well to point out that in no 
event may analogy be drawn with  sound1 
as many have tried to do4 ,in the first 
place sound requires  a material medium 
and its velocity with regard to this    
rather than  relative to the observer we
 know to be fixed2 in the second place1 
requiring a material medium1 sound is   
not a universal signalling  agent2 in   

the third place1 we know definitely that
 its velocity can  be exceeded1 and are 
therefore barred from making the        
assumption  necessary to establish the  
analogy4 ,the very extraordinary        
behavior  of light in presenting a      
velocity that is the same for all       
observers1  and in refusing to betray   
the least material evidence of any      
medium  for its transmission1 rather    
fortifies us in believing that          
,einstein's  assumption regarding the   
ultimate character of this velocity is  
in  accord with the nature of things4   
,,relativity ,,and ,,the ,,layman    ,a 
great deal can be said in the direction 
of general comment making the  ,special 
,theory and its surprising              
accompaniments easier of acceptance1    
and we shall conclude the present       
discussion by saying some of these      
things499 ,it has been objected that the
 various effects catalogued  above are  
only apparent1 due to the finite        
velocity of light--that  the real shape 

and size of a body or the real time of  
an event cannot  be affected by the     
point of view or the motion of an       
observer4 ,this  argument would be      
perfectly valid1 if there were real     
times and  distances2 but there are not4
 ,these are earth-bound notions1 due to 
our experience on an apparently         
motionless platform1 with slow-moving   
bodies4 ,under these circumstances      
different observations of the same      
thing or of the same event agree4 ,but  
when we no longer have the solid  earth 
to stand on1 and are dealing with       
velocities so high that the  relativity 
effects become appreciable1 there is no 
standard by which  to resolve the       
disagreements4 ,no one of the           
observations can claim  to be nearer    
reality than any other4 ,to demand the  
real size of a  thing is to demand a    
stationary observer or an instantaneous 
means  of information4 ,both are        
impossible4    ,when relativity asks us 
to give up our earth-bound notions of   

absolute  space and absolute time the   
sensation1 at first1 is that we have    
nothing left to stand on4 ,so must the  
contemporaries of ,columbus have  felt  
when told that the earth rested         
on--nothing4 ,the remedy too is         
similar4 ,just as they had to be taught 
that falling is a local affair1  that   
the earth is self-contained1 and needs  
no external support--so we  must be     
taught that space and time standards are
 local affairs4 ,each  moving body      
carries its own space and time standards
 with it2 it is  self-contained4 ,it    
does not need to reach out for eternal  
support1  for an absolute space and time
 that can never quite be attained4 ,all 
we ever need to know is the relation of 
the other fellow's space  and time      
standards to our own4 ,this is the first
 thing relativity  teaches us4#ada      
,the consequences of ,einstein's        
assumptions have led many to reject the 
theory of relativity1 on the ground that
 its conclusions are contrary  to common

 sense--as they undoubtedly are4 ,but to
 the contemporaries  of ,copernicus and 
,galileo the theory that the earth      
rotates on its  axis and revolves around
 the sun was contrary to common sense2  
yet  this theory prevailed4 ,there is   
nothing sacred about common sense2  in  
the last analysis its judgments are     
based on the accumulated  experience of 
the human race4 ,from the beginning of  
the world up to  the present generation1
 no bodies were known whose velocities  
were  not extremely small compared with 
that of light4 ,the development of      
modern physics has led to discovery of  
very much larger velocities1  some as   
high as #afe1jjj miles per second4 ,it  
is not to be wondered at  that such an  
enlargement of our experience requires a
 corresponding  enlargement or          
generalization of the concepts of space 
and time4 ,just  as the presupposition  
of primitive man that the earth was flat
 had  to be given up in the light of    
advancing knowledge1 so we are now      

called upon to give up our              
presupposition that space and time are  
absolute and independent in their       
nature4    ,the reader must not expect  
to understand the theory of relativity  
in  the sense of making it fit in with  
his previous ideas4 ,if the theory  be  
right these ideas are wrong and must be 
modified1 a process apt  to be          
painful4#bbc ,all the reader can do is  
to become familiar with  the new        
concepts1 just as a child gets used to  
the simple relations  and quantities he 
meets until he understands them4#bba    
,mr4 ,francis  has said something of the
 utmost significance when he points out 
that  understanding really means nothing
 in the world except familiarity  and   
accustomedness499 ,the one thing about  
the relativity doctrine  that we can    
hope thus to understand at once and     
without pain is the  logical process    
used in arriving at our results4#bba    
,particularly is  it hard to give a     
satisfactory explanation of the theory  

in popular  language1 because the       
language itself is based on the old     
concepts2  the only language which is   
really adequate is that of              
mathematics4#bbc  ,unless we have1 in   
addition to the terms of our ordinary   
knowledge1  a set of definitions that   
comes with a wide knowledge of          
mathematics  and a lively sense of the  
reality of mathematical constructions1  
we  are likely to view the theory of    
relativity through a fog of familiar    
terms suddenly become self-contradictory
 and deceptive4 ,not that we  are       
unfamiliar with the idea that some of   
our habitual notions may  be wrong2 but 
knowledge of their illusory nature      
arises and becomes  convincing only with
 time4 ,we may now be ready to grant    
that the earth1  seemingly so solid1 is 
really a whirling globe rushing through 
space2  but we are no more ready        
immediately to accept the bald assertion
 that  this space is not what it seems  
than our ancestors were to accept the   

idea that the earth was round or that it
 moved4#aef ,what we must have1  if we  
are to comprehend relativity with any   
degree of thoroughness1  is the         
mathematician's attitude toward his     
assumptions1 and his  complete readiness
 to swap one set of assumptions for     
another as a  mere part of the day's    
work1 the spirit of which ,i have       
endeavored  to convey in the chapter on 
non-,euclidean geometry499              
,,physics ,,vs4 ,,metaphysics    ,the   
ideas of relativity may seem1 at first  
sight1 to be giving us  a new and       
metaphysical theory of time and space4  
,new1 doubtless2 but  certainly the     
theory was meant by its author to be    
quite the opposite  of metaphysical4    
,our actual perception of space is by   
measurement1  real and imagined1 of     
distances between objects1 just as our  
actual  perception of time is by        
measurement4 ,is it not less            
metaphysical  to accept space and time  
as our measurements present them to us1 

than to invent hypotheses to force our  
perceptual space into an  absolute space
 that is forever hidden from us8#ahb ,in
 order not to  be metaphysical1 we must 
eliminate our preconceived notions of   
space  and time and motion1 and focus   
our attention upon the indications  of  
our instruments of observation1 as      
affording the only objective            
manifestations of these qualities and   
therefore the only attributes  which we 
can consider as functions of observed   
phenomena4#dg ,einstein  has            
consistently followed the teachings of  
experience1 and completely  freed       
himself from metaphysics4#aad ,that this
 is not always easy  to do is clear1 ,i 
think1 if we will recall the highly     
metaphysical  character often taken by  
the objections to action-at-a-distance  
theories and concepts2 and if we will   
remind ourselves that it was  on purely 
metaphysical grounds that ,newton       
refused to countenance  ,huyghens' wave 
theory of light4 ,whether1 as in the one

 case1 it  leads us to valid            
conclusions1 or1 as in the other1 to    
false ones1  metaphysical reasoning is  
something to avoid4 ,einstein1 ,i think1
  has avoided it about as thoroughly as 
anyone ever did499                    ,v
    ,,that ,,parallel ,,postulate       
,modern ,geometric ,methods2 the        
,dividing ,line ,between ,euclidean and 
,non-,euclidean2 and the ,significance  
of the ,latter    ,,by ,,the ,,editor   
,the science of geometry has undergone a
 revolution of which the  outsider is   
not informed1 but which it is necessary 
to understand  if we are to attain any  
comprehension of the geometric          
formulation  of ,einstein's results2 and
 especially if we are to appreciate why 
it is proper and desirable to formulate 
these results geometrically  at all4    
,the classical geometer regarded his    
science from a narrow  viewpoint1 as the
 study of a certain set of observed     
phenomena--those  of the space about us1
 considered as an entity in itself and  

divorced  from everything in it4 ,it is 
clear that some things about that space 
are not as they appear 7optical         
illusions71 and that other things  about
 it are true but by no means apparent   
7the sum-of-squares  property of a right
 triangle1 the formul for surface and   
volume  of a sphere1 etc474 ,while many 
things about space are obvious1  these  
need in the one case disproof and in the
 other discovery and  proof4 ,with all  
their love of mental processes for their
 own sake1  it is then not surprising   
that the ,greeks should have set        
themselves  the task of proving by      
logical process the properties of space1
  which a less thoughtful folk would    
have regarded as a subject only  for    
observational and experimental          
determination4    ,but1 abstract or     
concrete1 the logical structure must    
have a starting  point2 and it is fair  
to demand that this consist in a        
statement  of the terms we are going to 
use and the meanings we are going to    

attach to them4 ,in other words1 the    
first thing on the program will  be a   
definition1 or more probably1 several   
definitions4    ,now the modern         
scientist has a somewhat iconoclastic   
viewpoint toward  definitions1 and      
especially toward the definition of his 
very most  fundamental ideas4    ,we do 
not speak here in terms of dictionary   
definitions4 ,these have  for object the
 eminent necessity of explaining the    
meaning and use  of a word to some one  
who has just met it for the first time4 
,it  is easy enough to do this1 if the  
doer possesses a good command  of the   
language4 ,it is not even a matter of   
grave concern that the  words used in   
the definition be themselves known to   
the reader2 if  they are not1 he must   
make their acquaintance too4 ,dr4       
,johnson's  celebrated definition of a  
needle stands as perpetual evidence     
that when he cannot define a simple     
thing in terms of things still  simpler1
 the lexicographer is forced to define  

it in terms of things  more complex4 ,or
 we might demonstrate this by noting    
that the best  dictionaries are driven  
to define such words as and and but  by 
using such complex notions as are       
embodied in connective1  continuative1  
adversative1 and particle4    ,it is    
otherwise with the scientist who        
undertakes to lay down a  definition as 
the basis of further procedure in       
building up the  tissue of his science4 
,here a degree of rigorous logic is     
called for  which would be as           
superfluous in the dictionary as the    
effort there  to attain it would be out 
of place4 ,the scientist1 in building up
  a logical structure that will         
withstand every assault1 must define    
everything1 not in terms of something   
which he is more or less  warranted in  
supposing his audience to know about1   
but actually in  terms of things that   
have already been defined4 ,this really 
means that  he must explain what he is  
talking about in terms of simpler ideas 

and simpler things1 which is precisely  
what the lexicographer does  not have to
 worry about4 ,this is why it is quite  
trivial to quote a  dictionary          
definition of time or space or matter or
 force or motion  in settlement of a    
controversy of scientific or            
semi-scientific nature4          ,,terms
 ,,we ,,cannot ,,define    ,but the     
scientist who attempts to carry out this
 ideal system of  defining everything in
 terms of what precedes meets one       
obstacle  which he cannot surmount      
directly4 ,even a layman can construct a
  passable definition of a complex thing
 like a parallelopiped1 in  terms of    
simpler concepts like point1 line1 plane
 and parallel4 ,but  who shall define   
point in terms of something simpler and 
something  which precedes point in the  
formulation of geometry8 ,the scientist 
is  embarrassed1 not in handling the    
complicated later parts of his work1    
but in the very beginning1 in dealing   
with the simplest concepts with  which  

he has to deal4    ,suppose a dictionary
 were to be compiled with the           
definitions  arranged in logical rather 
than alphabetical order3 every word to  
be  defined by the use only of words    
that have already been defined4 ,the    
further back toward the beginning we    
push this project1 the harder it  gets4 
,obviously we can never define the first
 word1 or the second1  save as          
synonymous with the first4 ,in fact we  
should need a dozen  words1 more or     
less1 to start with--,god-given words   
which we cannot  define and shall not   
try to define1 but of which we must     
agree that  we know the significance4   
,then we have tools for further         
procedure2  we can start with1 say1 the 
thirteenth word and define all the rest 
of the words in the language1 in        
strictly logical fashion4    ,what we   
have said about definitions applies     
equally to statements  of fact1 of the  
sort which are going to constitute the  
body of our  science4 ,in the absence of

 simpler facts to cite as authority1 we 
shall  never be able to prove anything1 
however simple this may itself be2  and 
in fact the simpler it be1 the harder it
 is to find something  simpler to       
underlie it4 ,if we are to have a       
logical structure of any  sort1 we must 
begin by laying down certain terms which
 we shall not  attempt to define1 and   
certain statements which we shall not   
try to  prove4 ,mathematics1 physics1   
chemistry--in the large and in all their
  many minor fields--all these must     
start somewhere4 ,instead of deceiving  
ourselves as to the circumstances       
surrounding their start1 we prefer to   
be quite frank in recognizing that they 
start where we decide to start  them4   
,if we don't like one set of undefined  
terms as the foundation1  by all means  
let us try another4 ,but always we must 
have such a set4    ,the classical      
geometer sensed the difficulty of       
defining his first  terms4 ,but he      
supposed that he had met it when he     

defined these in  words free of         
technical significance4 ,a point is that
 which has  position without size seemed
 to him an adequate definition1 because 
position and size are words of the      
ordinary language with which  we may all
 be assumed familiar4 ,but today we feel
 that position  and size represent ideas
 that are not necessarily more          
fundamental  than those of line and     
point1 and that such a definition begs  
the question4 ,we get nowhere by        
replacing the undefined terms point  and
 line and plane1 which really everybody 
understands1 by other  undefined terms  
which nobody understands any better4    
,in handling the facts that it was      
inconvenient to prove1 the classical    
geometer came closer to modern practice4
 ,he laid down at the beginning  a few  
statements which he called axioms1 and  
which he considered to  be so           
self-evident that demonstration was     
superfluous4 ,that the term             
self-evident left room for a vast amount

 of ambiguity appears to  have escaped  
him altogether4 ,his axioms were axioms 
solely because  they were obviously     
true4          ,,laying ,,the           
,,foundation    ,the modern geometer    
falls in with ,euclid when he writes an 
elementary  text1 satisfying the        
beginner's demand for apparent rigor by 
defining  point and line in some        
fashion4 ,but when he addresses to his  
peers an  effort to clarify the         
foundations of geometry to a further    
degree of  rigor and lucidity than has  
ever before been attained1 he meets     
these  difficulties from another        
quarter4 ,in the first place he is      
always  in search of the utmost possible
 generality1 for he has found this to   
be his most effective tool1 enabling him
 as it does to make a single  general   
statement take the place and do the work
 of many particular  statements4 ,the   
classical geometer attained generality  
of a sort1  for all his statements were 
of any point or line or plane4 ,but the 

modern geometer1 confronted with a      
relation that holds among points  or    
between points and lines1 at once goes  
to speculating whether there  are not   
other elements among or between which it
 holds4 ,the classical  geometer isn't  
interested in this question at all1     
because he is  seeking the absolute     
truth about the points and lines and    
planes which  he sees as the elements of
 space2 to him it is actually an object 
so to  circumscribe his statements that 
they may by no possibility refer to     
anything other than these elements4     
,whereas the modern geometer feels  that
 his primary concern is with the fabric 
of logical propositions  that he is     
building up1 and not at all with the    
elements about which  those propositions
 revolve4    ,it is of obvious value if 
the mathematician can lay down a        
proposition  true of points1 lines and  
planes4 ,but he would much rather lay   
down  a proposition true at once of     
these and of numerous other things2  for

 such a proposition will group more     
phenomena under a single  principle4 ,he
 feels that on pure scientific grounds  
there is quite  as much interest in any 
one set of elements to which his        
proposition  applies as there is in any 
other2 that if any person is to confine 
his attention to the set that stands for
 the physicist's space1  that person    
ought to be the physicist1 not the      
geometer4 ,if he has  produced a tool   
which the physicist can use1 the        
physicist is welcome  to use it2 but the
 geometer cannot understand why1 on that
 ground1  he should be asked to confine 
his attention to the materials on which 
the physicist employs that tool4    ,it 
will be alleged that points and lines   
and planes lie in the  mathematician's  
domain1 and that the other things to    
which his  propositions may apply may   
not so lie--and especially that if he   
will  not name them in advance he cannot
 expect that they will so lie4 ,but  the
 mathematician will not admit this4 ,if 

mathematics is defined on  narrow       
grounds as the science of number1 even  
the point and line  and plane may be    
excluded from its field4 ,if any wider  
definition  be sought--and of course one
 must be--there is just one definition  
that the mathematician will accept3 ,dr4
 ,keyser's statement that  mathematics  
is the art or science of rigorous       
thinking4    ,the immediate concern of  
this science is the means of rigorous   
thinking--undefined terms and           
definitions1 axioms and propositions4   
,its  collateral concern is the things  
to which these may apply1 the things    
which may be thought about              
rigorously--everything4 ,but now the    
mathematician's domain is so vastly     
extended that it becomes more  than ever
 important for him to attain the utmost 
generality in all  his pronouncements4  
,one barrier to such generalization is  
the very name geometry1  with the       
restricted significance which its       
derivation and long usage  carry4 ,the  

geometer therefore must have it         
distinctly understood  that for him     
geometry means simply the process of    
deducing a set  of propositions from a  
set of undefined primitive terms and    
axioms2  and that when he speaks of a   
geometry he means some particular set   
of propositions so deduced1 together    
with the axioms1 etc41 on which  they   
are based4 ,if you take a new set of    
axioms you get a new geometry4    ,the  
geometer will1 if you insist1 go on     
calling his undefined terms by  the     
familiar names point1 line1 plane4 ,but 
you must distinctly  understand that    
this is a concession to usage1 and that 
you are not  for a moment to restrict   
the application of his statements in any
  way4 ,he would much prefer1 however1  
to be allowed new names for his         
elements1 to say ,we start with three   
elements of different sorts1  which we  
assume to exist1 and to which we attach 
the names ,a1 ,b and  ,c--or if you     
prefer1 primary1 secondary and tertiary 

elements--or  yet again1 names          
possessing no intrinsic significance at 
all1 such as  ching1 chang and chung4   
,he will then lay down whatever         
statements he  requires to serve the    
purposes of the ancient axioms1 all of  
these  referring to some one or more of 
his elements4 ,then he is ready for  the
 serious business of proving that1 all  
his hypotheses being granted1  his      
elements ,a1 ,b and ,c1 or ,i1 ,,ii and 
,,iii1 or ching1 chang and chung1  are  
subject to this and that and the other  
propositions4    ,the objection will be 
urged that the mathematician who does   
all  this usurps the place of the       
logician4 ,a little reflection will     
show this not to be the case4 ,the      
logician in fact occupies the same      
position with reference to the geometer 
that the geometer occupies  with        
reference to the physicist1 the chemist1
 the arithmetician1  the engineer1 or   
anybody else whose primary interest lies
 with some  particular set of elements  

to which the geometer's system applies4 
,the  mathematician is the tool-maker of
 all science1 but he does not make  his 
own tools--these the logician supplies4 
,the logician in turn  never descends to
 the actual practice of rigorous        
thinking1 save as  he must necessarily  
do this in laying down the general      
procedures  which govern rigorous       
thinking4 ,he is interested in          
processes1 not  in their application4   
,he tells us that if a proposition is   
true its  converse may be true or false 
or ambiguous1 but its contrapositive  is
 always true1 while its negative is     
always false4 ,but he never1  from a    
particular proposition ,if ,a is ,b then
 ,c is ,d1 draws the  particular        
contrapositive inference ,if ,c is not  
,d then ,a is not  ,b4 ,that is the     
mathematician's business4          ,,the
 ,r,,le ,,of ,,geometry    ,the         
mathematician is the quantity-production
 man of science4 ,in  his absence1 the  
worker in each narrower field where the 

elements  under discussion take         
particular concrete forms could work    
out1  for himself1 the propositions of  
the logical structure that applies  to  
those elements4 ,but it would then be   
found that the engineer had  duplicated 
the work of the physicist1 and so for   
many other cases2  for the whole trend  
of modern science is toward showing that
 the  same background of principles lies
 at the root of all things4 ,so the     
mathematician develops the fabric of    
propositions that follows from  this1   
that and the other group of assumptions1
 and does this without  in the least    
concerning himself as to the nature of  
the elements of  which these            
propositions may be true4 ,he knows only
 that they are true  for any elements of
 which his assumptions are true1 and    
that is all he  needs to know4 ,whenever
 the worker in some particular field    
finds that  a certain group of the      
geometer's assumptions is true for his  
elements1  the geometry of those        

elements is ready at hand for him to    
use4    ,now it is all right purposely  
to avoid knowing what it is that we  are
 talking about1 so that the names of    
these things shall constitute  mere     
blank forms which may be filled in1 when
 and if we wish1 by the  names of any   
things in the universe of which our     
axioms turn out  to be true4 ,but what  
about these axioms themselves8 ,when we 
lay them  down in ignorance of the      
identity of the elements to which they  
may  eventually apply1 they cannot by   
any possibility be self-evident4 ,we    
may1 at pleasure1 accept as self-evident
 a statement about points and  lines and
 planes2 or one about electrons1        
centimeters and seconds2  or one about  
integers1 fractions1 and irrational     
numbers2 or one about  any other        
concrete thing or things whatever4 ,but 
we cannot accept as  self-evident a     
statement about chings1 changs and      
chungs4 ,so we must  base our axioms on 
some other ground than this2 and our    

modern  geometer has his ground ready   
and waiting4 ,he accepts his axioms  on 
the ground that it pleases him to do so4
 ,to avoid all suggestion  that they are
 supposed to be self-evident1 or even   
necessarily true1  he drops the term    
axiom and substitutes for it the more   
color-less  word postulate4 ,a postulate
 is merely something that we agreed to  
accept1 for the time being1 as a basis  
of further argument4 ,if it  turns out  
to be true1 or if we can find           
circumstances under which and  elements 
to which it applies1 any conclusions    
which we deduce from it  by trustworthy 
processes are valid within the same     
limitations4 ,and  the propositions     
which tell us that1 if our postulates   
are true1 such  and such conclusions are
 true--they1 too1 are valid1 but without
 any  reservation at all6    ,perhaps an
 illustration of just what this means   
will not be out of  place4 ,let it be   
admitted1 as a postulate1 that #gbj is  
greater1 by  #a1 than #gai4 ,let us then

 consider the statement3 ,if #g  #ai    
#fe1  then #g  #bj  #ff4 ,we know--at   
least we are quite certain--that #g     
#ai is not equal to #fe1 if by #g and   
#ai and #fe we mean what you  think we  
mean4 ,we are equally sure1 on the same 
grounds1 that #g   #bj is not equal to  
#ff4 ,but1 under the one assumption that
 we have  permitted ourselves1 it is    
unquestionable that if #g  #ai were     
equal  to #fe1 then #g  #bj certainly   
would be equal to #ff4 ,so1 while the   
conclusion of the proposition which ,i  
have put in quotation marks is          
altogether false1 the proposition       
itself1 under our assumption1 is        
entirely true4 ,i have taken an         
illustration designed to be striking    
rather than to possess scientific       
interest2 ,i could just as easily  have 
shown a true proposition leading to a   
false conclusion1 but of  such sort that
 it would be of decided scientific      
interest as telling  us one of the      
consequences of a certain assumption4   

,,what ,,may ,,we ,,take ,,for          
,,granted8    ,this is all very fine2   
but how does the geometer know what     
postulates  to lay down8 ,one is tempted
 to say that he is at liberty to        
postulate  anything he pleases1 and     
investigate the results2 and that       
whether or  not his postulate ever be   
realized1 the propositions that he      
deduces  from it1 being true1 are of    
scientific interest4 ,actually1 however1
  it is not quite as simple as all that4
 ,if it were sufficient to make  a      
single postulate it would be as simple  
as all that2 but it turns  out that this
 is not sufficient any more than it is  
sufficient to  have a single undefined  
term4 ,we must have several postulates2 
and  they must be such1 as a whole1 that
 a geometry flows out of them4 ,the     
requirements are three4    ,in the first
 place1 the system of postulates must be
 categorical  or complete--there must be
 enough of them1 and they must cover    
enough  ground1 for the support of a    

complete system of geometry4 ,in        
practice  the test for this is direct4  
,if we got to a point in the building up
  of a geometry where we could not prove
 whether a certain thing was  one way   
always1 or always the other way1 or     
sometimes one way and  sometimes the    
other1 we should conclude that we needed
 an additional  postulate covering this 
ground directly or indirectly4 ,and we  
should  make that postulate--because it 
is precisely the things that we can't   
prove which1 in practical work1 we agree
 to assume4 ,even ,euclid had  to adopt 
this philosophy4    ,in the second      
place1 the system of postulates must be 
consistent--no  one or more of them may 
lead1 individually or collectively1 to  
consequences that contradict the results
 or any other or others4 ,if  in the    
course of building up a geometry we find
 we have proved  two propositions that  
deny one another1 we search out the     
implied  contradiction in our postulates
 and remedy it4    ,finally1 the        

postulates ought to be independent4 ,it 
should not be  possible to prove any one
 of them as a consequence of the others4
 ,if  this property fails1 the geometry 
does not fail with it2 but it is        
seriously disfigured by the superfluity 
of assumptions1 and one of  them should 
be eliminated4 ,if we are to assume     
anything unnecessarily1  we may as well 
assume the whole geometry and be done   
with it4    ,the geometer's business    
then is to draw up a set of postulates4 
,this  he may do on any basis whatever4 
,they may be suggested to him by  the   
behavior of points1 lines and planes1 or
 by some other concrete  phenomena2 they
 may with equal propriety be the product
 of an  inventive imagination4 ,on      
proceeding to deduce their consequences1
  he will discover and remedy any lack  
of categoricity or consistence  or      
independence which his original system  
of postulates may have  lacked4 ,in the 
end he will have so large a body of     
propositions without  contradiction or  

failure that he will conclude the       
propriety of his  postulates to have    
been established1 and the geometry based
 on them  to be a valid one4            
,,and ,,what ,,is ,,it ,,all ,,about8   
,is this geometry ever realized8        
,strictly it is not the geometer's      
business to ask or answer this question4
 ,but research develops two  viewpoints4
 ,there is always the man who indulges  
in the pursuit of  facts for their sake 
alone1 and equally the man who wants to 
see his  new facts lead to something    
else4 ,one great mathematician is quoted
  as enunciating a new theory of        
surpassing mathematical beauty with  the
 climacteric remark ,and1 thank ,god1 no
 one will ever be able to  find any use 
for it4 ,an equally distinguished       
contemporary1 on being  interrogated    
concerning possible applications for one
 of his most  abstruse theorems1 replied
 that he knew no present use for it2 but
 that  long experience had made him     
confident that the mathematician would  

never develop any tool1 however remote  
from immediate utility1 for  which the  
delvers in other fields would not       
presently find some use4    ,if we wish1
 however1 we may inquire with perfect   
propriety1 from the  side lines1 whether
 a given geometry is ever realized4 ,we 
may learn  that so far as has yet been  
discovered there are no elements for    
which all its postulates are verified1  
and that there is therefore no          
realization known4 ,on the other hand1  
we may more likely find that  many      
different sets of elements are such that
 the postulates can be  interpreted as  
applying to them1 and that we therefore 
have numerous  realizations of the      
geometry4 ,as a human being the geometer
 may be  interested in all this1 but as 
a geometer it really makes little       
difference to him4    ,when we look at  
space about us1 we see it1 for some     
reason grounded  in the psychological   
history of the human race1 as made up in
 the  small of points1 which go to make 

up lines1 which in turn constitute      
planes4 ,or we can start at the other   
end and break space down first  into    
planes1 then into lines1 finally into   
points4 ,our perceptions  and           
conceptions of these points1 lines and  
planes are very definite  indeed2 it    
seems indeed1 as the ,greeks thought1   
that certain things  about them are     
self-evident4 ,if we wish to take these 
self-evident  properties of point1 line 
and plane1 and combine with them enough 
additional hair-splitting specifications
 to assure the modern geometer  that we 
have really a categorical system of     
assumptions1 we shall  have the basis of
 a perfectly good system of geometry4   
,this will be  what we unavoidably think
 of as the absolute truth with regard to
  the space about us2 but you mustn't   
say so in the presence of the  geometer4
 ,it will also be what we call the      
,euclidean geometry4 ,it  has been      
satisfactory in the last degree1 because
 not only space1  but pretty much every 

other system of two or three elements   
bearing  any relations to one another   
can be made1 by employing as a means  of
 interpretation the ,cartesian scheme of
 plotting1 to fit into the  framework of
 ,euclidean geometry4 ,but it is not the
 only thing in the  world of conceptual 
possibilities1 and it begins to appear  
that it  may not even be the only thing 
in the world of cold hard fact that     
surrounds us4 ,to see just how this is  
so we must return to ,euclid1  and      
survey the historical development of    
geometry from his day to  the present   
time4          ,,euclid',s ,,geometry   
,point1 line and plane ,euclid attempts 
to define4 ,modern objection  to these  
efforts was made clear above4 ,against  
,euclid's specific  performance we urge 
the further specific fault that his     
definitions  are really assumptions     
bestowing certain properties upon       
points1 lines  and planes4 ,these       
assumptions ,euclid supplements in his  
axioms2 and in  the process of proving  

propositions he unconsciously           
supplements them  still further4 ,this  
is to be expected from one whose        
justification for  laying down an axiom 
was the alleged obvious character of the
 statement  made4 ,if some things are   
too obvious to require demonstration1   
others  may be admitted as too obvious  
to demand explicit statement at all4    
,thus1 if ,euclid has two points ,a and 
,b in a plane1 on opposite sides of  a  
line ,m1 he will draw the line ,,ab and 
without further formality speak  of the 
point ,c in which it intersects ,m4     
,that it does so intercept ,m1  rather  
than in some way dodges it1 is really an
 assumption as to the  nature of lines  
and planes4 ,or again1 ,euclid will     
speak of a point ,d  on the line ,,ab1  
between or outside the points ,a and ,b1
 without making  the formal assumption  
necessary to insure that the line is    
full of  points so that such a point as 
,d must exist4 ,that such assumptions as
  these are necessary follows from our  

previous remarks4 ,if we think of  our  
geometry as dealing with chings1 changs1
 and chungs1 or with  elements ,i1 ,,ii 
and ,,iii1 it is no longer in the least 
degree obvious  that the simplest       
property in the world applies to these  
elements4 ,if  we wish any property to  
prevail we must state it explicitly4    
,with the postulates embodied in his    
definitions1 those stated in his        
axioms1 and those which he reads into   
his structure by his methods of  proof1 
,euclid has a categorical set--enough to
 serve as foundation  for a geometry4   
,we may then climb into ,euclid's shoes 
and take the  next step with him4 ,we   
follow him while he proves a number of  
things  about intersecting lines and    
about triangles4 ,to be sure1 when he   
proves that two triangles are           
identically constituted by moving one of
  them over on top of the other1 we may 
protest on the ground that the          
admission of motion1 especially of      
motion thus imposed from without1  into 

a geometry of things is not beyond      
dispute4 ,if ,euclid has caught  our    
modern viewpoint1 he will rejoin that if
 we have any doubts as to  the          
admissibility of motion he will lay down
 a postulate admitting it1  and we shall
 be silenced4    ,having exhausted for  
the present the interest of intersecting
 lines1  our guide now passes to a      
consideration of lines in the same plane
  that never meet4 ,he defines such     
lines as parallel4 ,if we object that   
he should show the existence of a       
derived concept like this before  laying
 down a definition that calls for it to 
exist1 he can show that  two lines drawn
 perpendicular to the same line never   
meet4 ,he will  execute this proof by a 
special sort of superposition1 which    
requires  that the plane be folded over 
on itself1 through the third dimension  
of surrounding space1 rather than merely
 slid along upon itself4    ,we remain  
quiet while ,euclid demonstrates that if
 two lines are cut by  any transversal  

in such a way as to make corresponding  
angles at the  two intersections equal1 
the lines are parallel4 ,it is then in  
order  to investigate the converse3 if  
the lines are parallel to begin with1   
are the angles equal8          ,,axioms 
,,made ,,to ,,order    ,this sounds     
innocent enough2 but in no way was      
,euclid able to devise a  proof--or1 for
 that matter1 a disproof4 ,so he took   
the only way out1  and said that if the 
lines were parallel1 obviously they     
extended in  the same direction and made
 the angles equal4 ,the thing was so    
obvious1  he argued1 that it was really 
an axiom and he didn't have to prove    
it2 so he stated it as an axiom and     
proceeded4 ,he didn't state it  in      
precisely the form ,i have used2 he     
apparently cast about for the  form in  
which it would appear most obvious1 and 
found a statement that  suited him      
better than this one1 and that comes to 
the same thing4 ,this  statement tells  
us that if the transversal makes two    

corresponding  angles unequal1 the lines
 that it cuts are not parallel and do   
meet  if sufficiently prolonged4 ,but   
wisely enough1 he did not transplant    
this axiom1 once he had arrived at it1  
to the beginning of the book  where the 
other axioms were grouped2 he left it   
right where it was1  following the      
proposition that if the angles were     
equal the lines were  parallel4 ,this of
 course was so that it might appeal     
back1 for its  claim to obviousness1 to 
its demonstrated converse of the        
proposition4    ,euclid must have been  
dissatisfied with this cutting of the   
,gordian  knot2 his successors were     
acutely so4 ,for twenty centuries the   
parallel  axiom was regarded as the one 
blemish in an otherwise perfect work2   
every respectable mathematician had his 
shot at removing the defect by  proving 
the objectionable axiom4 ,the procedure 
was always the same3  expunge the       
parallel axiom1 in its place write      
another more or less  obvious           

assumption1 and from this derive the    
parallel statement more  or less        
directly4 ,thus if we may assume that   
the sum of the angles  of a triangle is 
always exactly #ahj degrees1 or that    
there can be  drawn only one line       
through a given point parallel to a     
given line1  we can prove ,euclid's     
axiom4 ,sometimes the substitute        
assumption was  openly made and stated1 
as in the two instances cited2 as often 
it  was admitted into the demonstration 
implicitly1 as when it is quietly       
assumed that we can draw a triangle     
similar to any given triangle and  with 
area as great as we please1 or when     
parallels are defined as  everywhere    
equidistant4 ,but such proofs never     
satisfied anyone other  than the man who
 made them2 the search went merrily on  
for a valid  proof that should not in   
substance assume the thing to be proved4
          ,,locating ,,the ,,discrepancy
    ,saccheri1 an ,italian ,jesuit1     
would have struck bottom if he had had a

  little more imagination4 ,he gave an  
exhaustive reductio ad absurdum1  on the
 basis of the angle-sum theorem4 ,this  
sum must be 7a7 greater  than or 7b7    
equal to or 7c7 less than #ahj degrees4 
,saccheri showed  that if one of these  
alternatives occurs in a single         
triangle1 it must  occur in every       
triangle4 ,the first case gave little   
trouble2 admitting  the possibility of  
superposing in the special manner       
mentioned above1  which he did          
implicitly1 he showed that this         
obtuse-angled hypothesis  contradicted  
itself4 ,he pursued the acute-angled    
hypothesis for  a long time before he   
satisfied himself that he had caught it1
 too1  in an inconsistency4 ,this left  
only the right-angled hypothesis1       
proving the ,euclidean angle-sum theory 
and through it the parallel  postulate4 
,but ,saccheri was wrong3 he had found  
no actual contradiction  in the         
acute-angled hypothesis--for none exists
 therein4    ,the full facts were       

probably first known to ,gauss1 who had 
a finger in  every mathematical pie that
 had to do with the transition to modern
  times4 ,they were first published by  
,lobatchewsky1 the ,russian1 who        
anticipated the ,hungarian ,john ,bolyai
 by a narrow margin4 ,all three  worked 
independently of ,saccheri1 whose book1 
though theoretically  available in      
,italian libraries1 was actually lost to
 sight and had  to be rediscovered in   
recent years4    ,like ,saccheri1       
,lobatchewsky investigated alternative  
possibilities4 ,but  he chose another   
point of attack3 through a given point  
it must be  possible to draw1 in the    
same plane with a given line 7a7 no     
lines or  7b7 one line or 7c7 a         
plurality of lines1 which shall not meet
 the  given line4 ,the word parallel is 
defined only in terms of the second  of 
these hypotheses1 so we avoid it here4  
,these three cases correspond1          
respectively1 to those of ,saccheri4    
,the first case ,lobatchewsky ruled out 

just as did ,saccheri1 but  accepting   
consciously the proviso attached to its 
elimination2  the third he could not    
rule out4 ,he developed the consequences
 of  this hypothesis as far as ,euclid  
develops those of the second one1       
sketching in a full outline for a system
 of geometry and trigonometry  based on 
a plurality of non-cutters4 ,this       
geometry constitutes a  coherent whole1 
without a logical flaw4    ,this made it
 plain what was the matter with         
,euclid's parallel  axiom4 ,nobody could
 prove it from his other assumptions    
because it  is not a consequence of     
these4 ,true or false1 it is independent
 of  them4 ,trinity ,church is in ,new  
,york1 ,faneuil ,hall is in ,boston1 but
  ,faneuil ,hall is not in ,boston      
because ,trinity is in ,new ,york2 and  
we  could not prove that ,faneuil ,hall 
was in ,boston if we knew nothing  about
 ,america save that ,trinity is in ,new 
,york4 ,the mathematicians  of #b1jjj   
years had been pursuing1 on a gigantic  

scale1 a delusion of  post hoc1 ergo    
propter hoc4          ,,what ,,the      
,,postulate ,,really ,,does             
,moreover1 in the absence of an         
assumption covering the ground1 we      
shall not know which of the alternatives
 7a71 7b71 7c7 holds4 ,but  when one    
holds in a single case it holds         
permanently1 as ,saccheri and           
,lobatchewsky both showed4 ,so we cannot
 proceed on this indefinite  basis2 we  
must know which one is to hold4 ,without
 the parallel  postulate or a substitute
 therefor that shall tell us the same   
thing  or tell us something different1  
we have not got a categorical set of    
assumptions--we cannot build a geometry 
at all4 ,that is why ,euclid  had to    
have his parallel postulate before he   
could proceed4 ,that is  why his        
successors had to have an assumption    
equivalent to his4    ,the reason why it
 took so long for this to percolate into
 the  understanding of the              
mathematicians was that they were       

thinking1  not in terms of the modern   
geometry and about undefined elements2  
but in terms of the old geometry and    
about strictly defined and              
circumscribed elements4 ,if we          
understand what is meant by ,euclidean  
line and plane1 of course the parallel  
postulate1 to use the old  geometer's   
word1 is true--of course1 to adopt the  
modern viewpoint1  if we agree to employ
 an element to which that assumption    
applies1  the assumption is realized4   
,the very fact of accepting the straight
  line and the flat plane of ,euclid    
constitutes acceptance of his  parallel 
postulate--the only thing that can      
separate his geometry  from other       
geometries4 ,but of course we can't     
prove it2 the prior  postulates which we
 would have to use in such an attempt   
apply where  it does not apply1 and     
hence it cannot possibly be consequences
 of4    ,to all this the classical      
,euclidean rejoins that we seem to have 
in mind elements of some sort to which1 

with one reservation1 his  postulates   
apply4 ,he wants to know what these     
elements look like4 ,we  can1 and must1 
produce them--else our talk about       
generality is mere  drivel4 ,but we must
 take care that the ,euclidean geometer 
does not  try to apply to our elements  
the notions of straightness and flatness
  which inhere in the parallel          
postulate4 ,we cannot satisfy and defy  
that postulate at the same time4 ,if we 
do not insist on this point1 we  shall  
find that we are reading non-,euclidean 
properties into ,euclidean  geometry1   
and interpreting the elements of the    
latter as straight  lines that are not  
straight1 flat planes that are not flat4
 ,it is not  the mission of             
non-,euclidean geometry thus to deny the
 possibility  of ,euclidean geometry2 it
 merely demands a place of equal honor4 
,,the ,,geometry ,,of ,,surfaces    ,let
 us ask the ,euclidean geometer whether 
he can recognize his plane  after we    
have crumpled it up like a piece of     

paper en route to the  waste basket4 ,he
 will hesitate only long enough to      
recall that in  the special case of     
superposition he has reserved for       
himself the  privilege of deforming his 
own plane1 and to realize that he can   
always iron his plane out smooth again  
after we are through with  it4 ,this    
emphasizes the true nature of the       
two-dimensionality which  is the        
fundamental characteristic of the plane 
7and of other things1  as we shall      
directly see74 ,the plane is            
two-dimensional in points not  because  
two sets of mutually perpendicular      
,euclidean straight lines  can be drawn 
in it defining directions of north-south
 and east-west1  but because a point in 
it can be located by means of two       
measures4 ,the  same statement may be   
made of anything whatever to which the  
term  surface is applicable2 anything1  
however crumpled or irregular it  be1   
that possesses length and breadth       
without thickness4 ,the surface  of a   

sphere1 of a cylinder1 of an ellipsoid1 
of a cone1 of a doughnut                
7mathematically known as a torus71 of a 
gear wheel1 of a ,french horn1  all     
these possess two-dimensionality in     
points2 on all of them we can  draw     
lines and curves and derive a geometry  
of these figures4 ,if we  get away from 
the notion that geometry of two         
dimensions must deal  with planes1 and  
adopt in place of this idea the broader 
restriction  that it shall deal with    
surfaces1 we shall have the             
generalization  which the ,euclidean has
 demanded that we produce1 and the one  
which  in the hands of the modern       
geometer has shown results4    ,in this 
two-dimensional geometry of surfaces in 
general1 that of the  plane is merely   
one special case4 ,certain of the       
features met in that  case are general4 
,if we agree that we know what we mean  
by distance1  we find that on every     
surface there is a shortest distance    
between  two points1 together with a    

series of lines or curves along which   
such distances are taken4 ,these lines  
or curves we call geodesics4 ,on  the   
plane the geodesic is the straight line4
 ,on surfaces in general  the geodesic1 
whatever its particular and peculiar    
shape1 plays  the same rle that is      
played by the straight line in the      
plane2  it is the secondary element of  
the geometry1 the surface itself  and   
all other surfaces of its type are the  
tertiary elements4 ,and  it is a fact   
that we can take all the possible       
spheres1 or all the  possible           
,french-horn surfaces1 and conceive of  
space as we know it  being broken down  
by analysis into these surfaces instead 
of into  planes4 ,the only reason we    
habitually decompose space into planes  
is  because it comes natural to us to   
think that way4 ,but geometric points1  
lines and surfaces must be recognized as
 abstractions without actual  existence1
 for all of them lack one or more of the
 three dimensions  which such existence 

implies4 ,these figures exist in our    
minds but  not in the external world    
about us4 ,so any decomposition of space
  into geometric elements is a          
phenomenon of the mind only2 it has no  
parallel and no significance in the     
external world1 and is made in  one way 
or in another purely at our pleasure4   
,there isn't a true1  honest-to-goodness
 geometrical plane in existence any more
 than there  is an honest-to-goodness   
spherical surface3 so on intrinsic      
grounds  one decomposition is as        
reasonable as another4    ,certain of   
the most fundamental postulates are     
obeyed by all  surfaces4 ,as we attempt 
to discriminate between surfaces of     
different  types1 and get1 for instance1
 a geometry that shall be valid for     
spheres and ellipsoids but not for      
conicoids in general1 we must do  so by 
bringing in additional postulates that  
embody the necessary  restrictions4 ,a  
characteristic shared by planes1        
spheres1 and various  other surfaces is 

that the geodesics can be freely slid   
along upon  themselves and will coincide
 with themselves in all positions when  
thus slid2 with a similar arrangement   
for the surface itself4 ,but  the plane 
stands almost unique among surfaces in  
that it does not  force us to           
distinguish between its two sides2 we   
can turn it over  and still it will     
coincide with itself2 and this property 
belongs  also to the straight line4 ,it 
does not belong to the sphere1 or to    
the great circles which are the         
geodesics of spherical geometry2  when  
we turn one of these over1 through the  
three-dimensional space  that surrounds 
it1 we find that the curvature lies in  
the wrong way  to make superposition    
possible4 ,if we postulate that         
superposition be  possible under such   
treatment1 we throw out the sphere and  
spherical  geometry2 if we postulate    
that superposition be only by sliding   
the  surface upon itself we admit that  
geometry--as ,saccheri failed to  see1  

as ,lobatchewsky realized1 and as       
,riemann showed at great length  in     
rehabilitating the obtuse-angled        
hypothesis4 ,lobatchewsky's             
acute-angled geometry is realized on a  
surface of the proper sort1  which      
admits of unrestricted superposition2   
but it is not the sort  of a surface    
that ,i care to discuss in an article of
 this scope4    ,euclidean geometry is  
the natural and easy one1 ,i suppose1   
because it  makes it easy to stop with  
three dimensions4 ,if we take a         
secondary  element1 a geodesic1 which is
 curved in the ,euclidean sense1 we  get
 a tertiary element1 a surface1 which is
 likewise curved4 ,then  unless we are  
to make an altogether abrupt and        
unreasonable break1  we shall find that 
just as the curved geodesic generated a 
curved  surface1 the curved surface must
 give rise to a curved space2 and  just 
as the curved geodesic needed a second  
dimension to curve into1  and the curved
 surface a third1 so the curved         

three-space requires  a fourth4 ,once   
started on this sort of thing1 there    
doesn't really  seem to be any end4     
,,euclidean ,,or ,,non-,,euclidean      
,nevertheless1 we must face the         
possibility that the space we live  in1 
or any other manifold of any sort       
whatever with which we deal on          
geometric principles1 may turn out to be
 non-,euclidean4 ,how shall we  finally 
determine this8 ,by measures--the       
,euclidean measures the angles  of an   
actual triangle and finds the sum to be 
exactly #ahj degrees2  or he draws      
parallel lines of indefinite extent and 
finds them to  be everywhere equally    
distant2 and from these data he         
concludes that  our space is really     
,euclidean4 ,but he is not necessarily  
right4    ,we ask him to level off a    
plot of ground by means of a plumb      
line4 ,since the line always points to  
the earth's center1 the level  plot is  
actually a very small piece of a        
spherical surface4 ,any test  conducted 

on this plot will exhibit the numerical 
characteristics of  the ,euclidean      
geometry2 yet we know the geometry of   
this surface is  ,riemannian4 ,the      
angle-sum is really greater than #ahj   
degrees2 lines  that are everywhere     
equidistant are not both geodesics4     
,the trouble1 of course1 is that on this
 plot we deal with  so minute a fraction
 of the whole sphere that we cannot make
  measurements sufficiently refined to  
detect the departure from  ,euclidean   
standards4 ,so it is altogether sensible
 for us to ask3  ,is the universe of    
space about us really ,euclidean in     
whatever of  realized geometry it       
presents to us8 ,or is it really        
non-,euclidean1  but so vast in size    
that we have never yet been able to     
extend our  measures to a sufficiently  
large portion of it to make the         
divergence  from the ,euclidean standard
 discernible to us8    ,this discussion 
is necessarily fragmentary1 leaving out 
much that  the writer would prefer to   

include4 ,but it is hoped that it will  
nevertheless make it clear that when the
 contestants in the ,einstein           
competition speak of a non-,euclidean   
universe as apparently having  been     
revealed by ,einstein1 they mean simply 
that to ,einstein has  occurred a happy 
expedient for testing ,euclideanism on a
 smaller  scale than has heretofore been
 supposed possible4 ,he has devised a   
new and ingenious sort of measure which1
 if his results be valid1  enables us to
 operate in a smaller region while yet  
anticipating that  any non-,euclidean   
characteristics of the manifold with    
which we deal  will rise above the      
threshold of measurement4 ,this does not
 mean that  ,euclidean lines and planes1
 as we picture them in our mind1 are no 
longer non-,euclidean1 but merely that  
these concepts do not quite so  closely 
correspond with the external reality as 
we had supposed4    ,as to the precise  
character of the non-,euclideanism which
 is revealed1  we may leave this to     

later chapters and to the competing     
essayists4 ,we  need only point out here
 that it will not necessarily be        
restricted  to the matter of            
parallelism4 ,the parallel postulate is 
of extreme  interest to us for two      
reasons2 first because historically it  
was the  means by which the             
possibilities and the importance of     
non-,euclidean  geometry were forced    
upon our attention2 and second because  
it happens  to be the immediate ground  
of distinction between ,euclidean       
geometry  and two of the most           
interesting alternatives4 ,but          
,euclidean geometry  is characterized1  
not by a single postulate1 but by a     
considerable  number of postulates4 ,we 
may attempt to omit any one of these so 
that  its ground is not specifically    
covered at all1 or to replace any one   
of them by a direct alternative4 ,we    
might conceivably do away with the      
superposition postulate entirely1 and   
demand that figures be proved           

equivalent1 if at all1 by some more     
drastic test4 ,we might do away  with   
the postulate1 first properly formulated
 by ,hilbert1 on which our  ideas of the
 property represented in the word       
between depend4 ,we  might do away with 
any single one of the ,euclidean        
postulates1 or  with any combination of 
two or more of them4 ,in some cases this
 would  lead to a lack of categoricity  
and we should get no geometry at all2   
in most cases1 provided we brought a    
proper degree of astuteness to  the     
formulation of alternatives for the     
rejected postulates1 we should  get a   
perfectly good system of non-,euclidean 
geometry3 one realized1  if at all1 by  
other elements than the ,euclidean      
point1 line and plane1  and one whose   
elements behave toward one another      
differently from  the ,euclidean point1 
line and plane4    ,merely to add       
definiteness to this chapter1 ,i annex  
here the  statement that in the geometry
 which ,einstein builds up as more      

nearly representing the true external   
world than does ,euclid's1 we  shall    
dispense with ,euclid's 7implicit7      
assumption1 underlying his  7explicitly 
stated7 superposition postulate1 to the 
effect that the  act of moving things   
about does not affect their lengths4 ,we
 shall  at the same time dispense with  
his parallel postulate4 ,and we shall   
add a fourth dimension to his           
three--not1 of course1 anything in the  
nature of a fourth ,euclidean straight  
line perpendicular1 in ,euclidean       
space1 to three lines that are already  
perpendicular to each other1  but       
something quite distinct from this1     
whose nature we shall see  more exactly 
in the next chapter4 ,if the present    
chapter has made it  clear that it is   
proper for us to do this1 and has       
prevented anyone  from supposing that   
the results of doing it must be         
visualized in a  ,euclidean space of    
three dimensions or of any number of    
dimensions1  it will have served its    

purpose4                    ,,vi        
,,the ,,space-,,time ,,continuum        
,minkowski's ,world of ,events1 and the 
,way ,it ,fits ,into ,einstein's        
,structure    ,,by ,,the ,,editor1      
,,except ,,as ,,noted      ,seeking a   
basis for the secure formulation of his 
results1 and  especially a means for    
expressing mathematically the facts of  
the  dependence which he had found to   
exist between time and space1 ,einstein 
fell back upon the prior work of        
,minkowski4 ,it may be stated right     
here that the idea of time as a fourth  
dimension is not particularly  a new    
one4 ,it has been a topic of abstract   
speculation for the best  part of a     
century1 even on the part of those whose
 notions of the  fourth dimension were  
pretty closely tied down to the idea of 
a  fourth dimension of ,euclidean       
point-space1 which would be marked by  a
 fourth real line1 perpendicular to the 
other three1 and visible  to us if we   
were only able to see it4 ,moreover1    

every mathematician1  whether or not he 
be inclined to this sort of mental      
exercise1 knows  well that whenever time
 enters his equations at all1 it does so
 on an  absolutely equal footing with   
each of his space coordinates1 so that  
as far as his algebra is concerned he   
could never distinguish between  them4  
,when the variables x1 y1 z1 t come to  
the mathematician  in connection with   
some physical investigation1 he knows   
before he  starts that the first three  
represent the dimensions of ,euclidean  
three-space and that the last stands for
 time4 ,but if the algebraic            
expressions of such a problem were      
handed to him independently of  all     
physical tie-up1 he would never be able 
to tell1 from them alone1  whether one  
of the four variables represented time1 
or if so1 which  one to pick out for    
this distinction4    ,it was ,minkowski 
who first formulated all this in a form 
susceptible  of use in connection with  
the theory of relativity4 ,his starting 

point  lies in the distinction between  
the point and the event4 ,mr4 ,francis  
has brought this out rather well in his 
essay1 being the only  competitor to    
present the ,euclidean geometry as a    
real predecessor  of ,newtonian science1
 rather than as a mere part of the      
,newtonian  system4 ,i think his point  
here is very well taken4 ,as he says1   
,euclid looked into the world about him 
and saw it composed of  points4         
,ignoring all dynamic considerations1 he
 built up in his  mind a static world of
 points1 and constructed his geometry as
 a  scientific machine for dealing with 
this world in which motion played  no   
part4 ,it could to be sure be introduced
 by the observer for his  own purposes1 
but when so introduced it was           
specifically postulated  to be a matter 
of no moment at all to the points or    
lines or figures  that were moved4 ,it  
was purely an observational device1     
intended  for the observer's            
convenience1 and in the bargain a mental

 device1  calling for no physical action
 and the play of no force4 ,so far as   
,euclid in his daily life was obliged to
 take cognizance of the fact  that in   
the world of work-a-day realities motion
 existed1 he must1  as a true ,greek1   
have looked upon this as a most         
unfortunate deviation  of the reality   
from his beautiful world of intellectual
 abstraction1  and as something to be   
deplored and ignored4 ,even in their    
statuary  the ,greeks clung to this     
idea4 ,a group of marvelous action1 like
  the ,laocoon1 they held to be         
distinctly a second rate production1  a 
prostitution of the noble art2 their    
ideal was a figure like the  majestic   
,zeus--not necessarily a mere bust1 be  
it understood1 but  always a figure in  
repose without action4 ,their statuary  
stood for  things1 not for action1 just 
as their geometry stood for points1  not
 for events4    ,galileo and ,newton    
took a different viewpoint4 ,they were  
interested  in the world as it is1 not  

as it ought to be2 and if motion appears
  to be a fundamental part of that      
world1 they were bound to include  it in
 their scheme4 ,this made it necessary  
for them to pay much more  attention to 
the concept of time and its place in the
 world than did  the ,greeks4 ,in the   
superposition process1 and even when he 
allowed  a curve to be generated by a   
moving point1 the sole interest which   
,euclid had in the motion was the effect
 which was to be observed upon  his     
static figures after its completion4 ,in
 this effect the rate of  the motion did
 not enter4 ,so all questions of        
velocity and time are  completely       
ignored1 and we have in fact the curious
 spectacle of  motion without time4     
,to ,galileo and ,newton1 on the other  
hand1 the time which it took a  body to 
pass from one point of its path to      
another was of paramount  importance4   
,the motion itself was the object of    
their study1 and they  recognized the   
part played by velocity4 ,but ,galileo  

and ,newton were  still sufficiently    
under the influence of ,euclid to fit   
the observed  phenomena of motion1 so   
far as they could1 upon ,euclid's static
 world  of points4 ,this they effected  
by falling in with the age-old procedure
  of regarding time and space as        
something entirely disassociated and    
distinct4 ,the motion of an object--in  
theory1 of a point--was to be  recorded 
by observing its successive positions4  
,with each of these  positions a time   
was to be associated1 marking the       
instant at which  the point attained    
that position4 ,but in the face of this 
association1  space and time were to be 
maintained as entirely separate         
entities4          ,,the                
,,four-,,dimensional ,,world ,,of       
,,events    ,this severe separation of  
time and space ,minkowski has now       
questioned1  with the statement that the
 elements of which the external  world  
is composed1 and which we observe1 are  
not points at all1  but are events4     

,this calls for a revision of our whole 
habit of  thought4 ,it means that the   
perceptual world is four-dimensioned1   
not three-dimensioned as we have always 
supposed2 and it means1 at  the very    
least1 that the distinction between time
 and space is not  so fundamental as we 
had supposed4    ,this should not       
impress us as strange or                
incomprehensible4 ,what  do we mean when
 we say that a plane is two-dimensional8
 ,simply that  two coordinates1 two     
numbers1 must be given to specify the   
position  of any point of the plane4    
,similarly for a point in the space of  
our  accustomed concepts we must give   
three numbers to fix the position--as   
by giving the latitude and longitude of 
a point on the earth and its  height    
above sea-level4 ,so we say this space  
is three-dimensional4 ,but  a material  
body is not merely somewhere2 it is     
somewhere now1#ahb or  was somewhere    
yesterday1 or will be somewhere         
tomorrow4 ,the statement  of position   

for a material object is meaningless    
unless we at the  same time specify the 
time at which it held that position4 ,if
 ,i  am considering the life-history of 
an object on a moving train1  ,i must   
give three space-coordinates and one    
time-coordinate to fix  each of its     
positions4#ahb ,and each of its         
positions1 with the time  pertaining to 
that position1 constitutes an event4    
,the dynamic1  ever-changing world about
 us1 that shows the same aspect at no   
two  different moments1 is a world of   
events2 and since four measures or      
coordinates are required to fix an      
event1 we say this world of events  is  
four-dimensional4 ,if we wish to test   
out the soundness of this  viewpoint1 we
 may well do so by asking whether the   
naming of values  for the four          
coordinates fixes the event uniquely1 as
 the naming of  three under the old     
system fixes the point uniquely4        
,suppose we take some particular event  
as the one from which to measure1  and  

agree upon the directions to be taken by
 our space axes1 and make  any          
convention about our time-axis which    
subsequent investigation  may show to be
 necessary4 ,certainly then the act of  
measuring so many  miles north1 and so  
many west1 and so many down1 and so many
 seconds  backward1 brings us to a      
definite time and place--which is to    
say1  to a definite event4 ,perhaps     
nothing happened there1 in the sense  in
 which we usually employ the word2 but  
that is no more serious  than if we were
 to locate a point with reference to our
 familiar  space coordinate system1 and 
find it to lie in the empty void of     
interstellar space1 with no material    
body occupying it4 ,in this  second case
 we still have a point1 which requires1 
to insure its  existence and location1  
three coordinates and nothing more2 in  
the  first case we still have an event1 
which requires for its existence  and   
definition four coordinates and nothing 
more4 ,it is not an event  about which  

the headline writers are likely to get  
greatly excited2  but what of that8 ,it 
is there1 ready and waiting to define   
any physical  happening that falls upon 
it1 just as the geometer's point is     
ready  and waiting to define any        
physical body that chances to fall upon 
it4          ,a ,,continuum ,,of        
,,points    ,it is now in order to      
introduce a word1 which ,i shall have to
 confess  the great majority of the     
essayists introduce1 somewhat           
improperly1  without explanation4 ,but  
when ,i attempt to explain it1 ,i       
realize  quite well why they did this4  
,they had to have it2 and they didn't   
have space in their three thousand words
 to talk adequately about  it and about 
anything else besides4 ,the             
mathematician knows very  well indeed   
what he means by a continuum2 but it is 
far from easy  to explain it in ordinary
 language4 ,i think ,i may do best by   
talking  first at some length about a   
straight line1 and the points on it4    

,if the line contains only the points   
corresponding to the integral  distances
 #a1 #b1 #c1 etc41 from the starting    
point1 it is obviously  not             
continuous--there are gaps in it vastly 
more inclusive than the  few            
7comparatively speaking7 points that are
 present4 ,if we extend  the limitations
 so that the line includes all points   
corresponding  to ordinary proper and   
improper fractions like #a_/d and       
#ag_/bi and  #afcc_/g--what the         
mathematician calls the rational        
numbers--we shall  apparently fill in   
these gaps2 and ,i think the layman's   
first impulse  would be to say that the 
line is now continuous4 ,certainly we   
cannot  stand now at one point on the   
line and name the next point1 as we     
could a moment ago4 ,there is no next   
rational number to #aaf_/abe1  for      
instance2 #aae_/abd comes before it and 
#aag_/abf comes after it1  but between  
it and either1 or between it and any    
other rational  number we might name1   

lie many others of the same sort4 ,yet  
in spite  of the fact that the line     
containing all these rational points is 
now  dense 7the technical term for the  
property ,i have just indicated71  it is
 still not continuous2 for ,i can easily
 define numbers that  are not contained 
in it--irrational numbers in infinite   
variety  like sqrt#b2 or1 even worse1   
the number pi  #c4adaeib 444 which      
defines the ratio of the circumference  
of a circle to the diameter1  and many  
other numbers of similar sort4    ,if   
the line is to be continuous1 there may 
be no holes in it at all2  it must have 
a point corresponding to every number ,i
 can possibly  name4 ,similarly for the 
plane1 and for our three-space2 if they 
are  to be continuous1 the one must     
contain a point for every possible  pair
 of numbers x and y1 and the other for  
every possible set  of three numbers x1 
y and z1 that ,i can name4 ,there may be
 no  holes in them at all4    ,a line is
 a continuum of points4 ,a plane is a   

continuum of points4 ,a  three-space is 
a continuum of points4 ,these three     
cases differ only in  their             
dimensionality2 it requires but one     
number to determine a point  of the     
first continuum1 two and three          
respectively in the second and  third   
cases4 ,but the essential feature is not
 that a continuum shall  consist of     
points1 or that we shall be able to     
visualize a pseudo-real  existence for  
it of just the sort that we can         
visualize in the case  of line1 plane   
and point4 ,the essential thing is      
merely that it shall  be an aggregate of
 elements numerically determined in such
 a way as to  leave no holes1 but to be 
just as continuous as the real number   
system  itself4 ,examples1 however1     
aside from the three which ,i have used1
  are difficult to construct of such    
sort that the layman shall grasp  them  
readily2 so perhaps1 fortified with the 
background of example  already          
presented1 ,i may venture first upon a  

general statement4          ,,the       
,,continuum ,,in ,,general    ,suppose  
we have a set of elements of some       
sort--any sort4 ,suppose  that these    
elements possess one or more fundamental
 identifying  characteristics1 analogous
 to the coordinates of a point1 and     
which1 like these coordinates1 are      
capable of being given numerical        
values4 ,suppose we find that no two    
elements of the set possess  identically
 the same set of defining values4       
,suppose finally--and  this is the      
critical test--that the elements of the 
set are such that1  no matter what      
numerical values we may specify1 it we  
do specify the  proper number of        
defining magnitudes we define by these  
an actual  element of the set1 that     
corresponds to this particular          
collection  of values4 ,our elements    
then share with the real number system  
the property of leaving no holes1 of    
constituting a continuous  succession in
 every dimension which they possess4 ,we

 have then a  continuum4 ,whatever its  
elements1 whatever the character of     
their  numerical identifiers1 whatever  
the number n of these which stands for  
its dimension1 there may be no holes or 
we have no continuum4 ,there  must be an
 element for every possible combination 
of n numbers  we can name1 and no two of
 these combinations may give the same   
element4 ,granted this condition1 our   
elements constitute a continuum4    ,as 
,i have remarked1 it is not easy to cite
 examples of continua which  shall mean 
anything to the person unaccustomed to  
the term4 ,the  totality of             
carbon-oxygen-nitrogen-hydrogen         
compounds suggested  by one essayist as 
an example is not a continuum at all1   
for the  set contains elements          
corresponding only to integer values of 
the  numbers which tell us how many     
atoms of each substance occur in the    
molecule4 ,we cannot have a compound    
containing sqrt#b carbon atoms1  or #cpi
 oxygen atoms4 ,perhaps the most        

satisfactory of the continua1  outside  
the three ,euclidean space-continua     
already cited1 is the  manifold of music
 notes4 ,this is four-dimensional2 each 
note has  four distinctions--length1    
pitch1 intensity1 timbre--to distinguish
  it perfectly1 to tell how long1 how   
high1 how loud1 how rich4#bfc  ,we might
 have a little difficulty in reducing   
the characteristic of  richness to      
numerical expression1 but presumably it 
could be done2  and we should then be   
satisfied that every possible           
combination of four  values1 l1 p1 i1 t 
for these four identifying              
characteristics  would give us a musical
 effect1 and one to be confused with no 
other4    ,there is in the physical     
world a vast quantity of continua of one
  sort or another4 ,the music-note      
continuum brings attention to the  fact 
that not all of these are such that     
their elements make their  appeal to the
 visual sense4 ,this remark is a        
pertinent one2 for we are  by every     

right of heritage an eye-minded race1   
and it is frequently  necessary for us  
to be reminded that so far as the       
external world is  concerned1 the       
verdict of every other sense is entirely
 on a par with  that of sight4 ,the     
things which we really see1 like matter1
 and the  things which we abstract from 
these visual impressions1 like space1   
are by no means all there is to the     
world4          ,,euclidean ,,and       
,,non-,,euclidean ,,continua    ,if we  
are dealing with a continuum of any sort
 whatever having one  or two or three   
dimensions1 we are able to represent it 
graphically  by means of the line1 the  
plane1 or the three-space4 ,the same set
 of  numbers that defines an element of 
the given continuum likewise defines  an
 element of the ,euclidean continuum of 
the same dimensionality2  so the one    
continuum corresponds to the other1     
element for element1  and either may    
stand for the other4 ,but if we have a  
continuum of four  or more dimensions1  

this representation breaks down in the  
absence  of a real1 four-dimensional    
,euclidean point-space to serve as a    
picture4 ,this does not in the least    
detract from the reality of the         
continuum which we are thus prevented   
from representing graphically  in the   
accustomed fashion4    ,the ,euclidean  
representation1 in fact1 may in some    
cases be  unfortunate--it may be so     
entirely without significance as to be  
actually misleading4 ,for in the        
,euclidean continuum of points1 be  it  
line1 plane or three-space1 there are   
certain things which we  ordinarily     
regard as secondary derived properties1 
but which possess  a great deal of      
significance none the less4    ,in      
particular1 in the ,euclidean plane and 
in ,euclidean three-space1  there is the
 distance between two points4 ,i have   
indicated1 in  the chapter on           
non-,euclidean geometry1 that the       
parallel postulate  of ,euclid1 which   
distinguishes his geometry from others1 

could be  replaced by any one of        
numerous other postulates4 ,grant       
,euclid's  postulate and you can prove  
any of these substitutes2 grant any of  
the substitutes and you can prove       
,euclid's postulate4 ,now it happens    
that there is one of these substitutes  
to which modern analysis has  given a   
position of considerable importance4 ,it
 is merely our good  old friend the     
,pythagorean theorem1 that the square on
 the hypotenuse  equals the sum of the  
squares on the sides2 but it is dressed 
in new  clothes for the present         
occasion4    ,mr4 ,francis' discussion  
of this part of the subject1 and        
especially his  figure1 ought to make it
 clear that this theorem can be         
considered  as dealing with the distance
 between any two points4 ,when we so    
consider it1 and take it as the         
fundamental1 defining postulate of      
,euclidean geometry which distinguishes 
this geometry from others1  we have a   
statement of considerable content4 ,we  

have1 first1 that the  characteristic   
property of ,euclidean space is that the
 distance between  two points is given  
by the square root of the sum of the    
squares of  the coordinate-differences  
for these points--by the expression     
,d  sqrt7,x - x7#b  7,y - y7#b  7,z -   
z7#b11    where the large letters       
represent the coordinates of the one    
point  and the small ones those of the  
other4 ,we have more than this1         
however2 we have that this distance is  
the same for all observers1  no matter  
how different their values for the      
individual coordinates  of the          
individual points4 ,and we have1        
finally1 as a direct result of  looking 
upon the thing from this viewpoint1 that
 the expression for ,d  is an invariant2
 which simply means that every observer 
may use the  same expression in         
calculating the value of ,d in terms of 
his own  values for the coordinates     
involved4 ,the distance between two     
points  in our space is given           

numerically by the square root of the   
sum of the  squares of my               
coordinate-differences for the two      
points involved2 it  is given equally by
 the square root of the sum of the      
squares of your  coordinate-differences1
 or those of any other observer         
whatsoever4 ,we  have then a natural    
law--the fundamental natural law        
characterizing  ,euclidean space4 ,if we
 wish to apply it to the ,euclidean     
two-space 7the  plane7 we have only to  
drop out the superfluous                
coordinate-difference2  if we wish to   
see by analogy what would be the        
fundamental natural law  for a          
four-dimensional ,euclidean space1 we   
have only to introduce under  the       
radical a fourth coordinate-difference  
for the fourth dimension4    ,if we were
 not able to attach any concrete meaning
 to the expression  for ,d the value of 
all this would be materially lessened4  
,consider1  for instance1 the continuum 
of music notes4 ,there is no distance   

between different notes4 ,there is of   
course significance in talking  about   
the difference in pitch1 in intensity1  
in duration1 in timbre1  between two    
notes2 but there is none in a mode of   
speech that implies  a composite        
expression indicating how far one note  
escapes being  identical with another in
 all four respects at once4 ,the        
trouble1  of course1 is that the four   
dimensions of the music-note continuum  
are not measurable in terms of a common 
unit4 ,if they were1 we should  expect  
to measure their combination more or    
less absolutely in terms  of this same  
unit4 ,we can make measurements in all  
three dimensions  of ,euclidean space   
with the same unit1 with the same       
measuring  rod in fact4 ,this presents a
 peculiarity of our three-space which   
is not possessed by all                 
three-dimensional manifolds4 ,riemann   
has  given another illustration in the  
system of all possible colors1  composed
 of arbitrary proportions of the three  

primaries1 red1 green  and violet4 ,this
 system forms a three-dimensional       
continuum2 but we  cannot measure the   
distance or difference between two      
colors in  terms of the difference      
between two others4#acj    ,accordingly1
 in spite of the fact that the          
,euclidean three-space  gives us a      
formal representation of the color      
continuum1 and in  spite of the fact    
that the hypothetical four-dimensional  
,euclidean  space would perform a like  
office for the music-note continuum1    
this  representation would be without   
significance4 ,we should not say that   
the geometry of these two manifolds is  
,euclidean4 ,we should realize  that any
 set of numerical elements can be       
plotted in a ,euclidean  space of the   
appropriate dimensionality2 and that    
accordingly1 before  allowing such a    
plot to influence us to classify the    
geometry of the  given manifold as      
,euclidean1 we must pause long enough to
 ask whether  the rest of the ,euclidean

 system fits into the picture4 ,if the  
square  root of the sum of the squares  
of the coordinate-differences between   
two elements possesses significance in  
the given continuum1 and if it  is      
invariant between observers of that     
continuum who employ different  bases of
 reference1 then and only then may we   
allege the ,euclidean  character of the 
given continuum4    ,if under this test 
the given continuum fails of            
,euclideanism1 it is  in order to ask   
what type of geometry it does present4  
,if it is of  such character that the   
distance between two elements possesses 
significance1 we should answer this     
question by investigating that  distance
 in the hope of discovering a           
non-,euclidean expression for it  which 
will be invariant4 ,if it is not of such
 character1 we should seek  some other  
characteristic of single elements or    
groups of elements1  of real physical   
significance and of such sort that the  
numerical  expression for it would be   

invariant4    ,if the continuum with    
which we have to do is one in which the 
distance between two elements possesses 
significance1 and if it  turns out that 
the invariant expression for this       
distance is not  the ,pythagorean one1  
but one indicating the non-,euclideanism
 of  our continuum1 we say that this    
continuum has a curvature4 ,this  means 
that1 if we interpret the elements of   
our continuum as points  in space 7which
 of course we may properly do7 and if we
 then try to  superpose this            
point-continuum upon a ,euclidean       
continuum1 it will not  go2 we shall be 
caught in some such absurdity as trying 
to force a  sphere into coincidence with
 a plane4 ,and of course if it won't go1
  the only possible reason is that it is
 curved or distorted1 like  the sphere1 
in such a way as to prevent its going4  
,it is unfortunate  that the visualizing
 of such curvature requires the         
visualizing of  an additional dimension 
for the curved continuum to curve into2 

so  that while we can picture a curved  
surface easily enough1 we can't  picture
 a curved three-space or four-space4    
,but that is a barrier to  visualization
 alone1 and in no sense to              
understanding4          ,,our ,,world   
,,of ,,four ,,dimensions    ,it will be 
observed that we have now a much broader
 definition of  non-,euclideanism than  
the one which served us for the         
investigation  of ,euclid's parallel    
postulate4 ,if we may at pleasure accept
  this postulate or replace it by       
another and different one1 we  may      
presumably do the same for any other or 
any others of ,euclid's  postulates4    
,the very statement that the distance   
between elements of  the continuum shall
 possess significance1 and shall be     
measurable by  considering a path in the
 continuum which involves other         
elements1  is an assumption4 ,if we     
discard it altogether1 or replace it by 
one  postulating that some other joint  
property of the elements than  their    

distance be the center of interest1 we  
get a non-,euclidean  geometry4 ,so for 
any other of ,euclid's postulates2 they 
are all  necessary for a ,euclidean     
system1 and in the absence of any one of
  them we get a non-,euclidean system4  
,now the four-dimensional time-space    
continuum of ,minkowski is plainly  of a
 sort which ought to make susceptible of
 measurement the separation  between two
 of its events4 ,we can pass from one   
element to another  in this             
continuum--from one event to another--by
 traversing a path  involving successive
 events4 ,our very lives consist in     
doing just  this3 we pass from the      
initial event of our career to the final
  event by traversing a path leading us 
from event to event1 changing  our time 
and space coordinates continuously and  
simultaneously in the  process4 ,and    
while we have not been in the habit of  
measuring anything  except the space    
interval between two events and the time
 interval  between two events1          

separately1 ,i think it is clear enough 
that1  considered as events1 as elements
 in the world of four dimensions1  there
 is a less separation between two events
 that occur in my office  on the same   
day than between two which occur in my  
office a year  apart2 or between two    
events occurring #aj minutes apart when 
both  take place in my office than when 
one takes place there and one in        
,london or on ,betelgeuse4    ,it is not
 at all unreasonable1 a priori1 then1 to
 seek a numerical  measure for the      
separation1 in space-time of            
four-dimensions1 of  two events4 ,if we 
find it1 we shall doubtless be asked    
just what  its subjective significance  
to us is4 ,this must be answered with   
some circumspection4 ,it will presumably
 be something which we cannot  observe  
with the visual sense alone1 or it would
 have forced itself  upon our attention 
thousands of years ago4 ,it ought1 ,i   
should think1  to be something that we  
would sense by employing at the same    

time  the visual sense and the sense of 
time-passage4 ,in fact1 ,i might  very  
plausibly insist that1 by my very       
remarks about it in the above           
paragraph1 ,i have sensed it4           
,minkowski1 however1 was not worried    
about this phase of the matter4 ,he  had
 only to identify the invariant         
expression for distance2 sensing  it    
could wait4 ,he found1 of course1 that  
this expression was not the  ,euclidean 
expression for a four-dimensional       
interval4 ,he had discarded  several of 
the ,euclidean assumptions and could not
 expect that the  postulate governing   
the metric properties of ,euclid's space
 would  persist4 ,especially had he     
violated the ,euclidean canons in       
discarding1  with ,einstein1 the notion 
that nothing which may happen to a      
measuring  rod in the way of uniform    
translation at high velocity can affect 
its measures4 ,so he had to be prepared 
to find that his geometry was           
non-,euclidean2 yet it is surprising to 

learn how slightly it deviates  from    
that of ,euclid4 ,without any extended  
discussion to support the  statement1 we
 may say that he found that when two    
observers measure the  time- and the    
space-coordinates of two events1 using  
the assumptions  and therefore the      
methods of ,einstein and hence          
subjecting themselves  to the condition 
that their measures of the pure         
time-interval and of  the pure          
space-interval between these events will
 not necessarily be  the same1 they will
 discover that they both get the same   
value for  the expression        ,s     
sqrt7,x - x7#b  7,y - y7#b  7,z - z7#b -
 7,,ct - ,ct7#b14    ,if our acceptance 
of this as the numerical measure of the 
separation  in space-time between the   
two events should lead to contradiction 
we  could not so accept it4 ,no         
contradiction arises however and we may 
therefore accept it4 ,and at once the   
mathematician is ready with some        
interpretative remarks4          ,,the  

,,curvature ,,of ,,space-,,time    ,the 
invariant expression for separation1 it 
will be seen1 is in  the same form as   
that of the ,euclidean four-dimensional 
invariant  save for the minus sign      
before the time-difference 7the         
appearance  of the constant ,c in       
connection with the time coordinate t is
  merely an adjustment of units2 see    
page #aec74 ,this tells us that not     
alone is the geometry of the time-space 
continuum non-,euclidean in its  methods
 of measurement1 but also in its        
results1 to the extent that it          
possesses a curvature4 ,it compares with
 the ,euclidean four-dimensional        
continuum in much the same way that a   
spherical surface compares with  a      
plane4 ,as a matter of fact1 a more     
illuminating analogy here would  be that
 between the cylindrical surface and the
 plane1 though neither  is quite exact4 
,to make this clear requires a little   
discussion of  an elementary notion     
which we have not yet had to consider4  

,our three-dimensional existence often  
reduces1 for all practical  purposes1 to
 a two-dimensional one4 ,the objects and
 the events of  a certain room may quite
 satisfactorily be defined by thinking  
of them1 not as located in space1 but as
 lying in the floor of  the room4       
,mathematically the justification for   
this viewpoint  is got by saying that we
 have elected to consider a slice of our
  three-dimensional world of the sort   
which we know as a plane4 ,when we      
consider this plane and the points in   
it1 we find that we have taken  a       
cross-section of the three-dimensional  
world4 ,a line in that world  is now    
reduced1 for us1 to a single point--the 
point where it cuts  our plane2 a plane 
is reduced to a line--the line where it 
cuts our  plane2 the three-dimensional  
world itself is reduced to our plane    
itself4 ,everything three-dimensional   
falls down into its shadow in  our      
plane1 losing in the process that one of
 the three dimensions  which is not     

present in our plane4    ,for           
simplicity's sake it is usual to take a 
cross-section of space  parallel to one 
of our coordinate axes4 ,we think of our
 three  dimensions as extending in the  
directions of those axes2 and it is     
easier to take a horizontal or vertical 
section which shall simply  wipe out one
 of these dimensions than to take an    
oblique section which  shall wipe out a 
dimension that consists partly of our   
original length1  and partly of our     
original width1 and partly of our       
original height4    ,if we have a       
four-dimensional manifold to begin with1
 we may equally  shake out one of the   
four dimensions1 one of the four        
coordinates1  and consider the          
three-dimensional result of this process
 as a  cross-section of the original    
four-dimensional continuum4 ,and where1 
in  cross-sectioning a three-dimensioned
 world1 we have but three choices  of a 
coordinate to eliminate1 in             
cross-sectioning a world of four        

dimensions we have four choices4 ,by    
dropping out either the x1 or the  y1 or
 the z1 or the t1 we get a              
three-dimensioned cross-section4    ,now
 our accustomed three-dimensional space 
is strictly ,euclidean4 ,when  we       
cross-section it1 we get a ,euclidean   
plane no matter what the  direction in  
which we make the cut4 ,likewise the    
,euclidean plane is  wholly ,euclidean1 
because when we cross-section it in any 
direction  whatever we get a ,euclidean 
line4 ,a cylindrical surface1 on the    
other  hand1 is neither wholly          
,euclidean nor wholly non-,euclidean in 
this  matter of cross-sectioning4 ,if we
 take a section in one direction we  get
 a ,euclidean line and if we take a     
section in the other direction  we get a
 circle 7if the cylindrical surface be a
 circular one74 ,and  of course if we   
take an oblique section of any sort1 it 
is neither  line nor circle1 but a      
compromise between the two--the         
significant  thing being that it is     

still not a ,euclidean line4    ,the    
space-time continuum presents an        
analogous situation4 ,when  we          
cross-section it by dropping out any one
 of the three space  dimensions1 we get 
a three-dimensional complex in which the
 distance  formula is still             
non-,euclidean1 retaining the minus sign
 before  the time-difference and        
therefore retaining the geometric       
character  of its parent4 ,but if we    
take our cross-section in such a way as 
to  eliminate the time coordinate1 this 
peculiarity disappears4 ,the signs  in  
the invariant expression are then all   
plus1 and the cross-section  is in fact 
our familiar ,euclidean three-space4    
,if we set up a surface geometry on a   
sphere1 we find that the  elimination of
 one dimension leaves us with a         
line-geometry that is  still            
non-,euclidean since it pertains to the 
great circles of the  sphere rather than
 to ,euclidean straight lines4 ,in      
shaking ,minkowski's  continuum down    

into a three-dimensional one by         
eliminating any one of  his coordinates1
 if we eliminate either the x1 the y or 
the z1  we have left a three-dimensional
 geometry in which the disturbing minus 
sign still occurs in the                
distance-formula1 and which is therefore
 still  non-,euclidean4 ,if we omit the 
t1 this does not occur4 ,we see1 then1  
that the time dimension is the          
disturbing factor1 the one which gives  
to space-time its non-,euclidean        
character so far as the possession  of  
curvature is concerned4 ,and we see that
 this curvature is not the  same in all 
directions1 and in one direction is     
actually zero--whence  the attempted    
analogy with a cylinder instead of with 
a sphere4    ,many writers on relativity
 try to give the space-time continuum an
  appeal to our reason and a character  
of inevitableness by insisting on  the  
lack of any fundamental distinction     
between space and time4 ,the  very      
expression for the space-time invariant 

denies this4 ,time is  distinguishable  
from space4 ,the three dimensions of    
space are quite  indistinguishable--we  
can interchange them without affecting  
the  formula1 we can drop one out and   
never know which is gone4 ,but the  very
 formula singles out time as distinct   
from space1 as inherently  different in 
some way4 ,it is not so inherently      
different as we have  always supposed2  
it is not sufficiently different to     
offer any obstacle  to our thinking in  
terms of the four-dimensional continuum4
 ,but while  we can group space and time
 together in this way1 this does not    
mean  at all that space and time cease  
to differ4 ,a cook may combine meat     
with potatoes and call the product hash1
 but meat and potatoes do  not thereby  
become identical4#bbc          ,,the    
,,question ,,of ,,visualization    ,to  
the layman there is a great temptation  
to say that while1  mathematically      
speaking1 the space-time continuum may  
be a great  simplification1 it does not 

really represent the external world4 ,to
  be sure1 you can't see the space-time 
continuum in precisely the same  way    
that you can the three-dimensional space
 continuum1 but this is  only because   
,einstein finds the time dimension to be
 not quite freely  interchangeable with 
the space dimension4 ,yet you do        
perceive this  space-time continuum1 in 
the manner appropriate for its          
perception2  and it would be just as    
sensible to throw out the space         
continuum  itself on the ground that    
perception of the two is not of exactly 
the same sort1 as to throw out the      
space-time continuum on this  ground4   
,with appropriate conventions1 either   
may stand as the mental  picture of the 
external world2 it is for us to choose  
which is the  more convenient and useful
 image4 ,einstein tells us that his     
image  is the better1 and tells us why4 
,before we look into this1 we must let  
him tell us something more  about the   
geometry of his continuum4 ,what he     

tells us is1 in its  essentials1 just   
this4 ,the observer in a pure space     
continuum of three  dimensions finds    
that as he changes his position1 his    
right-and-left1  his                    
backward-and-forward1 and his           
up-and-down are not fixed directions    
inherent in nature1 but are fully       
interchangeable4 ,the observers1 in  the
 adjoined sketch1 whose verticals are as
 indicated by the arrows1  find very    
different vertical and horizontal       
components for the distance  between the
 points ,o and ,p2 a similar situation  
would prevail if  we used all three     
space directions4 ,the statement        
analogous to this  for ,einstein's      
four-dimensional continuum of space and 
time combined  is that1 as observers    
change their relative motion1 their time
 axes  take slightly different          
directions1 so that what is purely space
 or  purely time for the one becomes    
space with a small component in the     
time direction1 or time with a small    

component in the space direction1  for  
the other4 ,this it will be seen        
explains fully why observers in         
relative motion can differ about space  
and time measurements4 ,we  should not  
be at all surprised if the two observers
 of the figure  reported different      
values for horizontals and verticals2 we
 should  realize that what was vertical 
for one had become partly horizontal    
for the other4 ,it is just so1 says     
,einstein1 with his observers of  time  
and space who are in relative motion to 
one another2 what one  sees as space the
 other sees as partly time1 because     
their time axes  do not run quite       
parallel4    ,the natural question here1
 of course1 is ,well1 where are their   
time axes8 ,if you know what to look    
for1 of course1 you ought to  be able to
 perceive them in just the way you      
perceive ordinary time  intervals--with 
the reservation that they are imaginary1
 after all1  just like your space axes1 
and that you must only expect to see    

them  in imagination4 ,if you look for a
 fourth axis in ,euclidean three-space  
to represent your time axis1 you will of
 course not find it4 ,but you  will by  
all means agree with me that your time  
runs in a definite  direction2 and this 
it is that defines your time axis4      
,einstein adds  that if you and ,i are  
in relative motion1 my time does not run
 in  quite the same direction as yours4 
,how shall we prove it8 ,well1 how would
 we prove it if he told us that  our    
space axes did not run in precisely the 
same direction8 ,of course  we could not
 proceed through direct measures upon   
the axes themselves2  we know these are 
imaginary4 ,what we should do would be  
to strike out1  each of us1 a very long 
line indeed in what seemed the true     
horizontal  direction2 and we should    
hope that if we made them long enough1  
and measured them accurately enough1 we 
should be able to detect any  divergence
 that might exist4 ,this is precisely   
what we must do with  our time axes if  

we wish to verify ,einstein's statement 
that they are  not precisely parallel2  
and what better evidence could we demand
 of  the truth of this statement than   
the evidence already presented--that    
when we measure our respective time     
components between two events1  we get  
different results8          ,,what ,,it 
,,all ,,leads ,,to    ,the preceding    
chapters have been compiled and written 
with a view to  putting the reader in a 
state of mind and in a state of         
informedness  which shall enable him to 
derive profit from the reading of the   
actual  competing essays which make up  
the balance of the book4 ,for this      
purpose it has been profitable to take  
up in detail the preliminaries  of the  
,special ,theory of ,relativity1 and to 
allow the ,general ,theory  to go by    
default1 in spite of the fact that it is
 the latter which  constitutes          
,einstein's contribution of importance  
to science4 ,the  reason for this is    
precisely the same as that for taking up

 ,euclidean  geometry and mastering it  
before proceeding to the study of       
,newtonian  mechanics4 ,the fundamental 
ideas of the two theories1 while by no  
means identical1 are in general terms   
the same2 and the conditions            
surrounding their application to the    
,special ,theory are so very  much      
simpler than those which confront us    
when we apply them to the  more general 
case1 that this may be taken as the     
controlling factor  in a popular        
presentation4 ,we cannot omit the       
,general ,theory from  consideration1 of
 course2 but we can omit it from our    
preliminary  discussion1 and leave its  
development to the complete essays which
  follow1 and which in almost every case
 give it the larger half of their  space
 which its larger content demands4 ,in  
the process of the slow  and difficult  
preparation of the lay mind for the     
assimilation of an  altogether new set  
of fundamental ideas1 it is altogether  
desirable to  give the ,special ,theory1

 with its simpler applications of these 
ideas1  a place out of proportion to its
 importance in ,einstein's completed    
structure2 and this we have therefore   
done4    ,the ,special ,theory1         
postulating the relativity of uniform   
motion and  deducing the consequences of
 that relativity1 is often referred to  
as a special case of the ,general       
,theory1 in which this restriction  of  
uniformity is removed4 ,this is not     
strictly speaking correct4 ,the         
,general ,theory1 when we have          
formulated it1 will call our attention  
to  something which we really knew all  
the time1 but to which we chose not  to 
give heed--that in the regions of space 
to which we have access1  uniform motion
 does not exist4 ,all bodies in these   
regions are under  the gravitational    
influence of the other bodies therein1  
and this  influence leads to accelerated
 motion4 ,nothing in our universe can   
possibly travel at uniform velocity2 the
 interference of the rest of  the bodies

 in the universe prevents this4         
,obviously1 we ought not to apply the   
term special case to a case that  never 
occurs4 ,nevertheless1 this case is of  
extreme value to us in our  mental      
processes4 ,many of the motions with    
which we are concerned are  so nearly at
 constant velocity that we find it      
convenient to treat  them as though they
 were uniform1 either ignoring the      
resulting  error or correcting for it at
 the end of our work4 ,in many other    
cases we are able to learn what actually
 occurs under accelerated  motion by    
considering what would have occurred    
under uniform motion  were such a thing 
possible4 ,science is full of           
complications which  we unravel in this 
fashion4 ,the physicist deals with gas  
pressures  by assuming temperatures to  
be constant1 though he knows temperature
  never is constant2 and in turn he     
deals with temperatures by assuming     
pressures to be constant4 ,after this1  
he is able to predict what will  happen 

when1 as in nature1 pressures and       
temperatures are varying                
simultaneously4 ,by using as a channel  
of attack the artificially simple  case 
that never occurs1 we get a grip on the 
complex case that gives  us a true      
picture of the phenomenon4 ,and because 
in actual nature we  can come as close  
as we please to this artificial case1 by
 supposing  the variable factor to      
approach constancy1 so when we assume it
 to be  absolutely constant we speak of 
the result as the limiting case4 ,this  
situation does not occur1 but is the    
limiting case for those that  do occur4 
,when1 in the matter of motion1 we      
abandon the artificial1 limiting  case  
of uniform velocity and look into the   
general1 natural one of  unrestricted   
motion1 we find that the structure which
 we have built up  to deal with the     
limiting case provides us with many of  
the necessary  ideas and viewpoints4    
,this is what we expect--in it lies the 
value  of the limiting case4 ,we shall  

see that the relativity of time and     
space1 established for the limiting     
case1 holds good in the general  one4   
,we shall see that the idea of the      
four-dimensional space-time  continuum  
as representing the external world      
persists1 forming the  whole background 
of the ,general ,theory much more       
definitely than  in the ,special        
,theory4 ,incidentally we shall see that
 the greater  generality of the case    
under consideration will demand a       
greater degree  of generality in the    
geometry of this continuum1 a           
non-,euclideanism of  a much more       
whole-hearted type than that of the     
,special ,theory4 ,but  all the         
revisions of fundamental concepts which 
we have been at such  pains to make for 
the sake of the ,special ,theory will   
remain with us  in the ,general4 ,with  
this we may consider our preliminary    
background  as established1 and give our
 attention to the essayists1 who will   
try to take us more deeply into the     

subject than we have yet gone1  without 
losing us in its intricacies4           
,,vii    ,,relativity    ,the ,winning  
,essay in the ,contest for the ,eugene  
,higgins #e1jjj ,prize    ,,by ,,lyndon 
,,bolton  ,,senior ,,examiner1 ,,british
 ,,patent ,,office ,,london      ,the   
reader is probably acquainted with the  
method of specifying  positions of      
points in a plane by their distances    
from two mutually  perpendicular lines1 
or if the points are in space by their  
distances  from three mutually          
perpendicular planes like adjacent sides
 of a  flat-sided box4 ,the method is in
 fact in common use for exhibiting      
relations between quantities by graphs  
or diagrams4 ,these sets of axes1  as   
they are called1 together with any      
scales used for measuring1 must  be     
supposed rigid1 otherwise the events or 
points which they are used  to specify  
are indefinite4 ,the lengths which      
locate any point with  reference to a   
set of axes are called its coordinates4 

,when such systems are used for physical
 purposes1 they must be  supplemented by
 clocks to enable the times at which    
events occur to  be determined4 ,the    
clocks must be synchronized1 and must go
 at the  same rate1 but it must suffice 
here to state that this is possible     
without indicating how these conditions 
can be attained4 ,a system of  axes with
 its clocks will hereinafter be called a
 ,frame of ,reference1  and every       
observer will be supposed to be provided
 with such a frame  partaking of his    
motion4 ,all the objects which partake  
of an observer's  motion will be called 
his system4    ,it is a question whether
 among all possible frames of reference 
any one frame or class of frames is more
 suited than another for  the           
mathematical statement of physical laws4
 ,this is for experience  to decide1 and
 a ,principle of ,relativity is a       
statement embodying  the answer4        
,,the ,,mechanical ,,principle ,,of     
,,relativity    ,it has been ascertained

 that all such frames are equally       
suitable for  the mathematical statement
 of general mechanical laws1 provided   
that  their motion is rectilinear and   
uniform and without rotation4 ,this     
fact is comprehended in the general     
statement that all unaccelerated  frames
 of reference are equivalent for the    
statement of the general  laws of       
mechanics4 ,this is the mechanical      
principle of relativity4    ,it is well 
recognized however that the laws of     
dynamics as hitherto  stated involve the
 assumptions that the lengths of rigid  
bodies are  unaffected by the motion of 
the frame of reference1 and that        
measured  times are likewise unaffected2
 that is to say that any length measured
  on his own system by either of two    
relatively moving observers appears  the
 same to both observers1 or that lengths
 of objects and rates of  clocks do not 
alter whatever the motion relative to an
 observer4 ,these  assumptions seem so  
obvious that it is scarcely perceived   

that they  are assumptions at all4 ,yet 
this is the case1 and as a matter of    
fact  they are both untrue4             
,,the ,,special ,,principle ,,of        
,,relativity    ,although all           
unaccelerated frames of reference are   
equivalent for  the purposes of         
mechanical laws1 this is not the case   
for physical  laws generally as long as 
the above suppositions are adhered  to4 
,electromagnetic laws do alter their    
form according to the motion  of the    
frame of reference2 that is to say1 if  
these suppositions are  true1           
electromagnetic agencies act in         
different ways according to the  motion 
of the system in which they occur4      
,there is nothing a priori  impossible  
in this1 but it does not agree with     
experiment4 ,the motion  of each        
locality on the earth is continually    
changing from hour to  hour but no      
corresponding changes occur in          
electromagnetic actions4 ,it  has       
however been ascertained that on        

discarding these suppositions  the      
difficulty disappears1 and              
electromagnetic laws retain their  form 
under all circumstances of unaccelerated
 motion4 ,according  to the theory of   
relativity1 the correct view which      
replaces these  suppositions is         
deducible from the following postulates3
      7#a7 ,by no experiment conducted  
on his own system can an observer       
detect the unaccelerated motion of his  
system4    7#b7 ,the measure of the     
velocity of light in vacuo is unaffected
        by relative motion between the  
observer and the source of light4       
,both these postulates are well         
established by experiment4 ,the first   
may be illustrated by the familiar      
difficulty of determining whether a     
slowly moving train one happens to be   
sitting in1 or an adjacent one1  is in  
motion4 ,the passenger has either to    
wait for bumps 7that is1  accelerations7
 or else he has to look out at some     
adjacent object which  he knows to be   

fixed1 such as a building 7that is1 he  
has to perform  an experiment on        
something outside his system71 before he
 can decide4    ,the second postulate is
 an obvious consequence of the wave     
theory of  light4 ,just as waves in     
water1 once started by a ship1 travel   
through  the water with a velocity      
independent of the ship1 so waves in    
space  travel onward with a speed       
bearing no relation to that of the body 
which originated them4 ,the statement   
however is based on experiment1  and can
 be proved independently of any theory  
of light4    ,it is not difficult to    
deduce from these postulates certain    
remarkable  conclusions relating to the 
systems of two observers1 ,a and ,b1 in 
relative motion1 among them the         
following3      7#a7 ,objects on ,b's   
system appear to ,a to be shorter in the
        direction of relative motion    
than they appear to ,b4    7#b7 ,this   
opinion is reciprocal4 ,b thinks that   
,a's measurements        on ,a's system 

are too great4    7#c7 ,similarly for   
times3 each observer thinks that the    
other's        clocks have a slower rate
 than his own1 so that ,b's durations   
of time appear shorter to ,b than to ,a1
 and conversely4    7#d7 ,events which  
appear simultaneous to ,a do not in     
general appear        so to ,b1 and     
conversely4    7#e7 ,lengths at right   
angles to the direction of motion are   
unaffected4    7#f7 ,these effects vary 
with the ratio of the relative velocity 
to that of light4 ,the greater the      
relative velocity1 the        greater   
the effects4 ,they vanish if there is no
 relative        velocity4    7#g7 ,for 
ordinary velocities the effects are so  
small as to escape        notice4 ,the  
remarkable point however is their       
occurrence rather        than their     
magnitude4    7#h7 ,the observers       
similarly form different estimates of   
the        velocities of bodies on each 
other's systems4 ,the velocity of       
light however appears the same to all   

observers4    ,taking into account these
 revised views of lengths and times the 
mechanical principle of relativity may  
be extended to physical laws  generally 
as follows3 ,all unaccelerated frames of
 reference are  equivalent for the      
statement of the general laws of        
physics4 ,in this  form the statement is
 called the ,special1 or ,restricted1   
,principle  of ,relativity1 because it  
is restricted to unaccelerated frames of
  reference4 ,naturally the laws of     
classical mechanics now require some    
modification1 since the suppositions of 
unalterable lengths and times  no longer
 apply4          ,,the ,,four           
,,dimensional ,,continuum    ,lengths   
and times therefore have not the        
absolute character formerly  attributed 
to them4 ,as they present themselves to 
us they are relations  between the      
object and the observer which change as 
their motion  relative to him changes4  
,time can no longer be regarded as      
something  independent of position and  

motion1 and the question is what is the 
reality8 ,the only possible answer is   
that objects must be regarded as        
existing in four dimensions1 three of   
these being the ordinary ones of        
length1 breadth and thickness1 and the  
fourth1 time4 ,the term space  is       
applicable only by analogy to such a    
region2 it has been called a  continuum1
 and the analogue of a point in ordinary
 three-dimensional  space has been      
appropriately called an event4 ,by      
dimension must be  understood merely one
 of four independent quantities which   
locate an  event in this continuum4 ,in 
the nature of the case any clear mental 
picture of such a continuum is          
impossible2 mankind does not possess    
the requisite faculties4 ,in this       
respect the mathematician enjoys  a     
great advantage4 ,not that he can       
picture the thing mentally any  better  
than other people1 but his symbols      
enable him to abstract the  relevant    
properties from it and to express them  

in a form suitable  for exact treatment 
without the necessity of picturing      
anything1  or troubling whether or not  
the properties are those on which others
  rely for their conceptions4           
,,gravitation ,,and ,,acceleration      
,the limitation of statements of general
 law to uniformly moving systems  is    
hardly satisfactory4 ,the very concept  
of general law is opposed  to the notion
 of limitation4 ,but the difficulties of
 formulating a  law so that the         
statement of it shall hold good for all 
observers1  whose systems may be moving 
with different and possibly variable    
accelerations1 are very great4          
,accelerations imply forces which  might
 be expected to upset the formulation of
 any general dynamical  principles1 and 
besides1 the behavior of measuring rods 
and clocks  would be so erratic as to   
render unmeaning such terms as rigidity 
and measured time1 and therefore to     
preclude the use of rigid scales1  or of
 a rigid frame of reference which is the

 basis of the foregoing  investigation4 
,the following example taken from       
,einstein will make this clear1  and    
also indicate a way out of the          
difficulty4 ,a rotating system is       
chosen1 but since rotation is only a    
particular case of acceleration  it will
 serve as an example of the method of   
treating accelerated  systems generally4
 ,moreover1 as it will be seen1 the     
attribution of  acceleration to the     
system is simply a piece of scaffolding 
which  can be discarded when the general
 theory has been further developed4     
,let us note the experiences of an      
observer on a rotating disk which  is   
isolated so that the observer has no    
direct means of perceiving  the         
rotation4 ,he will therefore refer all  
the occurrences on the disk  to a frame 
of reference fixed with respect to it1  
and partaking of  its motion4    ,he    
will notice as he walks about on the    
disk that he himself and all  the       
objects on it1 whatever their           

constitution or state1 are acted upon   
by a force directed away from a certain 
point upon it and increasing  with the  
distance from that point4 ,this point is
 actually the center  of rotation1      
though the observer does not recognize  
it as such4 ,the  space on the disk in  
fact presents the characteristic        
properties of a  gravitational field4   
,the force differs from gravity as we   
know it by  the fact that it is directed
 away from instead of toward a center1  
and it obeys a different law of         
distance2 but this does not affect  the 
characteristic properties that it acts  
on all bodies alike1 and  cannot be     
screened from one body by the           
interposition of another4 ,an  observer 
aware of the rotation of the disk would 
say that the force  was centrifugal     
force2 that is1 the force due to inertia
 which a body  always exerts when it is 
accelerated4    ,next suppose the       
observer to stand at the point of the   
disk where  he feels no force1 and to   

watch someone else comparing1 by        
repeated  applications of a small       
measuring rod1 the circumference of a   
circle  having its center at that point1
 with its diameter4 ,the measuring rod  
when laid along the circumference is    
moving lengthwise relatively  to the    
observer1 and is therefore subject to   
contraction by his  reckoning4 ,when    
laid radially to measure the diameter   
this contraction  does not occur4 ,the  
rod will therefore require a greater    
proportional  number of applications to 
the circumference than to the diameter1 
and the number representing the ratio of
 the circumference of the  circle to the
 diameter thus measured will therefore  
be greater than  #c4adaei1 which is its 
normal value4 ,moreover the relative    
velocity  decreases as the center is    
approached1 so that the contraction of  
the  measuring rod is less when applied 
to a smaller circle2 and the ratio  of  
the circumference to the diameter1 while
 still greater than the  normal1 will be

 nearer to it than before1 and the      
smaller the circle  the less the        
difference from the normal4 ,for circles
 whose centers  are not at the point of 
zero force the confusion is still       
greater1  since the velocities relative 
to the observer of points on them now   
change from point to point4 ,the whole  
scheme of geometry as we know  it is    
thus disorganized4 ,rigidity becomes an 
unmeaning term since  the standards by  
which alone rigidity can be tested are  
themselves  subject to alteration4      
,these facts are expressed by the       
statement  that the observer's measured 
space is non-,euclidean2 that is to say1
  in the region under consideration     
measurements do not conform to the      
system of ,euclid4    ,the same         
confusion arises in regard to clocks4   
,no two clocks will in  general go at   
the same rate1 and the same clock will  
alter its rate  when moved about4       
,,the ,,general ,,principle ,,of        
,,relativity    ,the region therefore   

requires a space-time geometry of its   
own1 and  be it noted that with this    
special geometry is associated a        
definite  gravitational field1 and if   
the gravitational field ceases to exist1
  for example if the disk were brought  
to rest1 all the irregularities  of     
measurement disappear1 and the geometry 
of the region becomes  ,euclidean4 ,this
 particular case illustrates the        
following propositions  which form the  
basis of this part of the theory of     
relativity3      7#a7 ,associated with  
every gravitational field is a system of
        geometry1 that is1 a structure  
of measured space peculiar to           
that field4    7#b7 ,inertial mass and  
gravitational mass are one and the same4
    7#c7 ,since in such regions ordinary
 methods of measurement fail1           
owing to the indefiniteness of the      
standards1 the systems of               
geometry must be independent of any     
particular measurements4    7#d7 ,the   
geometry of space in which no           

gravitational field exists        in    
,euclidean4 #a    ,the connection       
between a gravitational field and its   
appropriate  geometry suggested by a    
case in which acceleration was their    
common  cause is thus assumed to exist  
from whatever cause the gravitational   
field arises4 ,this of course is pure   
hypothesis1 to be tested by             
experimental trial of the results       
derived therefrom4    ,gravitational    
fields arise in the presence of matter4 
,matter is  therefore presumed to be    
accompanied by a special geometry1 as   
though  it imparted some peculiar kink  
or twist to space which renders the     
methods of ,euclid inapplicable1 or     
rather we should say that the  geometry 
of ,euclid is the particular form which 
the more general  geometry assumes when 
matter is either absent or so remote as 
to  have no influence4 ,the dropping of 
the notion of acceleration is  after all
 not a very violent change in point of  
view1 since under  any circumstances the

 observer is supposed to be unaware of  
the  acceleration4 ,all that he is aware
 of is that a gravitational field  and  
his geometry coexist4    ,the prospect  
of constructing a system of geometry    
which does not depend  upon measurement 
may not at first sight seem hopeful4    
,nevertheless this  has been done4 ,the 
system consists in defining points not  
by their  distances from lines or planes
 7for this would involve measurement7   
but by assigning to them arbitrary      
numbers which serve as labels  bearing  
no relation to measured distances1 very 
much as a house  is located in a town by
 its number and street4 ,if this        
labeling  be done systematically1 regard
 being had to the condition that the    
label-numbers of points which are close 
together should differ from  one another
 by infinitesimal amounts only1 it has  
been found that  a system of geometry   
can actually be worked out4 ,perhaps    
this will  appear less artificial when  
the fact is called to mind that even    

when standards of length are available  
no more can be done to render  lengths  
of objects amenable to calculation than 
to assign numbers to  them1 and this is 
precisely what is done in the present   
case4 ,this  system of labeling goes by 
the name of ,gaussian coordinates after 
the mathematician ,gauss who proposed   
it4    ,it is in terms of ,gaussian     
coordinates that physical laws must be  
formulated if they are to have their    
widest generality1 and the general      
principle of relativity is that all     
,gaussian systems are equivalent for    
the statement of general physical laws4 
,for this purpose the labeling  process 
is applied not to ordinary space but to 
the four dimensional  space-time        
continuum4 ,the concept is somewhat     
difficult and it may  easily be         
aggravated into impossibility by anyone 
who thinks that he  is expected to      
visualize it4 ,fortunately this is not  
necessary2 it is  merely one of these   
irrelevancies to which those who are    

unaccustomed  to think in symbols are   
liable4    ,it will now be seen that    
among physical laws the law of          
gravitation  stands pre-eminent1 for it 
is gravitating matter which determines  
the geometry1 and the geometry          
determines the form of every other  law4
 ,the connection between the geometry   
and gravitation is the law  of          
gravitation4 ,this law has been worked  
out1 with the result that  ,newton's law
 of the inverse square is found to be   
approximate only1  but so closely       
approximate as to account for nearly all
 the motions of  the heavenly bodies    
within the limits of observation4 ,it   
has already  been seen that departure   
from the ,euclidean system is           
intensified  by rapidity of motion1 and 
the movements of these bodies are       
usually  too slow for this departure to 
be manifest4 ,in the case of the planet 
,mercury the motion is sufficiently     
rapid1 and an irregularity in its       
motion which long puzzled astronomers   

has been explained by the more  general 
law4    ,another deduction is that light
 is subject to gravitation4 ,this has   
given rise to two predictions1 one of   
which has been verified4 ,the           
verification of the other is as yet     
uncertain1 though the extreme           
difficulty of the necessary observations
 may account for this4    ,since light  
is subject to gravitation it follows    
that the constancy  of the velocity of  
light assumed in the earlier part of    
this  paper does not obtain in a        
gravitational field4 ,there is really no
  inconsistency4 ,the velocity of light 
is constant in the absence of           
gravitation1 a condition which          
unaccelerated motion implies4 ,the      
special principle of relativity is      
therefore a limiting case of the        
general principle4                      
,,viii    ,,the ,,new ,,concepts ,,of   
,,time ,,and ,,space    ,the ,essay in  
,behalf of ,which the ,greatest ,number 
of ,dissenting ,opinions  ,have ,been   

,recorded    ,,by ,,montgomery ,,francis
 ,,new ,,york      ,we have all had     
experiences1 on trains and boats1       
illustrating our  inability to tell1    
without looking off to some external    
body1 whether  we are at rest or moving 
uniformly2 and when we do so look1 to   
tell1  without reference to the ground  
or some other point external to both    
systems1 whether ours or the other be   
the seat of motion4 ,uniform  motion    
must be relative1 because we find       
nowhere in the universe a  body in the  
unique state of absolute rest from which
 alone absolute  motion might be        
measured4    ,true1 the wave theory of  
light with its homogeneous space-filling
  ether seemed to provide a reference   
standard for the concept of  absolute   
motion1 and for its measurement by      
experiment with light  rays4 ,but when  
,michelson and ,morley looked for this  
absolute motion  they found no trace of 
it4 ,to the physicist1 observational    
student  of the external world1 nothing 

exists save observationally2 what he    
can never observe is not there4 ,so3 ,i4
 ,by no means whatever may we  regard   
uniform straight-line motion as other   
than relative4    ,as a further direct  
consequence of the ,michelson-,morley   
experiment  we have3 ,,ii4 ,light in a  
vacuum presents the same velocity1 ,c   
#ahf1ccj miles per second1 to all       
observers whatever their velocity  of   
relative motion4 ,in addition to being  
experimentally established1  this is    
necessary to support ,i1 for if light   
will distinguish between  our           
velocities1 its medium is necessarily a 
universal standard for  absolute motion4
 ,but it is contrary to common sense to 
suppose that  if ,i pass you at #ajj    
miles per hour1 the same light impulse  
can pass  us both at the same speed1 ,c4
 ,we feel1 instinctively1 that space and
  time are not so constituted as to make
 this possible4 ,but the fact has  been 
repeatedly demonstrated4 ,and when      
common sense and fundamental  concepts  

clash with facts1 it is not the facts   
that must yield4 ,we  have survived such
 crises1 notably one where we had to    
change the  fundamental concept of      
up-and-down2 if another one is here1    
says  ,einstein1 let us meet it4        
,this the ,special ,theory of           
,relativity does4 ,it accepts           
,postulates  ,i and ,,ii above2 their   
consequences it deduces and interprets4 
,for  extensive demonstration of these  
,i lack space1 and this has been        
satisfactorily done by others so it is  
not my chief duty2 but clearly  they    
will be startling4 ,for the very ray of 
light which refuses to  recognize our   
relative motion is the medium through   
which ,i must  observe your system and  
you mine4    ,it turns out that ,i get  
different values for lengths and time   
intervals  in your system than you get1 
and vice versa4 ,and we are both right6 
,for  me to accept your correction were 
for me to admit that you are  at        
absolute rest and ,i in absolute motion1

 that your measure of  light velocity is
 right and mine wrong3 admissions barred
 by the  postulates4 ,we have nothing to
 correct2 we can only recognize the     
reason for the discrepancy2 and knowing 
our relative velocity1 each  can        
calculate from his own results what the 
other's will be4 ,we find1  of course1  
that at ordinary velocities the         
discrepancy is many times  too small for
 detection2 but at relative velocities  
at all comparable  with that of light it
 rises above the observational horizon4 
,to inquire the true length is          
meaningless4 ,chicago is east of        
,denver1  west of ,pittsburgh1 south of 
,milwaukee2 we do not consider this     
contradictory1 or demand the true       
direction of ,chicago4 ,einstein  finds 
that the concept of length1 between     
points in space or events  in time1 does
 not as we had supposed represent an    
intrinsic property  of the points or the
 events4 ,like direction1 it is merely a
 relation  between these and the        

observer--a relation whose value changes
 with  the observer's velocity relative 
to the object4 ,if our ideas of the     
part played in the world by time and    
space do not permit us to believe  this1
 we must alter these ideas4 ,let us see 
how we may do this4          ,a ,,world 
,,of ,,points    ,to deal with points in
 a plane the mathematician draws two    
perpendicular lines1 and locates any    
point1 as ,p1 by measuring its          
distances1 ,x and ,y1 from these        
coordinate axes4 ,the directions  of his
 axes acquire for him a peculiar        
significance1 standing out  above other 
directions2 he is apt to measure the    
distances ,x-x and  ,y-y between the    
points ,p and ,q in these directions1   
instead of  measuring the single        
distance ,,pq4 ,we do the same thing    
when we say  that the railroad station  
is five blocks north and two east4      
,the mathematician visualizes himself as
 an observer1 located on his  coordinate
 framework4 ,for another observer on    

another framework1 the  horizontal and  
vertical distances ,x'-x' and ,y'-y'    
between ,p and  ,q are different4 ,but  
for both1 the distance from ,p direct to
 ,q  is the same4 ,in each case the     
right triangle tells us that3           
,,pq  sqrt7,x-x7#b  7,y-y7#b            
sqrt7,x'-x'7#b  7,y'-y'7#b    ,imagine  
an observer so dominated by his         
coordinate system that he  knows no way 
of relating ,p with ,q save by their    
horizontal and  vertical separation4    
,his whole scheme of things would be    
shattered  by the suggestion that other 
observers on other reference frames find
  different horizontal and vertical     
components4 ,we have to show him the    
line ,,pq4 ,we have to convince him that
 this length is the absolute  property  
enjoyed by his pair of points2 that     
horizontals and verticals  are merely   
relations between the points and the    
observer1 result of  the observer's     
having analyzed the distance ,,pq into  
two components2  that different         

observers effect this decomposition     
differently2 that  this seems not to    
make sense to him only because of his   
erroneous  concept of a fundamental     
difference between verticals and        
horizontals4          ,,the             
,,four-,,dimensional ,,world ,,of       
,,events    ,we too have created a      
distinction in our minds corresponding  
to no  sufficient reality4 ,our minds   
seize on time as inherently separable   
from space4 ,we see the world made up of
 things in a continuum of three  space  
dimensions2 to make this dead world live
 there runs through it  a               
one-dimensional time continuum1 imposed 
from without1 unrelated4    ,but did you
 ever observe anything suggesting the   
presence of time  in the absence of     
space1 or vice versa8 ,no2 these vessels
 of the  universe always occur together4
 ,association of the space dimensions   
into a manifold from which time is      
excluded is purely a phenomenon  of the 
mind4 ,the space continuum cannot begin 

to exist until the time  dimension is   
supplied1 nor can time exist without a  
place to exist in4    ,the external     
world that we observe is composed1 not  
of points1  but of events4 ,if a point  
lacks position in time it does not      
exist2  give it this position and it    
becomes an event4 ,this world of events 
is four-dimensional--which means nothing
 more terrifying than that  you must    
make four measures to locate an event4  
,it does not mean1  at all1 that you    
must visualize four mutually            
perpendicular lines in  your accustomed 
three-space or in a four-space analogous
 to it4 ,if  this world of four         
dimensions seems to lack reality you    
will be  able to exhibit no better      
reality for your old ideas4 ,time       
belongs1  without question2 and not as  
an afterthought1 but as part of the     
world of events4    ,to locate an event 
we use four measures3 ,x1 ,y and ,z for 
space1  ,t for time4 ,using the same    
reference frame for time and space1 we  

locate a second event by the measures x1
 y1 z1 t4 ,minkowski  showed that the   
quantity        sqrt7,x-x7#b  7,y-y7#b  
7,z-z7#b - 7,,ct-,ct7#b    is the same  
for all observers1 no matter how        
different their x's1  y's1 z's and t's2 
just as in the plane the quantity       
sqrt7,x-x7#b  7,y-y7#b    is the same   
for all observers1 no matter how        
different their x's  and y's4    ,such a
 quantity1 having the same value for all
 observers1 is  absolute4 ,in the plane 
it represents the true1 absolute        
distance  between the points--their     
intrinsic property4 ,in dealing with    
events it  represents the true1 absolute
 interval1 in time and space together   
between the events4 ,it is not space1   
nor time1 but a combination of  the two4
 ,we have always broken it down into    
separate space and time  components4 ,in
 this we are as naive as the plane      
observer who could  not visualize the   
distance ,,pq until it was split into   
separate  horizontals and verticals4 ,he

 understood with difficulty that another
  observer1 employing a different       
reference frame because in different    
position1 would make the decomposition  
differently4 ,we understand with        
difficulty that another observer1       
employing a different reference  frame  
because in uniform motion relative to   
us1 will decompose the  interval between
 events into time and space components  
different  from ours4 ,time and space   
are relative to the observer2 only the  
interval representing space-time is     
absolute4 ,so common sense stands       
reconciled to the ,special ,theory of   
,relativity4          ,,successive      
,,steps ,,toward ,,generality    ,is    
then our laboriously acquired geometry  
of points in a  three-dimensional space 
to go into the discard8 ,by no means4   
,jeans1  investigating the equilibrium  
of gaseous masses1 found the general    
case too difficult for direct attack4   
,so he considered the case where  the   
masses involved are homogeneous and     

incompressible4 ,this never  occurs2 but
 it throws such light on the general    
case as to point the  way toward attack 
on it4    ,euclidean geometry excludes  
motion1 save that engineered by the     
observer2 and then the time is          
immaterial4 ,time does not enter at all2
  the three space dimensions suffice4   
,this simple case never occurs  where   
matter exists2 but its conclusions are  
of value in dealing with  more general  
cases4    ,when we look into a world    
alleged to be that of ,euclid and find  
motion1  we may retain the ,euclidean   
concept of what constitutes the world   
and  invent a machinery to account for  
the motion2 or we may abandon the       
,euclidean world1 as inadequate1 in     
favor of a more general one4 ,we  have  
adopted the second alternative4         
,newton's laws tells us that a body free
 to move will do so1 proceeding  in a   
straight line at uniform velocity until 
interfered with4 ,we  do not ask1 nor   
does the theory tell us1 whence comes   

the initial  motion4 ,there is no       
machinery to produce it2 it is an       
inherent property  of ,newton's         
world--assured by the superposition of  
the time continuum  upon ,euclid's world
 to make ,newton's1 accepted without    
question along  with that world itself4 
,but ,newton saw that his world of      
uniform motion1 like ,euclid's1  was    
never realized4 ,in the neighborhood of 
one particle a second is  interfered    
with1 forced to give up its uniform     
motion and acquire a  constant          
acceleration4 ,this ,newton explained by
 employing the first  of the            
alternatives mentioned above4 ,he tells 
us that in connection  with all matter  
there exists a force which acts on other
 matter in a  certain way4 ,he does not 
display the actual machinery through    
which  this force works1 because he     
could not discover any machinery2  he   
had to stop with his brilliant          
generalization of the observed  facts4  
,and all his successors have failed to  

detect the slightest  trace of a        
machinery of gravitation4    ,einstein  
asks whether this is not because the    
machinery is  absent--because           
gravitation1 like position in ,euclid's 
world and  motion in ,newton's1 is a    
fundamental property of the world in    
which it  occurs4 ,his point of attack  
here lay in precise formulation of      
certain  familiar facts that had never  
been adequately appreciated4 ,these     
facts indicate that even accelerated    
motion is relative1 in spite of  its    
apparently real and absolute effects4   
,,gravitation ,,and ,,acceleration      
,an observer in a closed compartment1   
moving with constant acceleration       
through empty space1 finds that the     
bottom of his cage catches up  with     
objects that he releases2 that it       
presses on his feet to give  him the    
sensation of weight1 etc4 ,it displays  
all the effects that  he would expect if
 it were at rest in a gravitational     
field4 ,on the  other hand1 if it were  

falling freely under gravitational      
influence1  its occupant would sense no 
weight1 objects released would not leave
  his hand1 the reaction from his every 
motion would change his every  position 
in his cage1 and he could equally well  
assume himself at  rest in a region of  
space free from gravitational action4   
,accelerated  motion may always be      
interpreted1 by the observer on the     
system1  as ordinary force effects on   
his moving system1 or as gravitational  
effects on his system at rest4    ,an   
alternative statement of the ,special   
,theory is that the observed  phenomena 
of uniform motion may equally be        
accounted for by supposing  the object  
in motion and the observer with his     
reference frame at  rest1 or vice versa4
 ,we may similarly state the ,general   
,theory3 ,the  observed phenomena of    
uniformly accelerated motion may in     
every case be  explained on a basis of  
stationary observer and accelerated     
objective1  or of stationary objective  

with the observer and his reference     
system  in accelerated motion4          
,gravitation is one of these phenomena4 
,it  follows that if the observer enjoy 
properly accelerated axes 7in           
time-space1 of course71 the absolute    
character of the world about him  must  
be such as to present to him the        
phenomenon of gravitation4 ,it  remains 
only to identify the sort of world1 of  
which gravitation as  it is observed    
would be a fundamental characteristic4  
,euclid's and ,newton's systems stand as
 first and second approximations  to    
that world4 ,the ,special ,relativity   
,theory constitutes a correction  of    
,newton1 presumably because it is a     
third approximation4 ,we must  seek in  
it those features which we may most     
hopefully carry along1  into the still  
more general case4    ,newton's system  
retained the geometry of ,euclid4 ,but  
,minkowski's  invariant expression tells
 us that ,einstein has had to abandon   
this2  for in ,euclidean geometry of    

four dimensions the invariant takes  the
 form3        sqrt7,x-x7#b  7,y-y7#b    
7,z-z7#b  7,t-t7#b11    analogous to    
that of two and three dimensions4 ,it is
 not the presence  of the constant ,c in
 ,minkowski's formula that counts2 this 
is merely  an adjustment so that we may 
measure space in miles and time in the  
unit that corresponds to a mile4 ,it is 
the minus sign where ,euclidean         
geometry demands a plus that makes      
,minkowski's continuum non-,euclidean4  
,the editor has told us what this       
statement means4 ,i think he has made   
it clear that when we speak of the      
geometry of the four-dimensional  world1
 we must not read into this term the    
restrictions surrounding  the kind of   
geometry we are best acquainted         
with--that of the  three-dimensional    
,euclidean continuum4 ,so ,i need only  
point out that  if we are to make a     
fourth 7and we hope1 final7             
approximation to the  reality1 its      
geometry must preserve the generality   

attained by that  of the third step1 if 
it goes no further4                     
,,einstein',s ,,time-,,space ,,world    
,einstein accordingly examined the      
possible non-,euclidean geometries  of  
four dimensions1 in search of one       
displaying fundamental  characteristics 
which1 interpreted in terms of          
space-time1 would  lead to the observed 
facts of gravitation4 ,the mathematics  
of this  investigation is that part of  
his work which1 we are told1 but twelve 
men can follow2 so we may only outline  
his conclusions4    ,if we assume that  
in the neighborhood of matter the world 
of  space-time is non-,euclidean1 and   
that its curvature or distortion or     
non-,euclideanism is of a certain type  
already known to mathematicians2  that  
the curvature of this world in the      
neighborhood of matter  increases with  
the mass1 and decreases as the distance 
from the matter  increases2 and that    
every particle of matter that is not    
interfered  with travels through        

space-time in the most direct path      
possible in  that continuum2 then the   
observed facts of gravitation are       
accounted  for as an inherent geometric 
property of this space-time world4 ,we  
usually say that the presence of matter 
distorts this world1 and  that this     
distortion gives the track of particles 
through the region  affected its        
non-uniform character4    ,gravitation  
then is not a force at all2 it is the   
fundamental nature of  things4 ,a body  
free to move through the world must     
follow some definite  path4 ,euclid says
 it will stand still2 ,newton that it   
will traverse  a straight line in       
three-space at uniform time-rate2       
,einstein that it  will move in a       
geodesic through time-space--in         
every-day language1  that it will fall4 
,the numerical consequences of          
,einstein's theory are1 within the      
limits of observation1 the same as those
 of ,newton's for all bodies  save      
one--,mercury4 ,this planet shows a     

small deviation from the  path predicted
 by ,newton's law2 ,einstein's theory   
gives its motion  exactly4 ,again1 when 
modern research showed that light must  
be  affected by gravitation1 ,einstein's
 theory1 because of the extreme         
velocity of light1 deviates from        
,newton's1 where the speed is less a    
determining factor2 and observations of 
starlight deflected by the  sun during  
the eclipse were in much better accord  
with ,einstein's  theory than ,newton's4
 ,moreover1 the ,special ,theory        
predicts that mass  is an observational 
variable like length and duration4      
,radioactive  emanations have a velocity
 high enough to give appreciable results
  here1 and the prediction is verified1 
tending to support the general  theory  
by supporting its limiting case4    ,we 
like always to unify our science2 and   
seldom1 after effecting  a unification1 
are we forced to give it up4 ,einstein  
for the first  time brings mechanical1  
electromagnetic and gravitational       

phenomena  within one structure4 ,this  
is one reason why physicists are so open
  minded toward his theory--they want it
 to be true4          ,,the ,,layman',s 
,,last ,,doubt    ,the final answer to  
any series of questions is inevitably   
because the  world is so constructed4   
,the things we are content to leave on  
that  basis are those to which we are   
accustomed1 and which we therefore      
think we understand2 those for which    
this explanation leaves us  unsatisfied 
are those which are new and unfamiliar4 
,newton told us  that the world of      
three-dimensional space with            
one-dimensional time  superposed was so 
constructed that bodies left to         
themselves would go  on forever in a    
straight line at constant speed4 ,we    
think we understand  this1 but our      
understanding consists merely of the    
unspoken query1  ,why1 of course2 what  
is there to prevent8 ,the ,greeks1 an   
intelligent  people1 looked at this     
differently2 they would have met ,newton

 with  the unanimous demand ,why so2    
what is there to keep them going8 ,so   
if1 in seeking an explanation of        
anything1 we come sooner than we  had   
expected to the finality ,because the   
world is so constructed1  let us not    
feel that we have been cheated4         
,,ix    ,,the ,,principle ,,of          
,,relativity    ,a ,statement of ,what  
it is ,all ,about1 in ,ideas of ,one    
,syllable    ,,by ,,hugh ,,elliot       
,,chislehurst1 ,,kent1 ,,england        
,the invariance of the laws of nature   
was one of the most popular themes  of  
nineteenth century philosophy4 ,for it  
was not till last century that  general 
acceptance was accorded to the doctrine 
of the ,uniformity  of ,law1 adumbrated 
in ancient times by ,epicurus and       
,lucretius4 ,it  is now a cardinal axiom
 of science that the same cause in the  
same  conditions is always followed by  
the same effect4 ,there exists in       
nature no indeterminate element2 all    
things are governed by fixed laws1  and 

the discovery of these laws is the main 
business of science4          ,it is    
necessary to guard against reading into 
this statement an      erroneous idea of
 the content of a law of nature4 ,such a
 law      is of course not an enactment 
of any sort2 and it is not even to      
be thought of as an actual explanation  
of the how and why of the      phenomena
 with which it has to do4 ,it really is 
nothing but an      expression of our   
belief in the pronouncement of the      
preceding      paragraph1 that like     
conditions do produce like results4 ,it 
is      a prediction based on past      
experience1 and is of value merely in   
that past experience leads us to credit 
its accuracy4 ,the composite      essay 
beginning on page #ai discusses this    
question of the reality      of natural 
laws1 and should be consulted in        
connection with the      present        
contribution4--,editor4      ,this great
 philosophic principle was derived of   
course from the study  of natural       

science2 i4e41 from observations and    
experiments conducted  upon the earth4  
,their comprehensiveness is therefore   
limited by the  fact that the observer  
is always in a state of rest1 or nearly 
so1  as compared with the earth4 ,all   
observers upon the earth are moving     
through space at the same velocity2 and 
it was possible to argue that  the      
uniformity of law might only hold good1 
when experiments were  conducted at this
 velocity4 ,an observer moving at very  
different  velocity might discover that 
the laws of nature under these new      
conditions were somewhat different4     
,such a view could indeed never be very 
plausible1 for motion is only a         
relative conception4 ,imagine a universe
 consisting of infinite empty  space1 in
 which there is poised a single material
 body4 ,how shall  we determine whether 
this body is at rest1 or whether it is  
moving  at high or low velocity through 
space8 ,it is never getting nearer  to  
anything or farther from anything1 since

 there is no other body  for it to get  
nearer to or farther from4 ,if we say it
 is moving at a  uniform velocity of a  
thousand miles a second1 our statement  
really  has no significance4 ,we have no
 more reason for affirming that it is   
in motion than we have for affirming    
that it is at rest4 ,in short1  there is
 no such thing as absolute motion2 the  
conception of motion  only arises when  
there are two or more bodies changing   
their position  relatively to one       
another4 ,this is what is meant by the  
relativity  of motion4 ,it seemed       
therefore improbable that the laws of   
nature  would be different if the       
observer were moving at high velocity2  
for the movement of the observer is not 
an absolute quantity1 but  merely a     
statement of his relation to other      
bodies1 and if there were  no other     
bodies1 the statement itself would be   
meaningless4          ,,the ,,behavior  
,,of ,,light    ,now among the          
established laws of nature is that which

 specifies  the velocity of light moving
 through a vacuum4 ,if the laws of      
nature  are invariable1 this velocity   
will always be the same4 ,but consider  
what would happen under the following   
circumstances3 ,suppose that  we are at 
rest1 and that an observer on another   
body flies past us  at #aej1jjj miles a 
second4 ,suppose that at the moment he  
passes1 a  piece of flint projecting    
from him grazes a piece of steel        
projecting  from us1 giving rise to a   
spark2 and that we both thereupon set   
about  to measure the velocity of the   
light so produced4 ,after one second1   
we should find that the light had       
traveled about #ahf1jjj miles away1  and
 since during this second the other     
observer had traveled #aej1jjj  miles1  
we should infer that the light traveling
 in his direction was  only about       
#cf1jjj miles ahead of him4 ,we should  
also infer that he  would find this out 
by his experiment1 and that he would    
estimate the  velocity of light as only 

#cf1jjj miles a second in his own       
direction1  and #ccf1jjj miles a second 
in the opposite direction4 ,but if this 
is  so1 then that law of nature which   
specifies the velocity of light  is     
quite different for him and for us3 the 
laws of nature must be  dependent upon  
the observer's motion--a conclusion     
which appears  incompatible with the    
idea of the relativity of motion4       
,and it so happens that it is also      
contradictory to experimental           
conclusions4 ,experiments undertaken to 
settle the point show that  each        
observer finds the same velocity for the
 light of the spark2  and after one     
second1 each observer finds that the    
light has traveled  #ahf1jjj miles from 
himself4 ,but how is it possible that   
when it has  traveled #ahf1jjj miles in 
the same direction as the other observer
  who himself has moved #aej1jjj miles  
meanwhile1 he should still think  it    
#ahf1jjj miles ahead of him8 ,that is   
the initial paradox2 and since  there   

has been no room for error in the       
experiments1 we are forced  to conclude 
that there was something wrong in the   
assumptions and  preconceptions with    
which we started4          ,,space ,,and
 ,,time    ,there can in fact be only   
one interpretation4 ,if we each find    
that the  light has moved the same      
number of miles in the same number of   
seconds1  then we must be meaning       
something different when we speak of    
miles and  seconds4 ,we are speaking in 
different languages4 ,some subsidence   
has  occurred in the foundations of our 
systems of measurement4 ,we are each    
referring to one and the same objective 
fact2 but since we describe  it quite   
differently1 and at first sight         
incompatibly1 some profound  alteration 
must have occurred in our               
perceptions--all unsuspected by         
ourselves4 ,it has been shown precisely 
what this alteration is4 ,a body  moving
 at high velocity must become flattened 
in the direction of its  motion2 all its

 measuring apparatus1 when turned in    
that direction1  is shortened1 so that  
no hint of the flattening can be        
obtained from  it4 ,furthermore1 the    
standards of time are lengthened out1   
and clocks  go slower4 ,the extent of   
this alteration in standards of space   
and time  is stated in the equations of 
the so-called ,lorentz transformation4  
,objection might be urged to the above  
paragraph on the ground that the        
connection of the observer with the     
variability of measured lengths  and    
times is not sufficiently indicated1 and
 that this variability  therefore might 
be taken as an intrinsic property of the
 observed  body--which of course it is  
not4--,editor4      ,we are accustomed  
to describe space as being of three     
dimensions1 and  time as being of one   
dimension4 ,as a matter of fact1 both   
space and  time are ideas1 and not      
immediate sense-perceptions4 ,we        
perceive  matter2 we then infer a       
universal continuum filled by it1 which 

we call space4 ,if we had no knowledge  
of matter1 we should have no  conception
 of space4 ,similarly in the case of    
time3 we perceive one  event following  
another1 and we then invent a continuum 
which we call  time1 as an abstraction  
based on the sequence of events4 ,we do 
not see  space1 and we do not see time4 
,they are not real things1 in the sense 
that matter is real1 and that events are
 real4 ,they are products of            
imagination3 useful enough in common    
life1 but misleading when we  try to    
look on the universe as a whole1 free   
from the artificial  divisions and      
landmarks which we introduce into it for
 practical  convenience4 ,hence it is   
perhaps not so surprising after all that
  in certain highly transcendental      
investigations1 these artificial        
divisions should cease to be a          
convenience1 and become a hindrance4    
,take for instance our conception of    
time4 ,it differs from our  conception  
of space in that it has only one        

dimension4 ,in space1 there  is a right 
and left1 an up and down1 a before and  
after4 ,but in time  there is only      
before and after4 ,why should there be  
this limitation of  the time-factor8    
,merely because that is the verdict of  
all our human  experience4 ,but is our  
human experience based on a sufficiently
 broad  foundation to enable us to say  
that1 under all conditions and in all   
parts of the universe1 there can be only
 one time-direction8 ,may not  our      
belief in the uniformity of time be due 
to the uniformity of the  motion of all 
observers on the earth8 ,such in fact is
 the postulate of  relativity4 ,we now  
believe that1 at velocities very        
different from our  own1 the standard of
 time would also be different from ours4
 ,from our  point of view1 that         
different standard of time would not be 
confined  to the single direction fore  
and aft1 as we know it1 but would also  
have in it an element of what we might  
call right and left4 ,true1  it would   

still be of only one dimension1 but its 
direction would  differ from the        
direction of our time4 ,it would still  
run like a  thread through the universe1
 but not in the direction which we call 
straight forward4 ,it would have a slant
 in it1 and the angle of the  slant     
depends upon the velocity of motion4 ,it
 does not follow that  because we are   
all traveling in the same direction down
 the stream of  time1 therefore that    
stream can only flow in the direction   
which we  know4 ,before and after are   
expressions which1 like right and left1 
depend upon our personal situation4 ,if 
we were differently situated1  if to be 
precise we were moving at very high     
velocity1 we should1  so to speak1 be   
facing in a new direction and before and
 after  would imply a different         
direction of progress from that with    
which  we are now familiar4             
,,the ,,world ,,of ,,reality    ,but1   
after all1 the objective universe is the
 same old universe  however fast we are 

moving about in it1 and whatever way we 
are  facing4 ,these details merely      
determine the way we divide it up into  
space and time4 ,the universe is not    
affected by any arbitrary lines  which  
we draw through it for our personal     
convenience4 ,for practical  purposes1  
we ascribe to it four dimensions1 three 
in space and one in  time4 ,clearly if  
the time direction is altered1 all      
dimensions both  of space and time must 
have different readings4 ,if1 for       
instance1  the time direction slopes    
away to the left1 as compared with ours1
  then space measurements to right and  
left must be correspondingly  altered4  
,an analogy will simplify the matter4   
,suppose we desire to reach a point ten 
miles off in a roughly  northeasterly   
direction4 ,we might do so by walking   
six miles due east  and then eight miles
 due north4 ,we should then be precisely
 ten miles  from where we started4 ,but 
suppose our compass were out of order1  
so  that its north pole pointed somewhat

 to the west of north4 ,then in  order  
to get to our destination1 we might have
 to walk seven miles  in the direction  
which we thought was east1 and a little 
more than  seven miles in the direction 
which we thought was north4 ,we should  
then reach the same point as before4    
,both observers have walked  according  
to their lights1 first due east and then
 due north1 and  both have reached the  
same point3 the one observer is certain 
that  the finishing point is six miles  
east of the starting-point1 while  the  
other is sure it is seven miles4    ,now
 we on the earth are all using a compass
 which points in the same  direction as 
regards time4 ,but other observers1 on  
bodies moving with  very different      
velocity1 have a compass in which the   
time-direction is  displaced as compared
 with ours4 ,hence our judgments of     
distances will  not be alike4 ,in our   
analogy1 the northerly direction        
corresponds to  time1 and the easterly  
direction to space2 and so long as we   

use the  same compass we do not differ  
in our measurements of distances4 ,but  
for any one who has a different notion  
of the time-direction1 not  only time   
intervals but space distances will be   
judged differently4    ,in short1 the   
universe is regarded as a space-time    
continuum of four  dimensions4 ,a point 
in space-time is called an event--that  
which  occurs at a specified moment and 
at a specified place4 ,the distance     
between two points in space-time is     
called their interval4 ,all  observers  
will agree as to the magnitude of any   
interval1 since it  is a property of the
 objective universe2 but they will      
disagree as  to its composition in space
 and time separately4 ,in short1 space  
and time are relative conceptions2 their
 relativity is a necessary  consequence 
of the relativity of motion4 ,the       
paradox named at the  outset is         
overcome2 for the two observers         
measuring the velocity of  the light    
produced as they passed one another1    

were using different  units of space and
 time4 ,and hence emerges triumphant the
 ,special  ,principle of ,relativity1   
which states that the laws of nature are
  the same for all observers1 whether   
they are in a state of rest or  of      
uniform motion in a straight line4      
,,accelerated ,,motion    ,uniform      
motion in a straight line is however a  
very special kind of  motion4 ,our      
experience in ordinary life is of       
motions that are neither  uniform nor in
 a straight line2 both speed and        
direction of motion  are altering4 ,the 
moving body is then said to undergo     
acceleration3  which means either that  
its speed is increasing or diminishing1 
or  that its direction of motion is     
changing1 or both4 ,if we revert to our 
former supposition of a universe in     
which there is only a single body  in   
empty space1 we clearly cannot say      
whether it has acceleration  any more   
than whether it is moving1 there being  
no outside standard  of comparison2 and 

the ,general ,principle of ,relativity  
asserts the  invariance of the laws of  
nature for all states of motion of the  
observer4 ,in this case1 however1 a     
difference might be detected  by an     
observer on the moving body itself4 ,it 
would be manifested  to him as the      
action of a force2 such for instance as 
we feel when  a train in which we are   
traveling is increasing or reducing     
speed1  or when1 without changing speed1
 it is rounding a corner4 ,the force    
dies away as soon as the velocity       
becomes uniform4 ,thus acceleration     
reveals itself to us under the guise of 
action by a force4 ,force and           
acceleration go together1 and we may    
either say that the acceleration  is due
 to the force1 or the impression of     
force to the acceleration4    ,now when 
we are traveling with accelerated       
motion1 we have quite a  different idea 
of what constitutes a straight line from
 that which  we had when at rest or in  
uniform motion4 ,if we are moving at    

uniform  velocity in an airplane and    
drop a stone to the earth it will appear
  to us in the airplane to fall in a    
straight line downward1 while to  an    
observer on the earth it will appear to 
describe a parabola4 ,this  is due to   
the fact that the stone gathers speed as
 it falls2 it is  subject to the        
acceleration associated with gravity4   
,acceleration  obliterates the          
fundamental difference between a        
straight and  curved line4 ,unless we   
know what is the absolute motion of the 
stone1 and the two observers1 we cannot 
say whether the line is  really a       
straight or a curved line4 ,since       
absolute motion is an  illegitimate     
conception1 it follows that there is no 
such thing as  really straight or really
 curved4 ,these are only appearances    
set up as a consequence of our relative 
motions with respect to the  bodies     
concerned4 ,if there were no such thing 
as acceleration--if the  stone fell to  
the earth at uniform velocity--then an  

observer on the  earth or anywhere else 
would agree that it fell in a straight  
line2  and straight lines would always  
be straight lines4    ,under these      
circumstances1 ,euclidean geometry would
 be absolutely  true4 ,but if we are in 
a state of acceleration1 then what we   
think are  straight lines are really    
curved lines1 and ,euclidean geometry1  
based  on the assumption that its lines 
are straight1 must founder when tested  
by more accurate measurements4 ,and in  
point of fact we are in a state  of     
acceleration3 for we are being acted    
upon by a force--namely1 the  force of  
gravitation4 ,wherever there is matter1 
there is gravitation2  wherever there is
 gravitation there is acceleration2     
wherever there  is acceleration         
,euclidean geometry is inaccurate4      
,hence in the space  surrounding matter 
a different geometry holds the field2   
and bodies  in general move through such
 space in curved lines4    ,different   
parts of space are thus characterized by

 different  geometrical properties4 ,all
 bodies in the universe proceed on      
their established courses through space 
and time4 ,but when they  come to       
distorted geometrical areas1 their paths
 naturally seem  to us different from   
when they were moving through less      
disturbed  regions4 ,they exhibit the   
difference by acquiring an acceleration2
  and we explain the acceleration by    
alleging the existence of a force1      
which we call the force of gravitation4 
,but their motions can in fact  be      
perfectly predicted if we know the      
geometry of the space through  which    
they are traveling4 ,the predictions so 
based have in fact proved  more accurate
 than those based on the law of         
gravitation4                    ,x      
,,space1 ,,time ,,and ,,gravitation     
,an ,outline of ,einstein's ,theory of  
,general ,relativity    ,,by ,w4 ,,de   
,,sitter  ,,professor ,,of ,,astronomy  
,,in ,,the ,,university ,,of ,,leyden   
,henceforth space by itself and time by 

itself shall sink to mere  shadows1 and 
only a union of the two shall preserve  
reality4      ,the prophecy contained in
 the above-quoted words1 spoken by      
,minkowski  at the meeting of ,german   
,naturforscher und ,aerzte at ,cologne  
in #aijh1  has1 however1 only been      
completely fulfilled by ,einstein's     
,allgemeine  ,relativitts-theorie of    
#aiae1 which incorporated gravitation   
into  the union4 ,in the following pages
 an attempt is made to set forth1       
without using any technical language1   
the leading ideas of that theory3  ,i   
will confine myself to the theory as    
published by ,einstein in  ,november1   
#aiae1 which forms a consistent whole1  
complete in itself2  and ,i will not    
refer to later developments1 which are  
still more  or less tentative1 and not  
necessary for the understanding of the  
theory4 ,the mathematics used by        
,einstein is the so-called ,absolute    
,differential ,calculus4 ,it is not more
 difficult or recondite than that  used 

in other branches of theoretical        
physics1 but it is somewhat  unfamiliar 
to most of us1 because it is not        
generally taught in the  regular        
university courses4 ,i will1 however1 in
 this essay abstain  from using any     
mathematics at all1 at least1 ,i will   
not be using it  openly4 ,it is of      
course unavoidable to use at least the  
results of  the mathematical reasoning1 
if not the reasoning itself2 but so long
  as they are not put into formulas they
 will1 it is hoped1 not look  so        
formidable to the reader4    ,referring 
to the quoted words of ,minkowski1 we   
may ask what is meant by  reality4      
,physical science1 like common sense1   
takes for granted that  there is a      
reality behind the phenomena1 which is  
independent of the  person by whom1 and 
the particular methods by which it is   
observed1  and which is also there when 
it is not observed4 ,strictly speaking1 
all talk about what is not observed is  
metaphysics4 ,nevertheless  the         

physicist unhesitatingly believes that  
his laws are general1 and  that the     
phenomena continue to happen according  
to them when nobody  is looking4 ,and   
since it would be impossible to prove   
that they did  not1 he is fully entitled
 to his belief4 ,the observed phenomena 
are  the effects of the action of this  
reality1 of which we assume the         
existence1 on the observer's senses--or 
apparatus1 which are extended  and      
refined sense-organs4 ,the laws         
governing the phenomena therefore  must 
convey some information regarding this  
reality4 ,we shall never  by any means  
be able to know anything else about it  
but just these  laws4 ,to all intents   
and purposes the laws are the reality1  
if we  eliminate from them all that     
refers to the observer alone4 ,what     
refers  to the reality is called        
absolute1 and what involves reference to
  the observer relative4 ,the           
elimination of the relative is one of   
the things the theory of relativity has 

set out to do4          ,,the ,,external
 ,,world ,,and ,,its ,,geometry    ,to  
describe the phenomena and derive laws  
from them1 we locate them  in space and 
time4 ,to do this we use geometry4 ,here
 it is that the  part contributed by the
 observer comes in4 ,there are an       
infinite  number of geometries1 and a   
priori there seems to be no reason to   
choose one rather than the other4       
,taking geometry of two dimensions  as  
an example1 we can draw figures on a    
piece of paper1 and discuss  their      
properties1 and we can also do so on the
 shell of an egg4 ,but  we cannot draw  
the same figures on the egg as on the   
paper4 ,the  ones will be distorted as  
compared with the others3 the two       
surfaces  have a different geometry4    
,similarly it is not possible to draw an
  accurate map of the earth on a sheet  
of paper1 because the earth is          
spherical and its representation on the 
flat paper is always more or  less      
distorted4 ,the earth requires spherical

 geometry1 which differs  from the flat1
 or ,euclidean1 geometry of the paper4  
,up to a few years ago ,euclidean 7i4e4 
flat7 geometry of three  dimensions had 
been exclusively used in physical       
theories4 ,why8 ,because  it is the true
 one1 is the one answer generally given4
 ,now a statement  about facts can be   
true or false1 but a mathematical       
discipline is  neither true nor false2  
it can only be correct--i4e4 consistent 
in  itself--or incorrect1 and of course 
it always is correct4 ,the assertion    
that a certain geometry is the true one 
can thus only mean1 that  it is the     
geometry of true space1 and this again1 
if it is to have  any meaning at all1   
can only mean that it corresponds to the
 physical  reality4 ,leaving aside the  
question whether this reality has any   
geometry at all1 we are confronted with 
the more immediately practical          
consideration how we shall verify the   
asserted correspondence4 ,there is  no  
other way than by comparing the         

conclusions derived from the laws  based
 upon our geometry1 with observations4  
,it thus appears that the  only         
justification for the use of the        
,euclidean geometry is its success  in  
enabling us to draw an accurate map of  
the world4 ,as soon as any  other       
geometry is found to be more successful1
 that other must be used  in physical   
theories1 and we may1 if we like1 call  
it the true one4    ,accurate           
observations always consist of measures1
 determining the  position of material  
bodies in space4 ,but the positions     
change1  and for a complete description 
we also require measures of time4 ,an   
important remark must be made here4     
,nobody has ever measured a  pure       
space-distance1 nor a pure lapse of     
time4 ,the only thing that  can be      
measured is the distance from a body at 
a certain point of  space and a certain 
moment of time1 to a body 7either the   
same or  another7 at another point and  
another time4 ,we can even go further   

and say that time cannot be measured at 
all4 ,we profess to measure  it by      
clocks4 ,but a clock really measures    
space1 and we derive the  time from its 
space-measures by a fixed rule4 ,this   
rule depends  on the laws of motion of  
the mechanism of the clock4 ,thus       
finally  time is defined by these laws4 
,this is so1 whether as a clock  we use 
an ordinary chronometer1 or the rotating
 earth1 or an atom  emitting            
light-waves1 or anything else that may  
be suggested4 ,the  physical laws1 of   
course1 must be so adjusted that all    
these devices  give the same time4      
,about the reality of time1 if it has   
any1 we  know nothing4 ,all we know     
about time is that we want it4 ,we      
cannot  adequately describe nature with 
the three space-coordinates alone1  we  
require a fourth one1 which we call     
time4 ,we might thus say with  some     
reason that the physical world has four 
dimensions4 ,but so long  as it was     
found possible adequately to describe   

all known phenomena  by a space of three
 dimensions and an independent time1 the
 statement  did not convey any very     
important information4 ,only after it   
had been  found out that the            
space-coordinates and the time are not  
independent1  did it acquire a real     
meaning4    ,as is well known the       
observation by which this was found out 
is the  famous experiment of ,michelson 
and ,morley4 ,it led to the special     
theory of relativity1 which is the one  
referred to by ,minkowski  in #aijh4 ,in
 it a geometry of four dimensions is    
used1 not a mere  combination of a      
three-dimensional space and a           
one-dimensional time1  but a continuum  
of truly fourfold order4 ,this          
time-space is not  ,euclidean1 since the
 time-component and the three           
space-components  are not on the same   
footing1 but its fundamental formula has
 a  great resemblance to that of        
,euclidean geometry4 ,we may call it    
pseudo-,euclidean4    ,this theory1     

which we need not explain here1 was very
 satisfactory so  far as the laws of    
electromagnetism1 and especially the    
propagation  of light1 were concerned1  
but it did not include gravitation1 and 
mechanics generally4 ,we then had this  
curious state of affairs1 that          
physicists actually believed in two     
different realities4 ,when they  were   
thinking of light they believed in      
,minkowski's time-space2 when  they were
 thinking of gravitation they believed  
in the old ,euclidean  space and        
independent time4 ,this1 of course1     
could not last4 ,attempts  were made so 
to alter ,newton's law of gravitation   
that it would fit  into the             
four-dimensional world of the special   
relativity-theory1  but these only      
succeeded in making the law1 which had  
been a model  of simplicity1 extremely  
complicated1 and1 what was worse1 it    
became  ambiguous4    ,it is ,einstein's
 great merit to have perceived that     
gravitation is  of such fundamental     

importance1 that it must not be fitted  
into a  ready-made theory1 but must be  
woven into the space-time geometry from 
the beginning4 ,and that he not only saw
 the necessity of doing this1  but      
actually did it4          ,,gravitation 
,,and ,,its ,,place ,,in ,,the          
,,universe    ,to see the necessity we  
must go back to ,newton's system of     
mechanics4 ,newton did two things       
7amongst others74 ,he canonised         
,galileo's system of mechanics into his 
famous laws of motion1  the most        
important of which is the law of        
inertia1 which says that3          a    
body1 that is not interfered with1 moves
 in a straight line      with constant  
velocity4      ,the velocity1 of course1
 can be nil1 and the body at rest4 ,this
 is  a perfectly general law1 the same  
for all material bodies1 whatever  their
 physical or chemical status4 ,newton   
took good care exactly to  define what  
he meant by uniform motion in a straight
 line1 and for  this purpose he         

introduced the absolute ,euclidean space
 and absolute  time as an essential part
 of his system of laws at the very      
beginning  of his great work4 ,the other
 thing ,newton did was to formulate the 
law of gravitation4 ,gravitation was in 
his system considered as  an            
interference with the free1 or inertial1
 motion of bodies1 and  accordingly     
required a law of its own4    ,but      
gravitation has this in common with     
inertia1 and in this it  differs from   
all other interferences1 that it is     
perfectly general4 ,all  material bodies
 are equally subjected to it1 whatever  
their physical  or chemical status may  
be4 ,but there is more4 ,gravitation and
 inertia  are actually indistinguishable
 from each other1 and are measured  by  
the same number3 the mass4 ,this was    
already remarked by ,newton  himself1   
and from his point of view it was a most
 wonderful accidental  coincidence4 ,if 
an apple falls from the tree1 that which
 makes it fall  is its weight1 which is 

the gravitational attraction by the     
earth1  diminished by the centrifugal   
force due to the earth's rotation and   
the  apple's inertia4 ,in ,newton's     
system the gravitational attraction is a
  real force1 whereas the centrifugal   
force is only fictitious4 ,but  the one 
is as real as the other4 ,the most      
refined experiments1  already begun by  
,newton himself1 have not succeeded in  
distinguishing  between them4 ,their    
identity is actually one of the best    
established  facts in experimental      
physics4 ,from this identity of         
fictitious1 or  inertial1 and real1 or  
gravitational1 forces it follows that   
locally  a gravitational field can be   
artificially created or destroyed4 ,thus
  inside a closed room which is falling 
freely1 say a lift of which the  cable  
has been broken1 bodies have no weight3 
a balance could be in  equilibrium with 
different weights in the two scales4    
,having thus come to the conclusion that
 gravitation is not an  interference1   

but is identical with inertia1 we are   
tempted to restate  the law of motion1  
so as to include both1 thus3            
,bodies which are not interfered        
with--do not move in straight           
lines1 but--fall4      ,now this is     
exactly what ,einstein did4 ,only the   
falling of  course requires a precise   
mathematical definition 7like the       
uniform  motion in a straight line71 and
 the whole gist of his theory is the    
finding of that definition4 ,in our     
earthly experience the falling  never   
lasts long1 very soon something--the    
floor of the room1 or  the earth        
itself--interferes4 ,but in free space  
bodies go on falling  forever4 ,the     
motion of the planets is1 in fact1      
adequately described as  falling1 since 
it consists in nothing else but obeying 
,newton's law of  gravitation together  
with his law of inertia4 ,a body very   
far removed  from all other matter is   
not subjected to gravitation1           
consequently it  falls with constant    

velocity in a straight line according to
 the law  of inertia4 ,the problem was  
thus to find a mathematical definition  
of  falling1 which would embrace the    
uniform straight-line motion very  far  
from all matter as well as the complex  
paths of the planets around  the sun1   
and of an apple or a cannon-ball on     
earth4          ,,gravitation ,,and     
,,space-,,time    ,for the definition of
 the uniform rectilinear motion of pure 
inertia  ,newton's ,euclidean space and 
independent time were sufficient4 ,for  
the much more complicated falling under 
the influence of gravitation  and       
inertia together1 evidently a more      
complicated geometry  would be needed4  
,minkowski's pseudo-,euclidean          
time-space also was  insufficient4      
,einstein accordingly introduced a      
general non-,euclidean  four-dimensional
 time-space1 and enunciated his law of  
motion thus3          ,bodies which are 
not interfered with move in geodesics4  
,a geodesic in curved space is exactly  

the same thing as a straight  line in   
flat space4 ,we only call it by its     
technical name1 because  the name       
straight line would remind us too much  
of the old ,euclidean  space4 ,if the   
curvature gets very small1 or zero1 the 
geodesic becomes  very nearly1 or       
exactly1 a straight line4    ,the       
problem has now become to assign to     
time-space such curvatures  that the    
geodesics will exactly represent the    
tracks of falling  bodies4 ,space of two
 dimensions can just be flat1 like a    
sheet of  paper1 or curved1 like an egg4
 ,but in geometry of four dimensions    
there are several steps from perfect    
flatness1 or pseudo-flatness1 to        
complete curvature4 ,now the law        
governing the curvature of ,einstein's  
time-space1 i4e41 the law of            
gravitation1 is simply that it can      
never1  outside matter1 be curved more  
than just one step beyond perfect       
7pseudo-7flatness4    ,since ,i have    
promised not to use any mathematics ,i  

can hardly convey  to the reader an     
adequate idea of the difficulty of the  
problem1  nor do justice to the elegance
 and beauty of the solution4 ,it is1  in
 fact1 little short of miraculous that  
this solution1 which was  only adopted  
by ,einstein because it was the simplest
 he could find1  does so exactly        
coincide in all its effects with        
,newton's law4 ,thus the  remarkably    
accurate experimental verification of   
this law can at once  be transferred to 
the new law4 ,in only one instance do   
the two laws  differ so much that the   
difference can be observed1 and in this 
case  the observations confirm the new  
law exactly4 ,this is the well known    
case of the motion of the perihelion of 
,mercury1 whose disagreement  with      
,newton's law had puzzled astronomers   
for more than half a century4    ,since 
,einstein's time-space includes         
,minkowski's as a particular case1  it  
can do all that the other was designed  
to do for electro-magnetism  and light4 

,but it does more4 ,the track of a pulse
 of light is also a  geodesic1 and      
time-space being curved in the          
neighborhood of matter1  rays of light  
are no longer straight lines4 ,a ray of 
light from a star1  passing near the    
sun1 will be bent round1 and the star   
consequently  will be seen in a         
different direction from where it would 
be seen if  the sun had not been so     
nearly in the way4 ,this has been       
verified by  the observations of the    
eclipse of the sun of #aiai of ,may #bi4
    ,there is one other new phenomenon  
predicted by the theory1 which falls    
within the reach of observation with our
 present means4 ,gravitation  chiefly   
affects the time-component of the       
four-dimensional  continuum1 in such a  
way that natural clocks appear to run   
slower  in a strong gravitational field 
than in a weak one4 ,thus1 if we  make  
the hypothesis--which1 though extremely 
probable1 is still  a hypothesis--that  
an atom emitting or absorbing           

light-waves is a  natural clock1 and the
 further hypothesis--still very         
probable1 though  less so than the      
former--that there is nothing to        
interfere with its  perfect running1    
then an atom on the sun will give off   
light-waves  of smaller frequency than a
 similar atom in a terrestrial          
laboratory  emits4 ,opinions as yet     
differ as to whether this is confirmed  
or  contradicted by observations4       
,the great strength and the charm of    
,einstein's theory do however  not lie  
in verified predictions1 nor in the     
explanation of small  outstanding       
discrepancies1 but in the complete      
attainment of its  original aim3 the    
identification of gravitation and       
inertia1 and in  the wide range of      
formerly apparently unconnected subjects
 which it  embraces1 and the broad view 
of nature which it affords4             
,outside matter1 as has been explained1 
the law of gravitation restricts  the   
curvature of time-space4 ,inside        

continuous matter the curvature  can be 
of any arbitrary kind or amount2 the law
 of gravitation then  connects this     
curvature with measurable properties of 
the matter1  such as density1 velocity1 
stress1 etc4 ,thus these properties     
define  the curvature1 or1 if preferred1
 the curvature defines the properties   
of matter1 i4e4 matter itself4    ,from 
these definitions the laws of           
conservation of energy1 and of          
conservation of momentum1 can be deduced
 by a purely mathematical  process4     
,thus these laws1 which at one time used
 to be considered  as the most          
fundamental ones of mechanics1 now      
appear as simple  corollaries from the  
law of gravitation4 ,it must be pointed 
out that  such things as length1        
velocity1 energy1 momentum1 are not     
absolute1  but relative1 i4e4 they are  
not attributes of the physical reality1 
but relations between this reality and  
the observer4 ,consequently  the laws of
 conservation are not laws of the real  

world1 like the  law of gravitation1 but
 of the observed phenomena4 ,there is1  
however  one law which1 already before  
the days of relativity1 had come to be  
considered as the most fundamental of   
all1 viz3 the principle of least        
action4 ,now action is absolute4        
,accordingly this principle retains its 
central position in ,einstein's theory4 
,it is even more fundamental than  the  
law of gravitation1 since both this law1
 and the law of motion1  can be derived 
from it4 ,the principle of least action1
 so far as we  can see at present1      
appears to be the law of the real world4
                    ,,xi    ,,the       
,,principle ,,of ,,general ,,relativity 
,how ,einstein1 to a ,degree ,never     
,before ,equalled1 ,isolates the        
,external  ,reality from the ,observer's
 ,contribution    ,,by ,e4 ,t4 ,,bell   
,,university ,,of ,,washington          
,,seattle      ,einstein's general      
relativity is of such vast compass1     
being  coextensive with the realm of    

physical events1 that in any brief      
account a strict selection from its     
numerous aspects is prescribed4 ,the    
old1 restricted principle being         
contained in the general1 we shall      
treat the latter1 its close relations   
with gravitation1 and the  significance 
of both for our knowledge of space and  
time4 ,the essence  of ,einstein's      
generalization is its final             
disentanglement of that  part of any    
physical event which is contributed by  
the observer from  that which is        
inherent in the nature of things and    
independent of  all observers4    ,the  
argument turns upon the fact that an    
observer must describe any  event with  
reference to some framework from which  
he makes measurements  of time and      
distance4 ,thus1 suppose that at nine   
o'clock a ball is  tossed across the    
room4 ,at one second past nine the ball 
occupies a  definite position which we  
can specify by giving the three         
distances  from the centre of the ball  

to the north and west walls and the     
floor4 ,in this way1 refining our       
measurements1 we can give a precise     
description of the entire motion of the 
ball4 ,our final description  will      
consist of innumerable separate         
statements1 each of which contains  four
 numbers corresponding to four          
measurements1 and of these one  will be 
for time and three for distances at the 
time indicated4    ,imagine now that a  
man in an automobile looks in and       
observes the  moving ball4 ,suppose he  
records the motion4 ,to do so1 he must  
refer to  a timepiece and some body of  
reference4 ,say he selects his          
wrist-watch1  the floor of his auto and 
two sides meeting in a corner4 ,fancy   
that  just as he begins his series of   
observations his auto starts bucking    
and the main-spring of his watch breaks1
 so that he must measure  seconds by the
 crazy running-down of his watch1 and   
distances with  reference to the sides  
of his erratic auto4 ,despite these     

handicaps  he completes a set of        
observations1 each of which consists of 
a  time measured by his mad watch and   
three distances reckoned from  the sides
 of his bucking machine4 ,let us assume 
him to have been so  absorbed in his    
experiment that he noticed neither the  
disorders  of his watch nor the motion  
of his auto4 ,he gives us his sets of   
measurements4 ,we remark that his       
seconds are only small fractions  of    
ours1 also his norths and wests are     
badly mixed4 ,if we interpret  his sets 
in terms of our stationary walls and    
sober clock we find  the curious paradox
 that the ball zigzagged across the room
 like an  intoxicated bee4 ,he          
obstinately argues that we know no more 
than he  about how the ball actually    
moved4 ,for we got a smooth description1
  he asserts1 by choosing an            
artificially simple reference framework1
  having no necessary relations whatever
 to the ball4 ,the crooked  path plotted
 from his observations proves1 he       

declares1 that the  ball was subject to 
varying forces of which we in the room  
suspected  nothing4 ,he contends that   
our room was being jarred by a system of
  forces which exactly compensated and  
smoothed out the real jaggedness  of    
path observed by himself4 ,but if we    
know all about his watch and  auto we   
can easily apply necessary corrections  
to his measurements1  and1 fitting the  
corrected set to our reference-framework
 of walls  and clock1 recover our own   
smooth description4    ,for consistency 
we must carry our readjustments farther4
 ,the path  mapped from our measurements
 is a curve4 ,perhaps the curvature was 
introduced by some peculiarity of our   
reference framework8 ,possibly  our own 
room is being accelerated upward1 so    
that it makes the ball's  true          
path--whatever that may be--appear      
curved downward1 just as the  autoist's 
zigzags made the path he mapped appear  
jagged4 ,tradition  attributes the      
downward curving to the tug of gravity4 

,this force we  say accelerates the ball
 downward1 producing the curved path4   
,is this  the only possible explanation8
 ,let us see4          ,,gravitation    
,,and ,,acceleration    ,imagine a man  
in a room out of which he cannot see4   
,he notices that  when he releases      
anything it falls to the floor with a   
constant  acceleration4 ,further he     
observes that all his objects1          
independently  of their chemical and    
physical properties1 are affected in    
precisely  the same way4 ,now1 he       
previously has experimented with        
magnets1 and  has remarked that they    
attract certain bodies in essentially   
the same  way that the things which he  
drops are attracted to whatever is      
beneath the floor4 ,having explained    
magnetic attraction in terms of  forces1
 he makes his first hypothesis3 7,a7 ,he
 and his room are in  a strong field of 
force1 which he designates              
gravitational4 ,this  force pulls all   
things downward with a constant         

acceleration4 ,here  he notes a singular
 distinction between magnetic and       
gravitational  forces3 magnets attract  
only a few kinds of matter1 notably     
iron2  the novel force1 if indeed a     
force at all1 acts similarly upon all   
kinds of matter4 ,he makes another      
hypothesis3 7,b7 ,his room and he  are  
being accelerated upward4        ,either
 7,a7 or 7,b7 describes the facts       
perfectly4 ,by no experiment  can he    
discriminate between them4 ,so he takes 
the great step1 and  formulates the     
,equivalence ,hypothesis3      ,a       
gravitational field of force is         
precisely equivalent in its effects  to 
an artificial field of force introduced 
by accelerating the  framework of       
reference1 so that in any small region  
it is impossible  to distinguish between
 them by any experiment whatever4       
,next reconsidering his magnetic forces1
 he extends the equivalence  hypothesis 
to cover all manifestations of force3   
,the effects  attributed to forces of   

any kind whatever can be described      
equally  well by saying that our        
reference frameworks are accelerated2   
and  moreover there is possible no      
experiment which will discriminate      
between the descriptions4    ,if the    
accelerations are null1 the frameworks  
are at rest or in  uniform motion       
relatively to one another4 ,this special
 case is the  restricted principle of   
relativity1 which asserts that it is    
impossible experimentally to detect a   
uniform motion through the  ether4      
,being thus superfluous for descriptions
 of natural phenomena1  the ether may be
 abandoned1 at least temporarily4 ,the  
older physics  sought this absolute     
ether framework to which all motions    
could be  unambiguously referred1 and   
failed to find it4 ,the most exacting   
experiments1 notably that of            
,michelson-,morley1 revealed no trace of
  the earth's supposed motion through   
the ether4 ,fitzgerald accounted  for   
the failure by assuming that such motion

 would remain undetected if  every      
moving body contracted by an amount     
depending upon its velocity  in the     
direction of motion4 ,the contraction   
for ordinary velocities  is             
imperceptible4 ,only when as in the case
 of the beta particles1  the velocity is
 an appreciable fraction of the velocity
 of light1  is the contraction revealed4
 ,this contraction follows immediately  
from ,einstein's generalization         
constructed upon the equivalence        
hypothesis and the restricted relativity
 principle4 ,we shall see that  the     
contraction inevitably follows from the 
actual geometry of the  universe4 #b    
,let us return for a moment to the      
moving ball4 ,four measures1  three of  
distances and one of time1 are required 
in specifying its  position with        
reference to some framework at each     
point and at each  instant4 ,all of     
these measures can be summed up in one  
compendious  statement--the equations of
 motion showed how in changing from our 

room to his accelerated auto we found a 
new summary1 transformed  equations1    
which seemed to indicate that the ball  
had traversed  a strong1 variable field 
of force4 ,is there then in the chaos of
  observational disagreements anything  
which is independent of all  observers8 
,there is1 but it is hidden at the very 
heart of nature4          ,,paths       
,,through ,,the ,,world ,,of ,,four     
,,dimensions    ,to exhibit this1 we    
must recall a familiar proposition of   
geometry3  the square on the longest    
side of a right-angled triangle is equal
  to the sum of the squares on the other
 two sides4 ,it has long been  known    
that from this alone all the metrical   
properties of ,euclidean  space--the    
space in which for #b1jjj years we have 
imagined we were  living--can be        
deduced4 ,metrical properties are those 
depending upon  measurement4 ,now1 in   
the geometry of any space1 ,euclidean or
 not1  there is a single proposition of 
a similar sort which tells us how  to   

find the most direct distance between   
any two points that are  very close     
together4 ,this small distance is       
expressed in terms of  the two sets of  
distance measurements by which the      
end-points are  located1 just as two    
neighboring positions of our ball were  
located  by two sets of four            
measurements each4 ,we say by analogy   
that two  consecutive positions of the  
ball are separated by a small interval  
of  time-space4 ,from the formula for   
the very small interval of time-space   
we can calculate mathematically all the 
metrical properties of the time  and    
space in which measurements for the     
ball's motion must be made4 ,so  in any 
geometry mathematical analysis predicts 
infallibly the truth  about all facts   
depending upon measurements from the    
simple formula  of the interval between 
neighboring points4 ,thus1 on a sphere  
the sum  of the angles of any triangle  
formed by arcs of great circles exceeds 
#ahj1 and this follows from the formula 

for the shortest 7geodesic7  distance   
between neighboring points on the       
spherical surface4    ,we saw that it   
takes four measurements1 one for time   
and three for  distances1 to fix an     
elementary event1 viz41 the position of 
the  centre of our ball at any instant4 
,a system of all possible such sets  of 
four measurements each1 constitutes what
 mathematicians call a  four-dimensional
 space4 ,the study of the               
four-dimensional time-space  geometry1  
once its shortest-distance proposition  
is known1 reveals all  those relations  
in nature which can be ascertained by   
measurements1  that is1 experimentally4 
,we have then to find this indispensable
  proposition4    ,imagine the path     
taken by a particle moving solely under 
the influence  of gravitation4 ,this    
being the simplest possible motion of an
 actual  particle in the real world1 it 
is natural to guess that its path  will 
be such that the particle moves from one
 point of time-space to  another by the 

most direct route4 ,this in fact is     
verified by forming  the equations of   
the free particle's motion1 which turn  
out to be  precisely those that specify 
a geodesic 7most direct line7 joining   
the two points4 ,on the                 
7two-dimensional7 surface of a sphere   
such a  line is the position taken by a 
string stretched between two points  on 
the surface1 and this is the shortest   
distance on the surface  between them4  
,but in the time-space geometry we find 
a remarkable  distinction3 the interval 
between any two points of the path taken
 is  the longest possible1 and between  
any two points there is only one        
longest path4 ,translated into ordinary 
space and time this merely  asserts that
 the time taken between any two points  
on the natural  path is the longest     
possible4    ,recall now that when the  
line-formula for any kind of space is   
known  all the metrical properties of   
that space are completely determined1   
and combine with this what we have just 

found1 namely1 the equations  of motion 
of a particle subject only to           
gravitation are the same  equations as  
those which fix the line-formula for the
 four-dimensional  time-space4 ,since   
gravitation alone determines the motion 
of the  particle1 and since this motion 
is completely described by the very     
equations which fix all the metrical    
properties of time-space1 it  follows   
that the metrical 7experimentally       
determinable7 properties  of time-space 
are equivalent to those of gravitation1 
in the sense  that each set of          
properties implies the other4           
,,the ,,universe ,,of ,,space-,,time    
,we have found the thing in nature which
 is independent of all  observers1 and  
it turns out to be the very structure of
 time-space  itself4 ,the motion of the 
free particle obviously is a thing      
unconditioned by accidents of           
observation2 the particle under the     
influence of gravitation alone must go a
 way of its own4 ,and if some  observer 

in an artificial field of force produced
 by the acceleration  of his reference  
framework describes the path as knotted1
 he merely is  foisting eccentricities  
of his own motion upon the direct path  
of the  particle4 ,the conclusion is    
rational1 for we believe that time-space
  exists independently of any man's way 
of perceiving it4    ,incidentally note 
that this space is that of the physical 
world4 ,for  only by measurements of    
distances and times can we become aware 
of  our extension in time and space4 ,if
 beyond this time-space geometry  of    
measurements there is some absolute     
geometry1 science can have  no concern  
with it1 for never can it be revealed by
 the one exploring  device we           
possess--measurement4    ,we have       
followed a single particle4 ,let us now 
form a picture of  several4 ,any event  
can be analyzed into a multitude of     
coincidences  in time-space4 ,for       
consider two moving particles--say      
electrons4 ,if  they collide they both  

are in very approximately one place at  
the  same time4 ,we imagine the path of 
an electron through time-space  plotted 
by a line 7in four-dimensional space71  
which will deviate  from a most direct  
7geodesic7 path if the electron is      
subjected  to forces4 ,this is the      
world-line of the electron4 ,if the     
world  lines of several electrons       
intersect at one point in time-space1   
the intersection pictures the fact of   
their coincidence somewhere  and        
somewhen2 for all their world-lines     
having a time-space point in  common1 at
 some instant they must have been in    
collision4 ,each point  of a world-line 
pictures the position at a certain place
 at a certain  time2 and it is the      
intersections of world-lines which      
correspond to  physical events4 ,of what
 lies between the intersections we have 
no  experimental knowledge4    ,imagine 
the world-lines of all the electrons in 
the universe threading  time-space like 
threads in a jelly4 ,the intersections  

of the tangle  are a complete history of
 all physical events4 ,now distort the  
jelly4 ,clearly the mutual order of the 
intersections will be unchanged1  but   
the distances between them will be      
shortened or lengthened4 ,to a          
distortion of the jelly corresponds a   
special choice 7by some observer7  of a 
reference framework for describing the  
order of events4 ,he cannot  change the 
natural sequence of events4 ,again we   
have found something  which is          
independent of all observers4    ,we can
 now recapitulate our conclusions and   
state the principle of  relativity in   
its most general form4    7#a7          
,observers describe events by measures  
of times and distances made  with regard
 to their frameworks of reference4      
7#b7 ,the complete history of any event 
is summarized in a set of  equations    
giving the positions of all the         
particles involved at  every instant4   
7#c7 ,two possibilities arise4 7,a7     
,either these equations are the  same in

 form for all space-time reference      
frameworks1 persisting  formally        
unchanged for all shifts of the         
reference scheme2 or 7,b71  they subsist
 only when some special framework is    
used1 altering their  form as they are  
referred to different frameworks4 ,if   
7,b7 holds1  we naturally assume that   
the equations1 and the phenomena which  
they  profess to represent1 owe their   
existence to some peculiarity of the    
reference framework4 ,they do not1      
therefore1 describe anything which is   
inherent in the nature of things1 but   
merely some idiosyncrasy of the         
observer's way of regarding nature4 ,if 
7,a7 holds1 then obviously the          
equations describe some real relation in
 nature which is independent  of all    
possible ways of observing and recording
 it4    7#d7 ,in its most general form  
the principle of relativity states that 
those relations1 and those alone1 which 
persist unchanged in form  for all      
possible space-time reference frameworks

 are the inherent  laws of nature4      
,to find such relations ,einstein has   
applied a mathematical method  of great 
power--the calculus of tensors--with    
extraordinary  success4 ,this calculus  
threshes out the laws of nature1        
separating  the observer's              
eccentricities from what is independent 
of him1 with  the superb efficiency of a
 modern harvester4 ,the residue is a    
physical  geometry--or geometrical      
physics--of time-space1 in which it     
appears  that the times and spaces      
contributed by the several observers'   
reference frameworks are shadows of     
their own contrivings2 while  the real1 
enduring universe is a fourfold order of
 time and space  indissolubly bound     
together4 ,one observer separates this  
time-space  into his own time and space 
in one way1 determined by his path      
through the world of events2 another1   
moving relatively to the first1         
separates it differently1 and what for  
one is time shades into space  for      

another4    ,this time-space geometry is
 non-,euclidean4 ,it is warped 7curved71
  the amount of warping at any place    
being determined by the intensity  of   
the gravitational field there4 ,thus    
again gravitation is rooted in  the     
nature of things4 ,in this sense it is  
not a force1 but a property  of space4  
,wherever there is matter there is a    
gravitational field1  and hence a       
warping of space4 ,conversely1 as long  
ago imagined by  ,clifford1 wherever    
there is a warping of space1 there is   
matter2  and matter is resolved         
ultimately into wrinkles in time-space4 
,to visualize a warped space1 consider a
 simple analogy4 ,a man walks  away from
 a polished globe2 his image recedes    
into the mirror-space1  shortening and  
thinning as it goes1 and thinning 7in   
the direction of  motion7 faster than it
 shortens4 ,everything around him       
experiences  a like effect4 ,if he tries
 to discover this by a footrule it      
automatically shortens faster as he     

turns it into the horizontal  position1 
so his purpose eludes him4 ,the         
mirror-space is warped in  the direction
 of the image's motion4 ,so is our own4 
,for all bodies1  as evidenced by the   
,fitzgerald contraction1 shorten in the 
direction  of motion4 ,and just as the  
image can never penetrate the           
mirror-space  a greater distance than   
half its radius1 so probably time-space 
is  curved in such a way that our       
universe1 like the surface of a sphere1 
is finite in extent1 but unbounded4     
,,xii    ,,force ,,vs4 ,,geometry       
,how ,einstein ,has ,substituted the    
,second for the ,first in ,connection   
,with the ,cause of ,gravitation    ,,by
 ,,saul ,,dushman  ,,general ,,electric 
,,laboratories ,,schenectady1 ,n4 ,y4   
,the theory of relativity represents a  
most strikingly original  conception of 
time and space1 which was suggested by  
,einstein in order  to correlate with   
all our past experience certain         
observations made  in recent years4 ,it 

is therefore extremely comprehensive in 
its scope2  it demands from us a radical
 revision in our notions of time and    
space2  it throws new light on the      
nature of mass and energy1 and finally1 
it  furnishes a totally new conception  
of the old problem of gravitation4      
,the starting point of the theory is the
 familiar observation that  motion is   
always relative3 that is1 to define the 
motion of any object  we must always use
 some point of reference4 ,thus we speak
 of the  velocity of a train as #dj     
miles per hour with respect to the      
earth's  surface1 but would find it     
impossible to determine its absolute    
speed1  or motion in space1 since we    
know of no star whose position can be   
spoken of as absolutely fixed4 ,these   
and similar considerations have  led to 
the conclusion1 pointed out by ,newton  
and others1 that it is  impossible by   
any mechanical experiments on the earth 
to measure its  velocity in space4      
,however1 the results of observations on

 the phenomena of light and  electricity
 led to the revival of the same problem 
under another  form4 ,as well known1    
there was evolved from these            
discoveries1  the theory that light and 
electrical energy are of the same       
nature1  and are in each case           
manifestations of wave-disturbances     
propagated  through a hypothetical      
medium1 the ether1 with a velocity of   
#ahf1jjj  miles per second4    ,the     
problem therefore arose as to whether   
the earth and all stellar  bodies move  
through this ether4 ,in that case it    
ought to be possible  to measure the    
velocity of the earth with respect to   
this medium1 and  under these conditions
 we could speak1 in a sense1 of absolute
 motion4    ,a large number of          
experiments has been tried with this end
 in view4 ,the  most famous of these1   
and the one which stimulated the        
subsequent  development of the theory of
 relativity1 was that carried out by    
,michelson and ,morley in #ahhg4 ,to    

understand the significance of this     
experiment we shall refer briefly to an 
analogous observation which  is quite   
familiar4    ,does it take longer to    
swim to a point #a mile up a stream and 
back  or to a point #a mile across      
stream and back8 ,the experienced       
swimmer  will answer that the           
up-and-down journey takes longer4 ,if we
 assume  that the swimmer has a speed of
 #e miles an hour in still water and    
that the current is #c miles an hour1 we
 find that1 while it requires           
five-eighths hour to make the           
up-and-down journey1 it takes only      
one-half hour for the trip across stream
 and back4 ,the ratio between  the times
 required for the two journeys is thus  
five-fourths1 and if  this is written in
 the form        frac#asqrt#a -         
7frac#ce7b    it shows how the result   
depends upon the square of the ratio of 
the  speeds of the swimmer and the      
current4    ,now the earth is moving in 
its orbit about the sun with a velocity 

of #ah miles per second4 ,if the earth  
moves through the ether and a           
light-beam passes from one mirror to    
another and back again1 the time  taken 
for this journey ought to be longer when
 the light-path is in  the direction of 
the earth's motion than when it is at   
right angles  to this direction4 ,for we
 can consider the light as a swimmer    
having  a speed of #ahf1jjj miles per   
second and travelling in a stream whose 
current is #ah miles per second4        
,when ,michelson and ,morley tried the  
experiment they could not observe  any  
difference in the velocity of light in  
the two directions4 ,the  experiment has
 since been repeated under various      
conditions1 but  always with negative   
results4    ,einstein's contribution to 
science consists in interpreting this   
result as being in accord with ,newton's
 ideas on mechanical relativity  in that
 it demonstrates the impossibility of   
measuring absolute  motion1 not only by 
mechanical1 but also by optical or      

electrical  experiments4 ,consequently  
the velocity of light must be regarded  
as  constant and independent of the     
motion of either source or observer4    
,,the ,,relativity ,,of ,,uniform       
,,motion    ,let us consider some of the
 consequences which follow from this    
principle4 ,an observer travelling with 
say one-half the velocity  of light in  
the same direction as a ray of light    
would find that the  latter has the     
usual velocity of #ahf1jjj miles per    
second4 ,similarly an  observer         
travelling in the opposite direction to 
that of the light-ray1  with one-half   
the velocity of light1 would obtain the 
same result4    ,einstein has shown that
 these conclusions can be valid only if 
the  units of time and space used by the
 two observers depend upon their        
relative motions4 ,a careful calculation
 shows that the unit of length  used by 
either observer appears to the other    
observer contracted  when placed in the 
direction of their relative motion 7but 

not1  when placed at right angles to    
this direction71 and the unit of time   
used by either observer appears to the  
other too great4 ,moreover1 the  ratio  
of the units of length or of time varies
 with the square of the  relative speed 
of the two observers1 according to a    
relation which is  similar to that      
mentioned above for the swimmer in the  
current4 ,this  relation shows that as  
the relative speed approaches that of   
light  the discrepancy between the units
 increases4    ,thus1 for an observer   
moving past our earth with a velocity   
which is  nine-tenths that of light1 a  
meter stick on the earth would be #dd   
centimeters as measured by him1 while a 
second on our clocks would  be about two
 and a half seconds as marked by his    
clock4 ,similarly1  what he calls a     
meter length would1 for us1 be only #dd 
centimeters  and he would appear to us  
to be living about two and a half times 
slower than we are4 ,each observer is   
perfectly consistent in his             

measurements of time and space as long  
as he confines his observations  to his 
own system1 but when he tries to make   
observations on another  system moving  
past his1 he finds that the results     
which he obtains  do not agree with     
those obtained by the other observer4   
,it is not surprising that in accordance
 with this conclusion it also  follows  
that the mass of a body must increase   
with its velocity4 ,for  low velocities 
the increase is so small that we cannot 
ever hope to  measure it1 but as the    
velocity of light is approached the     
difference  becomes more and more       
appreciable and a body having the       
velocity of  light would possess        
infinite mass1 which simply means that  
such a  velocity cannot be attained by  
any material object4 ,this conclusion   
has been experimentally confirmed by    
observations on the mass of  the        
extremely small negatively charged      
particles which are emitted  by         
radioactive elements4 ,some of these    

particles are ejected with  velocities  
which are over nine-tenths that of      
light1 and measurements  show that the  
increase in mass is in accord with this 
theory4    ,the relativity theory also  
throws new light on the nature of mass  
itself4 ,according to this view1 mass   
and energy are equivalent4 ,the         
absolute destruction of #a gram of any  
substance1 if possible1 would  yield an 
amount of energy which is one hundred   
million times as much  as that obtained 
by burning the same mass of coal4       
,conversely1 energy  changes are        
accompanied by changes in mass4 ,the    
latter are ordinarily  so inappreciably 
small as to escape our most refined     
methods of  measurements1 but in the    
case of the radioactive elements we     
actually  observe this phenomenon4 ,from
 this standpoint1 also1 the laws of     
conservation of energy and of mass are  
shown to be intimately related4         
,,universal ,,relativity    ,so far we  
have dealt with what has been designated

 as the special  theory of relativity4  
,this1 as we have seen1 applies to      
uniform motion  only4 ,in extending the 
theory to include non-uniform or        
accelerated  motion1 ,einstein has at   
the same time deduced a law of          
gravitation  which is much more general 
than that of ,newton4    ,a body falling
 towards the earth increases in velocity
 as it falls4 ,the  motion is said to be
 accelerated4 ,we ascribe this increase 
in velocity  to a gravitational force   
exerted by the earth on all objects4 ,as
  shown by ,newton1 this force acts     
between all particles of matter in  the 
universe1 and varies inversely as the   
square of the distance1  and directly as
 the product of the masses4    ,of      
course1 we have had a number of theories
 of gravitation1 and  none of them have 
proven successful4 ,einstein1 however1  
was the  first one to suggest a         
conception of gravitation which has     
proven  extremely significant4 ,he      
points out that a gravitational force   

is non-existent for a person falling    
freely with the acceleration  due to    
gravity4 ,for this person there is no   
sensation of weight1  and if he were in 
a closed box which is also falling with 
the same  acceleration1 he would be     
unable to decide as to whether his      
system  were falling or situated in     
interplanetary space where there is  no 
gravitational field4 ,furthermore1 if he
 were to carry out any  optical or      
electrical experiments in this box he   
would observe the  same results as an   
experimenter on the earth4 ,a ray of    
light would  travel in a straight line  
so far as this observer can perceive1   
while an external observer would1 of    
course1 judge differently4    ,einstein 
shows that this is equally true for all 
kinds of acceleration  including that   
due to rotation4 ,in the case of a      
rotating body there  exists a           
centrifugal force which tends to make   
objects on the surface  fly outwards1   
but for an external observer this force 

does not exist  any more than gravity   
exists for the observer falling freely4 
,thus we can draw the general conclusion
 that a gravitational field or  any     
other field of force may be eliminated  
by choosing an observer  moving with the
 proper acceleration4 ,for this         
observer1 however1  the laws of optics  
and electricity must be just as valid as
 for an  observer on the earth4    ,in  
postulating this equivalence hypothesis 
,einstein merely makes use  of the very 
familiar observation that1 independently
 of the nature  of the material1 all    
bodies possess the same acceleration in 
a given  field of force4    ,the problem
 which ,einstein now sets out to solve  
is that of determining  the law which   
shall describe the motion of any system 
in a field of  force in such a general  
manner as to leave unaltered the        
fundamental  relations of electricity   
and optics4    ,in connection with the  
solution of this problem he finds it    
necessary  to discard the limitations   

placed on us by ordinary or ,euclidean  
geometry4 ,in this manner geometrical   
concepts as well as those of  force are 
completely robbed of all notions of     
absoluteness1 and the  goal of a general
 theory of relativity is attained4      
,,the ,,geometry ,,of ,,gravitation     
,let us consider a circular disc        
rotating with a uniform peripheral      
speed4 ,according to the deductions from
 the special theory of  relativity1 an  
observer situated near the edge of this 
disc1 but not  rotating with it1 will   
observe that units of length measured   
along  the circumference of the disc are
 contracted4 ,on the other hand1        
measurements along the diameter1 which  
is at right angles to the  direction of 
motion of the circumference1 will show  
no contraction  whatever1 and1          
consequently the observer will find that
 the ratio  of circumference to diameter
 has not the well known value #c4adaei  
444 but exceeds this value1 the         
difference being greater and greater  as

 the peripheral speed approaches that of
 light4 ,that is1 the laws  of ordinary 
geometry no longer hold true4           
,however1 we know other cases in which  
the ordinary or ,euclidean  geometry is 
not applicable4 ,thus suppose that on   
the surface of a  sphere we describe a  
series of concentric circles4 ,since the
 surface  is curved1 we are not         
surprised at finding that the           
circumference of  any one of these      
circles is less than #c4adaei 444 times 
the distance  across the circle as      
measured on the surface of the sphere4  
,what  this means1 therefore1 is that we
 cannot use ,euclidean geometry to      
describe measurements on the surface of 
a sphere1 and every schoolboy  knows    
this from comparing ,mercator's         
projection of the earth's surface  with 
the actual representation on a globe4   
,when we come to think of it1 the reason
 we realize all this is  because our    
sense of three dimensions enables us to 
differentiate  flat surfaces from those 

that are curved4 ,let us1 however1      
imagine  a two-dimensional being living 
on the surface of a large sphere4 ,so   
long as his measurements are confined to
 relatively small areas he  will find it
 possible to describe all his           
measurements in terms of  ,euclidean    
geometry4 ,as1 however1 his area of     
operation increases  he will begin to   
observe greater and greater             
discrepancies4 ,being  unfamiliar with  
the existence of such a                 
three-dimensional object as  a sphere1  
and therefore not realizing that he is  
on the surface of  one1 our intelligent 
two-dimensional being will conclude that
 the  disturbance in his geometry is due
 to the action of a force1 and by  means
 of plausible assumptions on the law of 
this force he will  reconcile his       
observations with the laws of plane     
geometry4    ,now since an acceleration 
in a gravitational field is identical   
with  that due to centrifugal force     
produced by rotation1 we concluded that 

the geometry in a gravitational field   
must also be non-,euclidean4 ,that  is1 
space in the neighborhood of matter is  
distorted or curved4 ,the  curvature of 
space bears the same relation to three  
dimensions that  the curvature of a     
spherical surface bears to two          
dimensions1 and  that is why we do not  
perceive it1 any more than the          
intelligent  two-dimensional being would
 be aware of the distortion of his space
  7or surface74 ,furthermore1 like this 
being1 we have assumed the  existence of
 a gravitational force to account for   
discrepancies in  our geometrical       
measurements4    ,the identification in 
this manner of gravitational effects    
with  geometrical curvature of space    
enables ,einstein to derive a general   
law for the path of any particle in a   
gravitational field1 with respect  both 
to space and to time4 ,furthermore1 the 
law expresses this motion  in terms     
which are independent of the relative   
motion and position  of the observer1   

and satisfies the condition that the    
fundamental  laws of physics be equally 
valid for all observers4 ,the solution  
of the problem involved the use of a new
 kind of higher calculus1  elaborated by
 two ,italian mathematicians1 ,ricci and
 ,levi-,civita4 ,the  result is a law of
 motion which is extremely general in   
its validity4    ,for low velocities it 
approximates to ,newton's solution1 and 
in the  absence of a gravitational field
 it leads to the same conclusions as    
the special theory of relativity4 ,there
 are three deductions from this  law    
which have aroused a great deal of      
interest1 and the confirmation  of two  
of these by actual observation must be  
regarded as striking  proof of          
,einstein's theory4                     
,,xiii    ,,an ,,introduction ,,to      
,,relativity    ,a ,treatment in ,which 
the ,mathematical ,connections of       
,einstein's ,work  are ,brought ,out    
,more ,strongly and ,more ,successfully 
,than ,usual in a  ,popular ,explanation

    ,,by ,,harold ,t4 ,,davis1          
,,university ,,of ,,wisconsin1 ,madison1
 ,wis4      ,one of the first questions 
which appears in philosophy is this3    
,what  is the great reality that        
underlies space and time and the        
phenomena  of the physical universe8    
,kant1 the philosopher1 dismissed it as 
a subjective problem1 affirming that    
space and time are a priori  concepts   
beyond which we can say no more4        
,then the world came upon some startling
 facts4 ,in #aije a paper appeared  by  
,professor ,albert ,einstein which      
asserted that the explanation of        
certain remarkable discoveries in       
physics gave us a new conception  of    
this strange four-dimensional manifold  
in which we live4 ,thus1  the great     
difference between the space and time of
 philosophy and the  new knowledge is   
the objective reality of the latter4 ,it
 rests upon  an amazing sequence of     
physical facts1 and the generalized     
theory1  which appeared several years   

later1 founded as it is upon the        
abstruse  differential calculus of      
,riemann1 ,christoffel1 ,ricci and      
,levi-,civita1  emerges from its maze of
 formulas with the prediction of real   
phenomena  to be sought for the in the  
world of facts4    ,we shall1 therefore1
 approach the subject from this         
objective point  of view4 ,let us go to 
the realm of actual physical events and 
see how  the ideas of relativity        
gradually unfolded themselves from the  
first  crude wonderings of science to   
the stately researches that first       
discovered the great ocean of ether and 
then penetrated in such a  marvelous    
manner into some of its most mysterious 
properties4          ,,the              
,,electromagnetic ,,theory ,,of ,,light 
,suppose that we go out on a summer     
night and look into the dark depths  of 
the sky4 ,a thousand bright specks are  
flashing there1 blue1 red1  yellow      
against the dark velvet of space4 ,and  
as we look we must all  be impressed by 

the fact that such remote objects as the
 stars can be  known to us at all4 ,how 
is it that light1 that curious thing    
which  falls upon the optic nerve and   
transmits its pictures to the brain1    
can ever reach us through the black     
regions of interstellar space8 ,that  is
 the question which has for its answer  
the electromagnetic theory  of light4   
,the first theory to be advanced was    
,newton's corpuscular theory  which     
supposed that the stars are sending off 
into space little pellets  of matter so 
infinitesimally small that they can move
 at the rate of  #ahf1jjj miles a second
 without injuring even so delicate a    
thing as  the eye when they strike      
against it4    ,but in #ahja1 when      
,thomas ,young made the very important  
discovery of  interference1 this had to 
give way to the wave theory1 first      
proposed  by ,huyghens in the #agth     
century4 ,the first great deduction from
 this1  of course1 was the luminiferous 
ether1 because a wave without some      

medium for its propagation was quite    
unthinkable4 ,certain peculiar          
properties of the ether were at once    
evident1 since we deduce that  it must  
fill all space and at the same time be  
so extremely tenuous  that it will not  
retard to any noticeable degree the     
motion through  it of material bodies   
like the planets4    ,but how light was 
propagated through the ether still      
remained a  perplexing problem and      
various theories were proposed1 most    
prominent  among them being the elastic 
solid theory which tried to ascribe to  
ether the properties of an elastic body4
 ,this theory1 however1 laid  itself    
open to serious objection on the ground 
that no longitudinal  waves had been    
detected in the ether1 so that it began 
to appear  that further insight into the
 nature of light had to be sought for   
in another direction4    ,this was soon 
forthcoming for in #ahfd a new theory   
was  proposed by ,james ,clerk ,maxwell 
which seemed to solve all of the        

difficulties4 ,maxwell had been working 
with the facts derived from  a study of 
electrical and magnetic phenomena and   
had shown that  electromagnetic         
disturbances were propagated through the
 ether at a  velocity identical with    
that of light4 ,this1 of course1 might  
have been  merely a strange coincidence1
 but ,maxwell went further and          
demonstrated  the interesting fact that 
an oscillating electric charge should   
give  rise to a wave that would behave  
in a manner identical with all of the   
known properties of a light wave4 ,one  
particularly impressive assertion  was  
that these waves1 consisting of an      
alternating electric field  accompanied 
by an alternating magnetic field at     
right angles to it1  and hence called   
electromagnetic waves1 would advance in 
a direction  perpendicular to the       
alternating fields4 ,this satisfied the 
first  essential property of light rays1
 i4e41 that they must be transverse     
waves1 and the ease with which it       

explained all of the fundamental        
phenomena of optics and predicted a most
 striking interrelation  between the    
electrical and optical properties of    
material bodies1  gave it at once a     
prominent place among the various       
theories4    ,the electromagnetic       
theory1 however1 had to wait until #ahhh
 for  verification when ,heinrich       
,hertz1 in a series of brilliant        
experiments1  succeeded in producing    
electromagnetic waves in the laboratory 
and  in showing that they possessed all 
of the properties predicted by          
,maxwell4 ,these waves moved with the   
velocity of light3 they could be        
reflected1 refracted1 and polarized3    
they exhibited the phenomenon  of       
interference and1 in short1 could not be
 distinguished from light  waves except 
for their difference in wave length4    
,,the ,,michelson-,,morley ,,experiment 
,with the final establishment of the    
electromagnetic theory of light as  a   
fact of physics1 we have at last endowed

 the ether with an actual               
substantiality4 ,the empty void is no   
longer empty1 but a great  ocean of     
ether through which the planets and the 
suns turn without  ever being aware that
 it is there4    ,in #ahha ,a4 ,a4      
,michelson undertook an experiment1     
originally suggested  by ,maxwell1 to   
determine the relative motion of our    
earth to the  ether ocean and six years 
later he repeated it with the assistance
  of ,e4 ,w4 ,morley4 ,the experiment is
 now known as the ,michelson-,morley    
experiment and since it is the great    
physical fact upon which the  theory of 
relativity rests1 it will be well for us
 to examine it  in detail4    ,since we 
can scarcely think that our earth is    
privileged in the  universe and that it 
is at rest with respect to this great   
ether ocean  that fills space1 we       
propose to discover how fast we are     
actually  moving4 ,but the startling    
fact is that the experiment devised for 
this  purpose failed to detect any      

motion whatever of the earth relative   
to the ether4 #c    ,the explanation of 
this very curious fact was given by both
  ,h4 ,a4 ,lorentz and ,g4 ,f4          
,fitzgerald in what is now widely known 
under  the name of the contraction      
hypothesis4 ,it is nothing more nor less
  than this3    ,every solid body       
undergoes a slight change in dimensions1
 of the order  of 7v#b_/;c#b71 when it  
moves with a velocity v through the     
ether4    ,the reason why the experiment
 failed1 then1 was not because the earth
  was not moving through the ether1 but 
because the instruments with  which the 
experiment was being conducted had      
shrunk just enough to  negative the     
effect that was being looked for4 #d    
,,the ,,lorentz ,,transformation    ,we 
can not at this point forebear          
introducing a little mathematics  to    
further emphasize the theory and the    
very logical nature of this  contraction
 hypothesis4    ,let us suppose that we 
were on a world that was absolutely     

motionless  with respect to the ether   
and were looking at a ray of light4 ,the
  magnetic and electric fields which    
form the ray can be described by  means 
of four mathematical expressions which  
have come to bear the  name of          
,maxwell's field equations4 ,now suppose
 that we ask ourselves  the question3   
,how must these equations be changed so 
that they will  apply to a ray of light 
which is being observed by people on a  
world  that is moving with a velocity v 
through the ether8    ,the answer is    
immediate4 ,from the ,michelson-,morley 
experiment we know  that we can not tell
 how fast or how slowly we are moving   
with respect  to the ether4 ,this means 
that no matter what world we may be     
upon1  the form of the ,maxwell field   
equations will always be the same1  even
 though the second set of axes 7or frame
 of reference7 may be  moving with high 
velocity with respect to the first4     
,starting from this hypothesis 7called  
in technical language the  covariance of

 the equations with respect to a        
transformation of  coordinates71        
,lorentz found that the transformation  
which leaves the  field equations       
unchanged in form was the following3    
x'  k7x - vt71 y'  y1 z'  z1 t'  k7t -  
vx_/c7    where k is as on page #ib4    
,and what1 now1 can be deduced from     
these very simple looking  equations8   
,in the first place we see that the     
space of x'1 y'1  z'1 t' is not our     
ordinary concept of space at all1 but a 
space in  which time is all tangled up  
with length4 ,to put it more concretely1
  we may deduce from them the           
interesting fact that whenever an       
aviator  moves with respect to our      
earth1 his shape changes1 and if he were
 to  compare his watch with one on the  
earth1 he would find that his time  had 
changed also4 ,a sphere would flatten   
into an ellipse1 a meter  stick would   
shorten up1 a watch would slow down and 
all because1 as  ,h4 ,minkowski has     
shown us from these very equations1 we  

are really  living in a physical world  
quite different from the world of       
,euclid's  geometry in which we are     
accustomed to think we live4    ,a      
variety of objections has very naturally
 been made to this rather  radical      
hypothesis in an attempt to discredit   
the entire theory1 but  it is easily    
seen that any result obtained through   
the field equations  must necessarily be
 in conformity with the theory of       
contraction1 since  this theory is only 
the physical interpretation of that     
transformation  which leaves the field  
equations unaltered4 ,indeed1 it is even
 possible  to postulate the ,lorentz    
transformation together with the        
assumption  that each element of charge 
is a center of uniformly diverging tubes
  of strain and derive the ,maxwell     
field equations from this1 which  shows 
from another point of view the truly    
fundamental nature of  the              
transformation4          ,,the ,,first  
,,theory ,,of ,,relativity    ,the whole

 question of the ether had arrived at   
this very  interesting point when       
,professor ,einstein in #aije stated the
 theory  of relativity4 ,he had noticed 
that the equations of dynamics as       
formulated by ,newton did not admit the 
,lorentz transformation1 but  only the  
simple ,galilean transformation3        
x'  x - vt1 y'  y1 z'  z1 t'  t14       
,here1 indeed1 was a curious situation4 
,two physical principles1  that of      
dynamics and that of electromagnetism1  
were coexistent and  yet each one       
admitted a different transformation when
 the system of  reference was           
transferred to axes moving with constant
 velocity with  respect to the ether4   
,now the electromagnetic equations and  
their transformation had been  shown to 
be in accord with experimental fact1    
whereas it had long  been felt that     
,newton's equations were only a first   
approximation  to the truth4 ,for       
example1 the elliptic orbit of a planet 
had been  observed by ,leverrier to     

exhibit a disquieting tendency to rotate
  in the direction of motion4 ,this     
precession1 which in the case of        
,mercury was as large as #dc'' per      
century1 could not be accounted  for in 
any way by the ordinary ,newtonian laws 
and was1 consequently1  a very          
celebrated case of discordance in       
gravitational astronomy4    ,with this  
example clearly before him1 ,einstein   
took the great step  and said that the  
laws of dynamics and all other physical 
laws had to  be remade so that they1    
also1 admit the ,lorentz transformation4
 ,that  is to say1    ,the laws of      
physical phenomena1 or rather the       
mathematical expressions  for these     
laws1 are covariant 7unchanged in form7 
when we apply the  ,lorentz             
transformation to them4    ,the         
deductions from the ,michelson-,morley  
experiment now seem to have  reached    
their ultimate conclusion4    ,one      
discordant fact in this new theory      
remained1 however4 ,that same           

precession of the perihelion of ,mercury
 which had first lead ,einstein  to his 
theory remained unsettled4 ,when the new
 approximations were  applied to the    
formula of orbital motion1 a precession 
was1 indeed1  obtained1 but the computed
 value fell considerably below that of  
the  observed #dc'' per century4        
,,the ,,inclusion ,,of ,,gravitation    
,with the idea of investigating the     
problem from the very bottom1  ,einstein
 now undertook a broader and more daring
 point of view4 ,in  the first place he 
said that there is no apparent reason in
 the great  scheme of world events why  
any one special system of coordinates   
should  be fundamental to the           
description of phenomena1 just as in the
 special  theory a ray of light would   
appear the same whether viewed from a   
fixed system or a system moving with    
constant velocity with respect  to the  
ether4 ,this makes the very broad       
assumption that no matter  what system  
of coordinates we may use1 the          

mathematical expressions  for the laws  
of nature must be the same4 ,in         
,einstein's own words1  then1 the first 
principle of this more general theory of
 relativity  must be the following3     
,the general laws of nature are         
expressed through equations which  hold 
for all systems of coordinates1 that is1
 they are covariant with  respect to    
arbitrary substitutions4 #e    ,but this
 was not enough to include gravitation  
so ,einstein  next formulated what he   
was pleased to call his equivalence     
hypothesis4 ,this is best illustrated by
 an example4 ,suppose that  we are      
mounting in an elevator and wish to     
investigate the world  of events from   
our moving platform4 ,we mount more and 
more rapidly1  that is with constant    
acceleration1 and we appear to be in a  
strong  gravitational field due to our  
own inertia4 ,suppose1 on the other     
hand1 that the elevator descends with an
 acceleration equal to that  of gravity4
 ,we would now feel certain that we were

 in empty space  because our own        
relative acceleration has entirely      
destroyed that of  the earth's          
gravitational field and all objects     
placed upon scales  in an elevator would
 apparently be without weight4          
,applying this idea1 then1 ,einstein    
decided to do away with gravitation     
entirely by referring all events in a   
gravitational field to a new set  of    
axes which should move with constant    
acceleration with respect to  the first4
 ,in other words we are going to deal   
with a system moving  with uniform      
acceleration with respect to the ether1 
just as we  considered a system moving  
with uniform velocity in the special    
theory4    ,the next step in the        
construction of this complicated theory 
is to  reduce these two hypotheses to   
the language of mathematics and this    
was accomplished by ,einstein with the  
help of ,m4 ,grossmann by means  of the 
theory of tensors4    ,on account of the
 very great intricacy of the details1 we

 must  content ourselves with the mere  
statement that this really involved  the
 generalization of the famous           
expressions known as ,laplace's  and    
,poisson's equations1 on the explicit   
assumption that these  two equations    
would still describe the gravitational  
field when we  are content to use a     
first approximation to the truth4 ,the  
set of  ten differential equations which
 ,einstein got as a result of his       
generalization he called his field      
equations of gravitation4 #f            
,,xiv    ,,new ,,concepts ,,for ,,old   
,what the ,world ,looks ,like ,after    
,einstein ,has ,had ,his ,way ,with ,it 
,,by ,,john ,g4 ,mc,h,a,r,d,y1          
,,commander ,r4,n41  ,,london           
,the new-created world1 which fame in   
heaven                      ,long had   
foretold1 a fabric wonderful1           
,of absolute perfection4                
,einstein's ,theory of ,relativity has  
led to determining a key law  of        
nature--the law of gravitation--which is

 also the basic law of  mechanics4 ,thus
 it embraces a whole realm of physics1  
and promises1  through the researches of
 ,professor ,weyl1 to embrace another   
realm--electro-dynamics4 ,its           
limitations are not yet reached1 for    
,einstein has already postulated        
therefrom a theory of a finite1 yet     
unbounded1 universe4 ,this essay1       
however1 is mainly concerned with       
mechanics1 and electrical forces are not
 considered4    ,to have synthesised    
,newton's two great principles--his law 
of motion  and law of                   
gravitation--interpreting in the process
 the empirical law  of equality of      
gravitational and inertial mass1 is     
alone an immense  achievement2 but      
,einstein's researches have opened up a 
new world  to the physicist and         
philosopher which is of greater         
importance4 ,he  has given us a vision  
of the immaterial world1 a geometrical  
or  mathematical vision1 which is more  
satisfying than the ether  conceptions  

hitherto presented4 ,the fabric of his  
vision is not  baseless4 ,it is this    
fabric we shall consider1 touching on   
certain  aspects of the ,einstein theory
 in the endeavor to present an image in 
miniature of his edifice of thought and 
to show the firmness of its             
foundations4 ,that they are well and    
truly laid was demonstrated by  the     
verification1 from observations made    
during the solar eclipse in  #aiai1 of  
,einstein's prediction of the           
displacement of a wave of light  in a   
gravitational field1 showing light to   
have the property of weight4    ,the    
physical world is shown by ,einstein to 
be a world of  relations4 ,underlying it
 there is an absolute world of which    
physical  phenomena are the             
manifestation4 ,give me matter and      
motion1 says  ,descartes1 and ,i will   
construct the world4 ,give me a world in
  which there are ordered relations1    
says the ,relativist1 and ,i will  show 
you the behavior of matter therein      

7mechanics74 ,we first view  this       
underlying world as an abstraction1     
abstracting energy 7bound  as in matter 
and electrons1 free as in light71 and   
its attribute  force4 ,this abstraction 
we will call the ,world-,frame4 ,later1 
we  will study the underlying world in  
connection with energy1 and will  call  
this absolute world the ,world-,fabric4 
,the connection between  the geometrical
 character of the ,world-,frame and the 
geometrical  characters of the          
,world-,fabric is the key to the law of 
gravitation4          ,,the             
,,world-,,frame    ,this is our         
conception of a world1 if such were     
possible1 entirely free  from the       
influence of energy4 ,we may conceive of
 it as an amorphous  immaterial         
something containing point-events 7a    
point-event  being an instant of time at
 a point in space--a conception1 not a  
definition74 ,these point-events have a 
fourfold order and definite  relation in
 this ,frame1 i4e4 they can be specified

 by four variables  or coordinates in   
reference to some base called a         
reference system1  with respect to which
 they are forward or backward1 right or 
left1  above or below1 sooner or later4 
,this shows the ,world-,frame to be     
four-dimensional4 ,thus an aggregate of 
point-events 7or an event1  which       
implies limited extension in space and  
limited duration  in time7 #g would have
 what we familiarly describe as length1 
breadth1 height and time4 ,to express   
these metrical properties most  simply  
we must choose a four-dimensional       
reference system having  a particular   
form--rectilinear axes 7,cartesian      
coordinates71 and a  particular         
motion--uniform and rectilinear1 i4e4   
unaccelerated1 and  non-rotating with   
respect to the path of a light ray4 ,we 
call this  an inertial system because   
,newton's ,law of ,inertia holds for    
such  a system alone4 ,this system      
indicates how observers partition the   
,world-,frame into space and time4 ,it  

restricts observers to uniform          
rectilinear motion1 and observations to 
bodies and light-pulses in  such motion4
 ,thus gravitational and other forces   
are discounted1  and we obtain          
,world-,frame conditions notwithstanding
 the fact that  observers are in the    
presence of energy4    ,now the         
separation between point-events which   
have a definite relation  to each other 
must be absolute4 ,the separation       
between two points in  a plane is       
defined by the unique distance between  
them 7the straight  line joining them74 
,between point-events the analogue of   
this unique  distance1 which we call the
 separation-interval 7to indicate its   
time-like and space-like nature71 is    
also unique4 ,its unique and  absolute  
character give it great importance as   
thereby it is the same  for all         
observers regardless of their reference 
system4    ,if1 in place of the rather  
cumbersome expression ,x-x to indicate  
the difference between the x-coordinates

 of two points1 we employ  the more     
compact expression dx2 if for the       
benefit of readers who  have a little   
algebra but no analysis we state        
explicitly that this  expression is a   
single symbol for a single quantity1 and
 has nothing  to do with any product of 
two quantities d and x2 and if we       
extend this notation to all our         
coordinates3 then it is clear from      
previous essays that the distance ,s    
between two points in a plane  referred 
to a rectilinear system ,,ox1 ,,oy1 is  
given by the simple  equation ,s#b      
7dx7#b  7dy7#b4 ,einstein and ,minkowski
 show that  the value for the separation
 interval ,omega1 the analogue of ,s1   
referred to an inertial system is given 
by the equation        ,omega#b  7dx7#b 
7dy7#b  7dz7#b - 7dt7#b11    which is   
seen to be a modified extension to four 
dimensions of the  equation for ,s4 ,we 
must measure t in the same units as x1  
y1  z4 ,by taking the constant velocity 
of light 7#cjj1jjj kilometres  per      

second7 as unit velocity1 we can measure
 in length or time  indiscriminately4 #h
    ,we will analyse briefly this       
equation as it epitomizes the ,special  
,theory of ,relativity4 ,if the         
,world-,frame had been ,euclidean the   
equation would have been        ,omega#b
  7dx7#b  7dy7#b  7dz7#b  7dt7#b    but 
this would not satisfy the              
transformation equations which  resulted
 from the ,special ,theory4 ,these      
transformation equations  arose directly
 from a reconciliation between two      
observed facts2 7a7  the observed       
agreement of all natural phenomena with 
the ,restricted  ,principle of          
,relativity--a principle which shows    
that absolute  rectilinear motion cannot
 be established--7as regards mechanics  
this  was recognized by ,newton2 the    
,michelson-,morley and other experiments
  showed this principle also applied to 
optical and electro-dynamical           
phenomena72 and 7b7 the observed        
disagreement of optical and             

electro-dynamical phenomena 7notably the
 constancy of light velocity7  with the 
laws of dynamics as given by classical  
mechanics1 e4g41 in  regard to the      
compounding of relative velocities4     
,einstein effected  this reconciliation 
by detecting a flaw in classical        
mechanics4 ,he  showed that by regarding
 space and time measurements as relative
  to the observer--not absolute as      
,newton defined them--there was  nothing
 incompatible between the ,principle of 
,relativity and the  laws of dynamics so
 modified4 ,newton's definitions were   
founded on  conception4 ,einstein's     
recognition of the relativity of space  
and  time is based on observation4      
,equation 7#a7 shows that the geometry  
of the ,world-,frame referred to  an    
inertial system is semi-,euclidean      
7hyperbolic71 and that space and  time  
measurements are relative to the        
observer's inertial reference  system4  
,the equation shows that the            
,world-,frame has a certain  geometrical

 character which we distinguish as      
four-dimensional  flatness4 ,it is      
everywhere alike 7homaloidal74 ,its flat
 character  is shown by the straight    
line nature of the separation-interval  
and  of the system to which it is most  
simply referred4    ,thus we have found 
two absolute features in the            
,world-,frame--7#a7 ,its  geometrical   
character--flatness2 7#b7 ,the          
separation-interval--which  can be      
expressed in terms of measurable        
variables called space and time         
partitions1 this partitioning being     
dependent on the observer's motion4     
,we are now in a position to explore the
 ,world-,fabric4 ,already we see  that1 
studied under inertial conditions 7free 
of force71 it agrees  with the          
,world-,frame4          ,,the           
,,world-,,fabric    ,the ,general       
,theory of relativity is largely        
concerned with the  investigation of the
 ,world-,fabric4 ,consider the          
,world-,frame to be  disturbed4 ,we may 

regard this disturbance1 which manifests
 itself in  physical phenomena1 as      
energy1 or more correctly action4       
,when energy is thwarted in its natural 
flow1 force is manifested1  with which  
are associated non-uniform motions such 
as accelerations  and rotations4 ,this  
disturbed ,world-,frame we distinguish  
as the  ,world-,fabric4 ,it is found to 
have various non-,euclidean characters  
differing from the simple flat character
 of the ,world-,frame according  to the 
degree of disturbance 7action7 in the   
region4 ,disturbance  gives the fabric a
 geometrical character of curvature2 the
 more  considerable the disturbance1 the
 greater the curvature4 ,thus an  empty 
region 7not containing energy1 but under
 its influence7 has  less curvature than
 a region in which free energy abounds4 
,our problem1 after showing the         
relativity of force 7especially         
gravitational force71 is to determine   
the law underlying the fabric's         
geometrical character2 to ascertain how 

the degree of curvature is  related to  
the energy influencing a region1 and how
 the curvature of  one region is linked 
by differential equations to that of    
neighboring  regions4 ,such a law will  
be seen to be the law of gravitation4   
,we study the ,world-,fabric by         
considering tracks on which material    
particles and light-pulses progress2 we 
find such tracks regulated  and defined 
by the ,fabric's curvature1 and not1 as 
hitherto supposed1  by attractive force 
inherent in matter4 ,as a track is      
measurable by  summing the              
separation-intervals between near-by    
point-events on it1  all observers will 
agree which is the unique track between 
two distant  point-events4 ,einstein    
postulates that freely progressing      
bodies  will follow unique tracks1 which
 are therefore called natural tracks    
7geodesics74    ,if material bodies are 
prevented from following natural tracks 
by  contact with matter or other causes1
 the phenomenon of gravitational  force 

is manifested relative to them4         
,whenever the natural flow  of energy is
 interrupted force is born4 ,for        
example1 when the  piston interrupts the
 flow of steam1 or golf ball flow of    
club1  force results--the interruption  
is mutual1 and the force relative  to   
both4 ,likewise when the earth          
interrupts the natural track of a       
particle 7or observer7 gravitational    
force is manifested relative  to both4  
,so long as a body moves freely no force
 is appreciated by it4 ,a  falling      
aviator 7neglecting air resistance7 will
 not appreciate any  gravitational      
force4 ,he follows a natural track1     
thereby freeing  himself from the force 
experienced in contact with matter4 ,he 
acquires an accelerating motion with    
respect to an inertial system4 ,by      
acquiring a particular accelerating     
motion an observer can annul any  force 
experienced in any small region where   
the field of force can  be considered   
constant4    ,thus ,einstein1           

interpreting the equality of            
gravitational and inertial  mass1 showed
 that the same quality manifests itself 
according to  circumstances as weight or
 as inertia1 and that all force is      
purely  relative and may be treated as  
one phenomenon 7an interruption in      
energy flow74 ,this ,principle of       
,equivalence shows that small portions  
of the ,world-,fabric1 observed from a  
freely moving particle 7free of  force71
 could be treated as small portions of  
the ,world-,frame4 #i    ,if such       
observations were practicable1 we could 
determine the ,fabric  curvature by     
referring point-event measurements to   
equation 7#a74 ,we  cannot observe from 
unique tracks but we can observe them   
from our  restrained situation4 ,their  
importance is now apparent1 because1 by 
tracing them over a region1 we are      
tracing something absolute in the       
,fabric--its geometrical character4 ,we 
study this curvature by exploring       
separation-intervals on the tracks of   

freely moving bodies1 relating  these   
separation-intervals to actual          
measurements in terms of space  and time
 components depending on the observer's 
reference system4 ,the  law of curvature
 must be the law of gravitation4 ,to    
illustrate the  lines on which ,einstein
 proceeded to survey the ,world-,fabric 
from  the earth we will consider a      
similar but more simple problem--the    
survey of the sea-surface curvature from
 an airship4 ,we study this  curvature  
by exploring small distances on the     
tracks of ships 7which we  must suppose 
can only move uniformly on unique       
tracks--arcs of great  circles71        
relating such distances to actual       
measurements in terms of  length and    
breadth components depending on the     
observer's reference  system4 ,this     
two-dimensional surface problem can be  
extended to the  four-dimensional       
,fabric one4    ,we consider the surface
 to be covered by two arbitrarily drawn 
intersecting series of curves3 curves in

 one series not intersecting  each      
other1 vide figure4 ,this ,gaussian     
system of coordinates is  appropriate   
only when the smaller the surface       
considered1 the  more nearly it         
approximates to ,euclidean conditions4  
,it admits of  defining any point on the
 surface by two numbers indicating the  
curves  intersecting at that point4 ,p  
is defined by x#a1 x#b4 ,p#a  7very near
 ,p7 is defined by x#a  dx#a1 x#b  dx#b4
 ,the equation  for the minute distance 
s between two adjacent points in such a 
system is given by the general formula  
s#b  g#aa;dx#ab  g#ab;dx#a;dx#b         
g#bb;dx#bb14    ,the g's may be         
constants or functions of x#a1 x#b4     
,their  value is dependent on the       
observer's reference system and on the  
geometrical character of the surface    
observed4 ,the curves being  arbitrary1 
the formula is appropriate for any      
reference system1  or even if the       
observer does not know exactly what his 
reference  system is4 7,the ,fabric     

observer does not know what his space   
and time  partitioning actually is      
because he is in a gravitational field74
 ,it  is the g's which disclose the     
geometry of an observer's partitions1   
and their values also contain a         
reflection of the character of the      
region observed4    ,we find s by direct
 exploration with a moving ship 7,omega 
is found  by direct exploration with a  
freely moving particle72 dx#a1 dx#b  are
 the observed length and breadth        
measurement differences which we  have  
to relate to s4 ,by making sufficient   
observations in a small  area and       
referring them to the general formula we
 can find the values  of the g's for the
 observer's particular reference system4
 ,different  values for g's will be     
found if the observer changes his       
reference  system1 but there is a       
limitation to the values so obtainable  
owing to  the part played by the surface
 itself1 which is diffidently expressing
  its intrinsic geometrical character in

 the g's in each observation4           
,,einstein',s ,,results    ,thus we     
approach the absolute character of the  
surface through  the relative nature of 
the observer's reference system4 ,there 
is a  relationship common to all values 
of the g's that belong to the  same     
curvature4 ,this relationship is        
expressed by a differential  equation4  
,it is this equation of curvature that  
the airship's observer  must find4      
,einstein's problem was similar1 but he 
was concerned with  four dimensions1    
which entailed a general formula with   
ten g's1  and he had to find a set of   
differential equations of the second    
order to determine the law of ,fabric   
curvature4 ,he divided the ,fabric  into
 regions3 ,i4 ,world-,frame--beyond     
influence of energy4 ,,ii4 ,empty       
region--free of energy1 but under its   
influence4 ,,iii4 ,region containing    
free energy only4 ,each region has a    
characteristic curvature4 ,by  means of 
an absolute differential calculus--a    

wonderful mathematical  scaffolding     
erected by ,riemann1 ,christoffel and   
others--involving the  theory of        
tensors1 he succeeded in finding such a 
set of equations4 ,he  kept the         
following points in view3 7#a7 ,the     
equations must not only  give the       
character of region ,,ii1 but must      
satisfy the special case  of region ,i2 
7#b7 ,they must be independent of any   
partitioning system1  because the       
,general ,theory of ,relativity demands 
that a law of nature  be in a form      
appropriate for all observers whatever  
their position  and motion2 7#c7 ,they  
must be concerned with energy which is  
conserved1  not mass which the ,special 
,theory showed dependent on velocity4   
,this  set of differential equations    
which shows how the curvature of the    
,fabric at any point links to the       
curvature at neighboring points is  the 
law of gravitation1 a law which has been
 severely tested by the  practical      
observation of the solar eclipse already

 referred to4 ,at a  first approximation
 these equations degenerate into        
,newton's ,law4 ,at  a second           
approximation they account for the      
motion of the perihelion  of ,mercury1  
which had hitherto baffled astronomers4 
,all the laws  of mechanics are         
deducible from this law of              
,world-,fabric curvature1  i4e4         
conservation of energy 7which includes  
conservation of mass since  we re-define
 mass as energy7 and conservation of    
momentum 7re-defined  by a relativist74 
,it must be noted that this law and the 
,general  ,theory show that the velocity
 of light is not absolutely constant1   
but1 like everything else1 a light-pulse
 is affected by the ,fabric  curvature  
in a gravitational field4 ,in conclusion
 we will contrast  some conspicuous     
differences in the old world view of    
classical  mechanics and the new view   
presented by ,einstein4    #a4 ,a       
three-dimensional ether medium with     
variously conceived properties  which   

communicated the supposed inherent      
attractive force in matter  in some     
unexplained way1 and transmitted        
electromagnetic waves1  has been        
replaced by a four-dimensional external 
,world-,fabric1 the  geometrical        
character of which controls the motion  
of matter 7energy7  and accounts for all
 mechanical laws4    #b4 ,after         
separating the observer's subjective    
share in definitions  from nature's     
share in the things defined1 space1     
time1 and force1  hitherto regarded as  
absolute1 have been shown to be purely  
relative  and dependent on the          
observer's track4 ,mass has also proved 
to be  relative to velocity unless      
re-defined as energy4 ,as classical     
mechanics bases all definitions on      
space1 time1 and mass units1  the       
relativity of such defined quantities is
 now apparent4    #c4 ,newton's laws of 
motion1 his law of gravitation1 and the 
laws of  conservation1 hitherto regarded
 as unrelated1 are now synthesised in  a

 basic law of mechanics4    ,einstein   
has not disturbed the electric theory of
 matter1 and both the  old and new      
physics have in common the ,principle of
 ,least ,action4 ,we  obtain a glimpse  
of this principle in the unique tracks  
pursued by  freely moving bodies1 which 
may be regarded as tracks of least      
effort1  force only being manifested as 
an expression of the ,fabric's          
resentment  when bodies depart from     
these natural tracks4 ,einstein has     
approached  nearer to the truth in      
regard to the laws underlying nature1   
and1  as always1 this means a           
simplification4 ,his theory1 which      
entails  a readjustment of such         
fundamental conceptions as space and    
time1  opens up fresh fields to         
scientific investigation and to         
philosophic  thought4 ,it reveals a     
bridge uniting the domains of physics   
and  philosophy1 and it heralds a new   
era in the history of science4          
,,xv    ,,the ,,new ,,world    ,a       

,universe in ,which ,geometry ,takes the
 ,place of ,physics1 and ,curvature     
,that of ,force    ,,by ,,george        
,,frederick ,,hemens1 ,m4,c41 ,b4,,sc41 
,,london      ,it is familiar knowledge 
that the line1 the surface and ordinary 
,euclidean space are to be regarded as  
spaces of one1 two and three  dimensions
 respectively and readers of this       
journal are aware  that a hypothetical  
space of four dimensions has been       
closely  investigated4 ,the most        
convenient space to study is the surface
 or  two-space1 since we can regard it  
as embedded in a three-space4 ,if a     
surface is curved it is generally       
impossible to draw a straight line  on  
it1 for as we see clearly1 the          
straightest line is changing its        
direction at every point4 ,to describe  
this property accurately it  is         
necessary to ascribe to each point a    
magnitude which expresses  what happens 
to the direction of a short line in the 
region when  displaced a short distance 

parallel to itself4 ,this is called the 
direction-defining magnitude4 ,different
 sets of values of this  magnitude      
relate to surfaces of different         
curvatures4    ,a second fundamental    
property has recently been pointed out4 
,there  is inherent in every part of a  
space a measure of length peculiar  to  
that particular region and which in     
general varies from region  to region4  
,to describe this variation accurately  
it is necessary to  ascribe to each     
point another magnitude called the      
length-defining  magnitude1 which       
expresses the change from each point to 
the next of  the unit of length4 ,these 
two magnitudes define the surface       
completely4    ,similarly1 a space of   
any number of dimensions is defined     
completely  by a similar pair of        
magnitudes4 ,a space is the field of    
such  a magnitude-pair and the nature of
 these magnitudes defines the           
dimensions of the space4 ,the four-space
 usually described is the  ,euclidean   

member of an infinity of four-spaces4   
,when we look into a mirror we see a    
space differing from ordinary  space in 
that right and left are interchanged and
 this is described  mathematically by   
saying that if we locate points as usual
 by  specifying three distances ,x#a1   
,x#b1 ,x#c of the point from three      
mutually perpendicular planes1 then a   
point ,x#a1 ,x#b1 ,x#c1 in  actual space
 corresponds with a point ,x#a1 ,x#b1   
-,x#c in the  mirrored space3 in other  
words the mirrored space is derived from
  the real space by multiplying the ,x#c
 coordinates by -#a4 ,if we  were to    
multiply by sqrt-#a instead of -#a we   
should derive a  different space2 in    
this case1 however1 we have no mirror to
 show us  what it looks like4 ,such a   
space is said to have one negative      
dimension  and it has the peculiar      
property that in the figure derived from
 the  right triangle of ordinary space  
the square of the hypotenuse equals  the
 difference and not the sum of the      

squares of the other two sides1  so that
 the length of a line may sometimes have
 to be represented by  the square-root  
of a negative number1 a complex number4 
,in considering what at first sight may 
appear to be fantastic  statements made 
by this theory1 it must be borne in mind
 that all our  knowledge of the external
 universe comes through our             
sense-impressions1  and our most        
confident statements about external     
things are really  of the nature of     
inferences from these sense-impressions 
and1 being  inferences1 liable to be    
wrong4 ,so that if the theory says that 
a  stone lying on the ground is not a   
simple three-dimensional object1 and    
that its substance is not the same as   
its substance a moment before1  the     
matter is one for due consideration and 
not immediate disbelief4    ,the idea   
that the universe extends in time as    
well as in space is not  new1 and       
fiction-writers have familiarized us    
with wonderful machines  in which       

travellers journey in time and are      
present at various stages  of the       
world's history4 ,this conception of the
 universe1 to which the  name space-time
 is usually applied1 is adopted by the  
new theory  and assigned the status of a
 physical reality4          ,,the       
,,world ,,geometry    ,the fundamental  
creed of the new theory is that the     
space-time universe  constitutes a true 
four-dimensional space of one negative  
dimension1  this dimension being time4  
,the variations from point to point of  
the direction-defining and              
length-defining magnitudes generate the 
geometrical properties of curvature1    
etc41 and these are cognised by  the    
human mind as physical phenomena3 our   
sense-impressions are nothing  more nor 
less than perceptions of the geometry of
 a fourspace4 ,so  instead of inferring 
from our sense-impressions the existence
 of  matter1 motion and the like as we  
are accustomed to do1 we should with    
equal justice infer the existence of a  

geometrical fourspace4 ,thus it  becomes
 necessary to prepare a dictionary in   
which the familiar things  of our world 
are identified with those geometrical   
properties of the  four-space which     
really constitute them1 and in so doing 
parts of  our geometrical knowledge     
assume the guise of new physical        
knowledge4    ,through the fourspace our
 consciousness travels1 cognising a     
changing  three-dimensional section of  
it as it goes and thus giving rise  to  
time4 ,it becomes aware that the        
fourspace is pleated or folded  along   
lines all running roughly in the same   
direction1 and possibly  because this is
 the easiest direction to follow1 it    
travels along the  lines4 ,the direction
 of this motion is the negative         
dimension4 ,thus  consciousness is      
always aware of the nearly constant     
forms of the  cross-sections of the     
pleats along which it travels4 ,these   
unvarying  forms constitute matter3     
matter is the form of a section through 

a uniform pleat of the fourspace--a     
three-dimensional aspect of a           
four-dimensional curvature2 so that in  
strict accuracy we should  say that a   
stone is the shape or form of a changing
 section of a  four-dimensional object1 
the complete object being a long fold in
 the  fourspace4 ,the physical          
interpretation of this conservation of  
form  of the cross-section is that      
matter is conserved4 ,it is thus seen   
that  the conscious mind1 by following  
these pleats1 has so determined time    
that the law of the conservation of     
matter must hold4 ,the mathematical     
treatment of the subject makes it clear 
that practically all other  physical    
laws similarly follow as a direct result
 of this choice of  time4 ,the type of  
order prevailing in the physical        
universe1 the laws  of gravitation1     
heat1 motion and the rest are not       
directly imposed  by some external      
power1 but are apparently chosen by mind
 itself4    ,in the neighborhood of     

these pleats the fourspace is still     
curved1  but to a smaller degree4 ,this 
we cognise as energy or as a field  of  
force4 ,thus energy is seen to be the   
same kind of thing as  matter and would 
therefore be expected to have weight4   
,this was  experimentally demonstrated  
in #aiai when light was in effect       
actually  weighed4 ,conversely1 matter  
consists of energy2 and it is calculated
  that one liter of water contains      
sufficient energy to develop a million  
horsepower for about four years4 ,it is 
now believed that the sun's  energy is  
derived from the disintegration of the  
matter of which it  is made4    ,the    
method of establishing these            
identifications will be clear from the  
following3 ,we already knew that matter 
is made up of electrons and that        
radiant energy is electromagnetic and   
before the advent of this theory  it was
 regarded as certain that practically   
all observed physical  phenomena except 
gravitation were manifestations of the  

electromagnetic  field4 ,the new theory 
has confirmed this belief4 ,it is found 
that  the gravitational and             
electromagnetic conditions of the       
universe are  completely defined if to  
each point of space-time a gravitational
 and  an electric potential are         
ascribed4 ,these are magnitudes of the  
same  nature as the direction-defining  
and length-defining magnitudes which    
must necessarily be associated with     
every point of space-time if it  is a   
true space1 and they are therefore      
identified with these4 ,by  performing  
ordinary mathematical operations on     
these magnitudes  statements of fact    
clothed in mathematical form are        
obtained1 which are  to be interpreted  
on the one hand as physical laws and on 
the other  as geometrical properties of 
the fourspace4 ,nearly all our physical 
laws are derivable mathematically in    
this way1 so that an extensive          
identification is effected which has    
been fruitful of results4    ,it has    

been mentioned that a slight curvature  
is sometimes cognised  as force and as  
this identification appeared originally 
as a postulate  its history is          
interesting4          ,,the ,,genesis   
,,of ,,the ,,theory    ,an experiment by
 ,michelson and ,morley 7#ahhg71 on     
which the whole theory  is based1 made  
it appear that if a man measures the    
velocity at which  light passes him he  
will get the same result whether he is  
stationary1  rushing to meet the light1 
or moving in the same direction as the  
light4 ,the solution was provided by    
,einstein in #aije4 ,he suggested that  
since we know the results of these      
determinations ought not to agree1      
something must have happened to the     
clocks and measuring-rods used in       
measuring the velocity so that the      
standards of length and time were  not  
the same in the three cases1 the        
alterations being exactly such as  to   
make the velocity of light constant4    
,this solution is universally  accepted 

as true and is the fundamental          
postulate4 ,thus the length of  a stick 
and the rate at which time passes will  
change as the velocity  of the person   
observing these things changes4 ,if a   
man measured the  length of an aeroplane
 going past him at #afa1jjj miles per   
second it  would measure only half the  
length observed when stationary4 ,if the
  aeroplane were going with the velocity
 of light1 its length would  vanish     
though its breadth and height would be  
unaltered4 ,similarly1  if of two twin  
brothers one were continually moving    
with reference  to the other their ages 
would gradually diverge1 for time would 
go at  different rates for the two4 ,if 
one moved with the velocity of light1   
time would stand still for him while for
 the other it would go on as  usual4 ,to
 get actually younger it would be       
necessary to move quicker  than light   
which is believed to be impossible4 ,the
 velocity of light  is assumed to be the
 greatest velocity occurring in nature4 

,evidently then if the distance in space
 and the interval in time  separating   
two given events1 such as the firing of 
a gun and the  bursting of the shell1   
are measured by two observers in uniform
  relative motion1 their estimates will 
not agree4 ,consider now the  simple    
problem of measuring the distance       
between two points on an  ordinary      
drawing-board4 ,if we draw two          
perpendicular axes1 we can  define this 
distance by specifying the lengths of   
the projections  on the two axes of the 
line joining the points4 ,if we choose  
two  different axes the projections will
 not be the same but will define  the   
same length4 ,similarly1 in a ,euclidean
 four-space the distance  between two   
points will be defined by the           
projections on the four  axes1 but if   
these axes be rotated slightly1 the     
projections will be  different1 but will
 define the same length4 ,now1 returning
 to the two  observers just mentioned1  
it was noticed by ,minkowski in #aijh   

that  if the space measurements between 
the two events are split into the  usual
 three components1 and if the time      
measurements are multiplied  by sqrt-#a1
 the difference between the two sets of 
measurements  is exactly the same as    
would have occurred had these two events
  been points in a ,euclidean fourspace1
 and two different observations  made of
 their distance apart using two sets of 
axes inclined to each  other4 ,the      
velocity of light is made equal to #a in
 this calculation by  a suitable choice 
of units4 ,this discovery threw a vivid 
light on the  problem of space-time1    
showing that it is probably a true      
four-space  of one negative dimension1 a
 simple derivative of the much-discussed
  and now familiar ,euclidean           
four-space4    ,although this discovery 
gave a tremendous impetus to the        
progress  of the theory1 it is probable 
that it holds a deeper significance  not
 yet revealed4 ,it is probably a        
statement of the stuff of which  the    

four-space is made1 and perhaps also of 
how it is made2 but the  problem remains
 unsolved4    ,it thus becomes plain    
that our two observers are merely       
looking at the  same thing from         
different viewpoints4 ,each has just as 
much right as  the other to regard      
himself as being at rest in ordinary    
space 7this is  the postulate of the    
relativity of uniform motion7 and to    
regard his  time direction as a straight
 line in the four-space4 ,the difference
  is merely that the two time axes are  
inclined to each other4 ,if1  however1  
one were moving with an acceleration    
with reference to the  other his path in
 the four-space will appear curved to   
the other1  though he himself1 since he 
regards it as his time axis1 will still 
assume it to be straight4 ,if there is a
 body moving in what one  observer sees 
to be a straight line1 the other will1  
of course1  in general see it as curved1
 and following the usual custom1 since  
this body1 without apparent reason1     

deviates from the straight path1  will  
say there must be some force acting on  
it4 ,thus the curvature  of his time    
axis1 due to his accelerated motion1    
makes it appear  that there is round him
 a field of force1 which causes freely  
moving  bodies to deviate from the      
straight path4 ,now if space-time is    
itself  inherently curved it is not     
generally possible for any line in it to
  be straight any more than it is       
possible for any line on the surface  of
 a sphere to be straight4 ,hence1 all   
axes must be curved1 and all  observers1
 whatever their states of motion1 must  
experience fields of  force which are of
 the same nature as those due to motion 
only4 ,the  extra force experienced when
 a lift begins to rise is an example of 
force due to pure motion3 gravitation is
 the similar force due to an  inherent  
curvature of the four-space1 and it was 
the postulate that  these forces were   
similar that made possible ,einstein's  
solution of  the general problem of     

gravitation4          ,,the ,,time      
,,diagram    ,the correlation of time   
with its geometrical analogue is of     
absorbing  interest4 ,representing      
velocity by the common method of        
plotting  a curve showing positions at  
various times and marking distances     
horizontally and times vertically1 the  
velocity of light being #a1  ,,mm' and  
,,nn' will both represent this velocity4
 ,since this is  assumed to be the      
greatest velocity occurring in nature1  
all other  possible velocities are      
represented by lines falling within the 
upper  and lower ,v's4 ,now this diagram
 correctly represents two dimensions  of
 ,minkowski's ,euclidean four-space so1 
transmuting to real but flat  four-space
 by multiplying times by sqrt-#a1 it is 
seen that there  is a region outside    
which no effect can be propagated from  
,o since  that would involve the        
existence of a velocity greater than    
that of  light4 ,this region represents 
the future of ,o4 ,similarly1 ,o can    

only be affected by events within the   
region derived from the                 
downward-opening ,v1 which therefore    
represents the past of ,o4  ,the region 
between the two represents events which 
may be either  simultaneous with ,o or  
not1 according to the velocity of the   
observer at ,o4 ,thus in this theory an 
event dictated by free-will1  could     
affect points in its future region1 but 
not in any other1  which agrees with    
experience and shows that the theory is 
not  essentially determinist4 ,if       
free-will is really free1 the future  is
 not yet determined1 and the fourspace  
must be in some way formed  by the will 
as time progresses4    ,the trains of   
thought inspired by ,einstein's         
postulates have already  carried us to a
 pinnacle of knowledge unprecedented in 
the history of  man4 ,on every hand1 as 
we look out upon the universe from our  
new and  lofty standpoint1 unexpected   
and enthralling vistas open up before   
us1  and we find ourselves confronting  

nature with an insight such as no  man  
has ever before dared aspire to4    ,it 
is completely unthinkable that this     
theory can ever be swept  aside4 ,apart 
from experimental verifications which1  
in point of fact1  lend it the strongest
 support1 no one could work through the 
theory  without feeling that here1 in   
truth1 the inner workings of the        
universe  were laid bare before him4    
,the harmony with nature is far too     
complete  for any doubt to arise of its 
truth4                    ,,xvi    ,,the
 ,,quest ,,of ,,the ,,absolute          
,modern ,developments in ,theoretical   
,physics1 and the ,climax ,supplied  by 
,einstein    ,,by ,,dr4 ,,francis ,d4   
,,murnaghan1  ,,johns ,,hopkins         
,,university  ,,baltimore      ,we shall
 discuss the more important aspects of  
the theory popularly  known as the      
,einstein ,theory of ,gravitation and   
shall try to show  clearly that this    
theory is a natural outcome of ideas    
long held by  physicists in general4    

,these ideas are3    7a7 ,the           
impossibility of action at a distance2  
in other words we  find an instinctive  
repugnance to admit that one body can   
affect  another1 remote from it1        
instantaneously and without the         
existence  of an intervening medium4    
7b7 ,the independence of natural1 i4e41 
physical1 laws of their  mathematical   
mode of expression4 ,thus1 when an      
equation is written down  as the        
expression of a physical law it must be 
satisfied1 no matter  what units we     
choose in order to measure the          
quantities occurring in  the equation4  
,as our physics teacher used to say the 
expression of the  law must have in     
every term the same dimensions4 ,more   
than this the  choice of the quantities 
used to express the law--if there be a  
choice  open--must have no effect on its
 correctness4 ,as we were told--all     
physical laws are capable of expression 
as relations between vectors or  else as
 relations between magnitudes of the    

same dimensions4 ,we shall  hope to make
 this clearer in its proper place in the
 essay1 as its  obvious generalization  
is ,einstein's cardinal principle of    
relativity4    ,the measurements which  
an experimental physicist makes are     
always the  expression of a coincidence 
of two points in space at the same time4
 ,if  we ask such an experimenter what  
he means by a point in space he tells   
us that1 for him1 the term has no       
meaning until he has a material  body   
with reference to which he can locate   
the point by measurements2  in general  
it requires three measurements and he   
expresses this by  saying that space has
 three dimensions4 ,he measures his     
distance1  as a rule1 parallel to three 
mutually perpendicular lines fixed in   
the material body--a ,cartesian         
reference-frame so-called4 ,so that a   
point in space is equivalent to a given 
material reference-frame  and three     
numbers or coordinates4 ,if1 for any    
reason1 we prefer to  use a new material

 reference-frame the coordinates or     
measurements  will change and1 if we    
know the relative positions of the two  
material  reference-frames1 there is a  
definite relation between the two sets  
of  three coordinates which is termed a 
transformation of coordinates4 ,but     
which particular material               
reference-frame shall we use8 ,the first
  choice would1 we think1 be that       
attached to the earth4 ,but1 even yet1  
we are in doubt as there are numberless 
,cartesian frameworks attached  to the  
earth 7as to any material body7 and it  
is here that our idea  7b7 begins to    
function4 ,we say it must be immaterial 
which of these  ,cartesian frames we    
use4 ,in each frame a vector has three  
components  and when we change from one 
frame to another the components change  
in such a way that if two vectors have  
their three components equal  in one    
framework they will be equal in any     
other attached to the  same material    
system4 ,so our idea 7b71 which says    

that our physical  equations must be    
vector equations1 is equivalent to      
saying that the  choice of the framework
 attached to any given material body can
 have  no effect on the mode of         
expression of a natural law4    ,shall  
we carry over our idea 7b7 to answer the
 next question3 ,to  which material body
 shall we attach our framework8 ,to this
 question  ,newton gave one answer and  
,einstein another4 ,we shall first      
consider  ,newton's position and then we
 may hope to see clearly where the new  
theory diverges from the classical or   
,newtonian mechanics4 ,newton's  answer 
was that there is a particular material 
frame with reference  to which the laws 
of mechanics have a remarkably simple   
form commonly  known as ,newton's laws  
of motion and so it is preferable to use
  this framework which is called an     
absolute frame4    ,what is the         
essential peculiarity of an absolute    
frame8 ,newton was  essentially an      
empiricist of ,bacon's school and he    

observed the  following facts4 ,let us  
suppose we have a framework of reference
  attached to the earth4 ,then a small  
particle of matter under the            
gravitational influence of surrounding  
bodies1 including the earth1  takes on a
 certain acceleration ,a#a4 ,now suppose
 the surrounding  bodies removed 7since 
we cannot remove the earth we shall have
 to  view the experiment as an          
abstraction71 and another set           
introduced2  the particle1 being again  
at its original position1 will begin to 
move with an acceleration ,a#b4 ,if both
 sets of surrounding bodies  are present
 simultaneously the particle begins to  
move with an  acceleration which is     
approximately but not quite the sum of  
,a#a  and ,a#b4 ,newton postulated there
 there is a certain absolute  reference 
frame in which the approximation would  
be an equality2  and so the             
acceleration1 relative to the material  
frame1 furnishes  a convenient measure  
of the effect of the surrounding        

bodies--which  effect we call their     
gravitational force4 ,notice that if the
 effect  of the surrounding bodies is   
small the acceleration is small and  so 
we obtain as a limiting case1 ,newton's 
law of inertia which says  that a body  
subject to no forces has no             
acceleration2 a law which1  as ,poincar 
justly observed1 can never be subjected 
to experimental  justification4 ,the    
natural questions then arise3 which is  
the  absolute and privileged            
reference-frame and how must the simple 
laws  be modified when we use a frame   
more convenient for us--one attached  to
 the earth let us say8 ,the absolute    
frame is one attached to the  fixed     
stars2 and to the absolute or real force
 defined as above1  we must add certain 
terms1 usually called centrifugal       
forces4 ,these  are referred to as      
fictitious forces because1 as it is     
explained1  they are due to the motion  
of the reference-frame with respect to  
the absolute frame and in no way depend 

on the distribution of the  surrounding 
bodies4 ,gravitational force and        
centrifugal forces have  in common the  
remarkable property that they depend in 
no way on the  material of the attracted
 body nor on its chemical state2 they   
act  on all matter and are in this way  
different from other forces met  with in
 nature1 such as magnetic or electric   
forces4 ,further ,newton  found that he 
could predict the facts of observation  
accurately on the  hypothesis that two  
small particles of matter attracted each
 other1  in the direction of the line   
joining them1 with a force varying      
inversely as the square of the distance 
between them4 ,this law is an  action at
 a distance law and so is opposed to the
 idea 7a74    ,we have tacitly supposed 
that the space in which we make our     
measurements is that made familiar to us
 by the study of ,euclid's  elements4   
,the characteristic property of this    
space is that  stated by the theorem of 
,pythagoras that the distance between   

two  points is found by extracting the  
square root of the sum of the  squares  
of the differences of the ,cartesian    
coordinates of the two  points4         
,mathematicians have long recognized the
 possibility of other  types of space   
and ,einstein has followed their lead4  
,he abandons  the empiricist method and 
when asked what he means by a point in  
space replies that to him a point in    
space is equivalent to four  numbers how
 obtained it is unnecessary to know a   
priori2 in certain  special cases they  
may be the three ,cartesian coordinates 
of the  experimenter 7measured with     
reference to a definite material        
framework7  together with the time4     
,accordingly he says his space is of    
four  dimensions4 ,between any two      
points we may insert a sequence of sets 
of four numbers1 varying continuously   
from the first set to the second1  thus 
forming what we call a curve joining the
 two points4 ,now we define  the length 
of this curve in a manner which involves

 all the points  on it and stipulate    
that this length has a physical reality1
 i4e41  according to our idea 7b7 its   
value is independent of the particular  
choice of coordinates we make in        
describing the space4 ,among all  the   
joining curves there will be one with   
the property of having  the smallest    
length2 this is called a geodesic and   
corresponds to  the straight line in    
,euclidean space4 ,we must now1 for lack
 of an  a priori description of the     
actual significance of our coordinates1 
extend the idea of vector so that we may
 speak of the components of  a vector no
 matter what our coordinates may        
actually signify4 ,in  this way are     
introduced what are known as tensors2 if
 two tensors  are equal1 i4e41 have all 
their components equal1 in any one set  
of  coordinates they are equal in any   
other and the fundamental demand of  the
 new physics is that all physical       
equations which are not merely  the     
expression of equality of magnitudes    

must state the equality of  tensors4 ,in
 this way no one system of coordinates  
is privileged above  any other and the  
laws of physics are expressed in a form 
independent  of the actual coordinates  
chosen2 they are written1 as we may say1
  in an absolute form4          ,,the   
,,gravitational ,,hypothesis            
,einstein flatly denies ,newton's       
hypothesis that there is an absolute    
system 7and1 indeed1 many others before 
him had found it difficult  to admit    
that so insignificant a part of the     
universe as our fixed  star system      
should have such a privileged position  
as that accorded  to it in the          
,newtonian ,mechanics74 ,in any system1 
he says1 we have no  reason to          
distinguish between the so-called real  
gravitational force  and the so-called  
fictitious centrifugal forces--if we    
wish so to  express it gravitational    
force is fictitious force4 #aj ,a       
particle  moving in the neighborhood of 
material bodies moves according to a law

  of inertia--a physical law            
expressible1 therefore1 in a manner     
quite  independent of the choice of     
coordinates4 ,the law of inertia is that
  a particle left to itself moves along 
the geodesics or shortest lines  in the 
space4 ,if the particle is remote from  
other bodies the space  has the         
,euclidean character and we have        
,newton's law of inertia2  otherwise the
 particle is in a space of a            
non-,euclidean character  7the space    
being always the four-dimensional space7
 and the path of  the particle is along 
a geodesic in that space4 ,einstein1 in 
order  to make the theory more concrete1
 makes a certain stipulation as to  the 
nature of the gravitational space which 
stipulation is expressed1  as are all   
physical laws1 by means of a tensor     
equation--and this is  sometimes called 
his law of gravitation4    ,perhaps it  
will be well1 in exemplification1 to    
explain why light  rays1 which pass     
close to the sun1 should be bent        

according to the new  theory4 ,it is    
assumed that light rays travel along    
certain geodesics  known as minimal     
geodesics4 ,the sun has an intense      
gravitational field  near it--or1 as we 
now say1 the departure of the           
four-dimensional  space from the        
,euclidean is very marked for points    
near the sun--but  for points so remote 
as the earth this departure is so small 
as  to be negligible4 ,hence the form of
 the geodesics near the sun is          
different from that near the earth4 ,if 
the space surrounding the  sun were     
,euclidean the actual paths of the light
 rays would appear  different from      
geodesics or straight-lines4 ,hence     
,einstein speaks  of the curvature of   
the light rays due to the gravitational 
field of  the sun2 but we must not be   
misled by a phrase4 ,light always       
travels  along geodesics 7or straight   
lines--the only definition we have of a 
straight line is that it is a geodesic72
 but1 owing to the distortion  of the   

space they traverse1 due to the sun1    
these geodesics reach us  with a        
direction different from that they would
 have if they did not  pass through the 
markedly non-,euclidean space near the  
sun4    ,the consideration of the       
fundamental four-dimensional space as   
being non-,euclidean where matter is    
present gives a possibility of  an      
answer to the world old question3 ,is   
space finite or infinite8 ,is  time     
eternal or finite8 ,the fascinating     
possibility arises that the  space may  
be like the two-dimensional surface of a
 sphere which to a  limited experience  
seems infinite in extent and flat or    
,euclidean in  character4 ,a new        
,columbus now asks us to consider other 
possibilities  in which we should have a
 finite universe--finite not only as to 
space  measurement but as to time 7for  
the space may be such that all of the   
four coordinates of its points are      
bounded in magnitude74 ,however1        
although ,einstein speaks of the        

possibility of a finite universe1 we  do
 not1 personally1 think his argument    
convincing4 ,points on a sphere  may be 
located by the ,cartesian coordinates of
 their stereographic  projections on the
 equatorial plane and these coordinates1
 which  might well be those actually    
measured1 are not bounded4              
,,the ,,special ,,relativity ,,theory   
,in our account of the ,einstein theory 
we have not followed its  historical    
order of development for two reasons4   
,firstly1 the  earlier ,special         
,relativity ,theory properly belongs to 
a school  of thought diametrically      
opposed to that furnishing the ,general 
,theory of ,relativity and1 secondly1   
the latter cannot be obtained  from the 
former by the process of generalization 
as commonly  understood4 ,einstein1 when
 proposing the earlier theory1 adopted  
the position of the empiricist so that  
to him the phrase1 a point in  space1   
had no meaning without a material       
framework of reference in  which to     

measure space distances4 ,when he came  
to investigate what  is meant by time   
and when he asked the question what is  
meant by  the statement that two remote 
events are simultaneous8 it became      
evident that some mode of communication 
between the two places is  necessary2   
the mode adopted was that by means of   
light-signals4 ,the  fundamental        
hypothesis was then made that the       
velocity of such signals  is independent
 of the velocity of their source 7some  
hypothesis is  necessary if we wish to  
compare the time associated with events1
 when  one material reference-system is 
used1 and the corresponding time when   
another in motion relative to the first 
is adopted74 ,it develops that  time and
 space measurements are inextricably    
interwoven2 there is no  such thing as  
the length of a body or the duration of 
an event but  rather these are relative 
to the reference-system4 #aa ,minkowski 
introduced the idea of the space of     
events--of four dimensions--but  this   

space was supposed ,euclidean like the  
three-dimensional space  of his         
predecessors4 ,to ,einstein belongs the 
credit of taking from  this             
representation a purely formal          
mathematical character and of  insisting
 that the real space--whose distances   
have a physical  significance--is the   
four-dimensional space4 ,but we cannot  
insist  too strongly on the fact that in
 the gravitational space of the  general
 theory there is no postulate of the    
constancy of velocity of a  light-signal
 and accordingly no method of assigning 
a time to events  corresponding to that 
adopted in the special theory4 ,in this 
latter  theory attention was confined to
 material systems moving with uniform   
velocity with respect to each other and 
it developed that the velocity  of light
 was the ultimate velocity faster than  
which no system could  move--a result   
surprising and a priori rather          
repugnant4 ,it is merely  a consequence 
of our mode of comparing times of       

events2 if some other  method--thought  
transference1 let us say--were possible 
the velocity  of this would be the      
limiting velocity4    ,in conclusion we 
should remark that the postulated       
equivalence of  gravitational and       
centrifugal forces demands that anything
  possessed of inertia will be acted    
upon by a gravitational field and  this 
leads to a possible identification of   
matter and energy4 ,further  our guiding
 idea 7a7 will prompt us to say1        
following the example  of ,faraday in   
his electrical researches1 that the     
geodesics of a  gravitational space have
 a physical existence as distinct from a
 mere  mathematical one4 ,the           
four-dimensional space we may call the  
ether1 and  so restore this bearer of   
physical forces to the position it lost 
when1  as a three-dimensional idea in   
the ,special ,relativity ,theory1 it had
 to  bear an identical relation to a    
multitude of relatively moving material 
systems4 ,the reason for our seemingly  

paradoxical title for an essay on       
,relativity will be clear when it is    
remembered that in the new theory  we   
consider those space-time properties    
which are absolute or devoid  of        
reference to any particular material    
reference-frame4 ,nevertheless1         
although the general characteristics of 
the theory are thus described1  without 
reference to experiment1 when the theory
 is to be tested it  is necessary to    
state what the four coordinates         
discussed actually  are--how they are   
determined by measurement4 ,it is our   
opinion that  much remains to be done to
 place this portion of the subject on a 
satisfactory basis4 ,for example1 in the
 derivation of the nature  of the       
gravitational space1 surrounding a      
single attracting body1  most of the    
accounts use ,cartesian coordinates as  
if the space were  ,euclidean and step  
from these to polar coordinates by the  
formul  familiar in ,euclidean geometry4
 ,but these details are1 perhaps1  like 

matters of elegance1 if we shall be     
allowed to give ,einstein's  quotation  
from ,boltzmann1 to be left to the      
tailor and the cobbler4                 
,,xvii    ,,the ,,physical ,,side ,,of  
,,relativity    ,the ,immediate         
,contacts ,between ,einstein's ,theories
 and ,current ,physics  and ,astronomy  
,,by ,,professor ,,william ,h4          
,,pickering  ,,harvard ,,college        
,,observatory1  ,,mandeville1 ,,jamaica 
,the ,theory of ,relativity will be     
treated first from the physical  side1  
leaving the three astronomical tests to 
which it has been put  to be discussed  
later4 ,there is one astronomical fact  
however that  must be mentioned in this 
connection1 and this is the discovery of
  the aberration of light by ,bradley in
 #agbf4 ,it is found that every  star in
 the heavens apparently describes a     
small annual ellipse1  whose major axis 
is #da'' in length4 ,this ,bradley      
showed to be due to a  combination of   
the velocity of the earth in its orbit1 

and the velocity  of light2 and it is so
 explained in all the elementary        
text-books on  astronomy4 ,it implies a 
stationary ether through which the earth
 is  moving4 ,the importance of this    
statement will appear presently4    ,the
 subject is usually illustrated by      
supposing a man to go out in a          
rainstorm carrying a vertical tube4 ,if 
the rain is falling vertically1  and the
 man stands still1 the sides of the tube
 will not be wet1  save by an occasional
 drop1 but if the tube is moved1 it must
 then be  inclined forward in order to  
keep it dry4 ,the angle of inclination1 
which corresponds to aberration1 will   
depend on the relative velocity  of the 
tube1 corresponding to the earth1 and   
the rain drops which  correspond to the 
waves of light4    ,if three lines are  
dropped upon a point in space1 each line
 being  perpendicular to the plane      
containing the other two1 we have what  
is known as a system of coordinates4    
,einstein's original theory  of         

relativity1 which he now designates as  
the special theory1  depends on two     
principles4 ,the first is that ,every   
law of nature  which holds good with    
respect to a coordinate system ,k must  
also  hold good for any other system    
,k'1 provided that ,k and ,k' are  in   
uniform movement of translation4 ,the   
second principle is that  ,light in a   
vacuum has a definite and constant      
velocity1 independent  of the velocity  
of its source4    ,these two sentences  
may be considered as authoritative1     
being quoted  in ,einstein's own        
words4#a ,the first of these principles 
need not  greatly surprise us4 ,the     
second is not well expressed1 because it
  is ambiguous4 ,he does not say how the
 first velocity is measured1  whether   
relatively to the ether or relatively to
 the observer4 ,in  fact this is the    
very gist of the whole matter1 as we    
shall presently  see4 ,in the case of   
sound the velocity is constant with     
regard to the  medium1 the air1 in the  

case of light it is supposed to be      
constant  with regard to the observer4  
,it reaches him with a constant         
velocity1  no matter how he moves4      
,in order to understand this statement  
clearly let us consider the  appended   
tabular diagram4 ,on a calm day imagine 
a source of sound  at ,s in line a4     
,this may be either a gun or a bell4    
,imagine an  observer #a1ajj feet       
distant1 located at ,o4 ,the velocity of
 sound in  air is #a1ajj feet per       
second4 ,this velocity we will take as  
unity1  as indicated in the third       
column1 and the velocity with which the 
sound reaches the observer is also #a1  
as shown in the fourth4 ,it will  reach 
him in a unit interval of #a second1 as 
shown in the fifth4 ,if  the bell is    
struck1 it will give its normal pitch or
 frequency1  which we will also call    
unity1 in the sixth column4    ,now     
imagine case b where the observer is on 
a train advancing toward  ,s4 ,when he  
is #a1ajj feet distant1 the gun is      

fired1 but as he is  advancing toward   
it1 he hears it at ,o in rather less    
than a second1  as shown in the fifth   
column4 ,the velocity of the sound with 
regard  to him is rather more than      
unity1 as shown in the fourth column4   
,if  the bell is sounded1 the pitch1    
that is the frequency1 is raised1       
because he receives more sound waves per
 second than before4    ,in case c the  
observer is stationary1 but the source  
of sound is  receding4 ,at a distance of
 #a1ajj feet the gun is fired1 and the  
observer hears it after an interval of  
just one second1 as in case  a4 ,the    
velocities with regard to the observer  
and through the medium  are also unity4 
,if the bell is struck the pitch is     
lowered1 since he  receives fewer sound 
waves per second1 the reverse of case b4
                          ,velocity     
,source     in       to                 
,interval   ,frequency   ,observer      
,medium   ,observer                     
,air    a       ,s     #a         #a    

#a          #a             ,o  b        
,s     #a         #a         #a-        
#a        ,o  c   ,s         #a         
#a          #a          #a-             
,o  d   ,s         #a         #a        
#a-         #a         ,o               
,ether    ,a       ,s     #a         #a 
#a         #a             ,o  ,b        
,s     #a-        #a           #a       
#a        ,o  ,c   ,s         #a        
#a           #a         #a-             
,o  ,d   ,s         #a -       #a       
#a         #a         ,o      ,in case d
 imagine the source and the observer    
#a1ajj feet apart1 and  advancing on the
 same train4 ,when the gun is fired1 the
 velocity of  the sound waves will be   
greater with regard to the observer1 and
 he  will hear the sound in less than a 
second1 as in case b4 ,when the  bell is
 struck it will have the normal pitch1  
the same as in case a4    ,we find      
therefore that for sound the velocity   
with regard to  the medium is always    
unity1 while the velocity with regard to

 the  observer1 and the interval        
elapsed1 depend only on the motion of   
the  observer himself1 and are          
independent of the motion of the source4
 ,the  frequency of the vibrations1 on  
the other hand1 depends only on the     
relative motion of the observer and the 
source1 but is independent of  their    
common motion in any direction4         
,further1 it makes no difference        
whether the source and the observer are 
moving on a train1 or whether  they are 
stationary1 and a uniform wind is       
blowing past them4    ,in the case of   
light waves we shall find a very        
different state of  affairs1 although   
the rules for frequency are the same as 
they are for  sound4 ,in case ,a we have
 the normal conditions1 where both the  
source  and observers are stationary4   
,in case ,b we have a representation of 
the ,michelson-,morley experiment as    
supplemented by that of ,majorana1#b    
where the source is stationary and the  
observer advances4 ,unlike the  case of 

sound1 the interval elapsed1 as shown by
 the experiment1 is now  the same as in 
case ,a1 and since the distance to the  
observer is less1  the velocity of light
 with respect to the ether must also be 
less than  unity4 ,since the observer is
 advancing against the light1 this will 
permit the velocity of light with regard
 to the observer to remain  unity1 in   
conformity with the second principle of 
relativity4 ,compare  with case b for   
sound4 ,as ,jeans expresses it1 ,the    
velocity of light  in all directions is 
the same1 whatever the motion of the    
observer4#c  ,that is to say it appears 
to be the same to him1 however he moves4
    ,case ,c represents ,einstein's     
statement1 as confirmed by ,majorana's  
experiment4 ,it does not differ from    
case c for sound4 ,case ,d is  more     
complex1 but accepting the statement    
above that the velocity  is constant    
with regard to the observer1 we see that
 the velocity  through the medium must  
be less1 and that the interval elapsed  

will  be constant1 as in case ,b4 ,could
 we use the brighter stars and planets  
as sources of light1 several of these   
cases could be further tested4    ,this 
brings us at once to statements that    
contradict our common  sense4 ,for      
instance1 ,jeans says no matter what the
 velocity of the  observer is1 the light
 surface1 as observed by that observer1 
is  invariably a sphere having that     
observer as center4#c ,that is to say   
the light surface1 or wave front1 is a  
contracting1 not an expanding1  sphere4 
,this1 if confirmed1 would go a long way
 toward making our  universe a          
subjective rather than an objective     
phenomenon4 ,again  imagine a flash of  
light1 such as an explosion1 to occur   
when an  observer is in a given         
position4 ,it makes no difference how   
the  observer may move while the light  
is approaching him1 whether several     
miles forward or backward1 the light    
will reach him in exactly the  same     
time1 as is shown by ,michelson's       

experiment4 ,or if two observers  are at
 the same spot when the explosion       
occurs1 and one moves forward1  and the 
other backward1 they will both see the  
explosion at exactly  the same instant4 
,this sounds ridiculous1 but not only is
 it what ,jeans says1 but it is  the    
logical interpretation of ,einstein's   
second principle1 if ,einstein  means by
 velocity1 velocity with regard to the  
observer4 ,if he means  velocity with   
regard to the medium1 then the case is  
exactly the same  as that of sound in   
air1 and ,michelson's experiment as well
 as the  ,maxwell-,lorentz theory of    
light are contradicted4 ,this theory is 
now  universally accepted1 and          
,michelson's experiment has been        
carefully  repeated by other observers1 
and fully confirmed4 ,this is the very  
heart of the relativity question4    ,if
 we state the matter objectively it     
comes to this4 ,the velocity  of light  
with regard to the ether is a variable  
quantity1 depending  merely on where the

 observer chooses to go4 ,as ,eddington 
well says1  these relations to the ether
 have no effect on the phenomena and can
  be disregarded--a step which appears  
to divest the ether of the last         
remnants of substantiality4#d    ,the   
only way of avoiding this apparent      
absurdity seems to be to  consider that 
the ether moves with the earth4         
,michelson's result would  then be fully
 explained4 ,of course this can only be 
true for a few  miles above the earth's 
surface4 ,beyond that the ether must    
either  be stationary or move with the  
sun4 ,the velocity of light with regard 
to the ether would then be a constant1  
just as the velocity of sound  is       
constant with regard to the air4 ,this  
would contradict ,einstein's  second    
principle as it is generally understood4
 ,the trouble with  this suggestion is  
that it fails to account for aberration1
 which1  as already explained1 appears  
to require that the earth should be     
moving through the ether4 ,to meet this 

emergency would involve some            
modification of the undulatory theory of
 light1 which apparently  would not be  
impossible1 but has not yet been made4  
,in #aiae ,einstein brought out an      
extension of his first principle4 ,this 
he calls the general theory of          
relativity4 ,it states that in our      
choice of coordinate systems we should  
not be limited in any way  so far as    
their state of motion is concerned4#a   
,this leads to the  three astronomical  
consequences mentioned later in this    
paper1 two  of which have been more or  
less confirmed1 and the third           
practically  contradicted as far as     
quantitative measures are concerned4#e  
,as is well known the kinetic energy of 
a moving body may be  expressed as e    
#a_/bmv#b1 but if the body is charged   
electrically1  the fraction becomes     
#a_/b7m  m'7v#b1 where m' is a quantity 
dependent on the square of the          
electrical charge4 ,that is to say1  we 
have the normal mass of the body1 and   

also what we may call its  electrical   
mass4 ,if when in this condition a      
portion of the mass is  electrical1 the 
question at once occurs to us1 why may  
not the whole  mass be electrical1 in   
other words1 a form of energy8 ,although
 this  has not been satisfactorily      
proved hitherto1 yet such is the general
  belief among physicists4 ,as ,einstein
 puts it inert mass is nothing  else    
than latent energy4#a ,the same idea is 
sometimes expressed as  the mass of     
ordinary matter is due to the           
electromagnetic energy of  its ultimate 
particles1 and electromagnetic energy   
wherever found  must possess mass1 i4e41
 inertia4#f ,if that is so1 since a ray 
of  light on the undulatory theory is a 
form of electromagnetic energy1  it too 
must possess mass4 ,since all mass with 
which we are familiar is  subject to the
 attraction of gravitation1 it seemed   
likely that a ray  of light would be    
bent out of its course in passing near  
the sun1 and  this as we have seen was  

proved to be true at the recent solar   
eclipse4    ,that portion of the mass of
 a body due to its electrical charge can
  be readily shown experimentally to    
vary with the velocity of the  body4    
,einstein has shown the same to be true 
of the normal mass1  as is illustrated  
in the advance of the perihelion of the 
orbit  of ,mercury4 ,he has also pointed
 out that gravitation1 inertia and      
centrifugal force are all closely       
related1 and obey similar laws4 ,thus   
if we rise from the earth with          
accelerated velocity1 we apparently     
increase our weight4 ,again if the      
velocity of rotation of the earth on    
its axis should be increased1 our weight
 would be diminished4 ,these  facts are 
suggestive when we come to consider the 
ultimate cause  of gravitation4         
,another fact which must be rather      
startling to the older school  of       
scientists is that momentum is no longer
 simply mv1 mass times  velocity1 but   
that the velocity of light c1 comes into

 the question1  and the formula for     
momentum now assumes the form of        
fracm vsqrt#a - fracv#b;c#b    ,for     
ordinary velocities this correction is  
extremely small1 but it  has been shown 
to be necessary1 both theoretically and 
experimentally1  when dealing with the  
high velocities with which we are now   
familiar4    ,the theory of relativity  
is so widespread in its application that
  several other theories have become    
more or less intimately combined  with  
it1 for which ,einstein is in no way    
responsible4 ,one of these is  known as 
the ,fitzgerald-,lorentz theory1 that   
all bodies are subject to  a contraction
 in the direction of their motions      
through space4 ,this was  first         
suggested in order to explain the       
,michelson-,morley experiment1  but has 
proved inadequate to do so1 particularly
 when the observer is  receding from the
 source4 ,this contraction is expressed 
by the same  factor used in the         
denominator of the revised expression   

for momentum1  given above4 ,again the  
quantity c is so enormous1 that even for
  large bodies at planetary velocities  
the contraction amounts to very  little4
 ,thus the earth moving at a speed of   
eighteen miles per second  in its orbit1
 is flattened only #a_/bjj1jjj1jjj1 or  
#b4e inches4 ,on the  other hand for    
high velocities of many thousand miles  
per second1  such as we have become     
familiar with in the case of the        
radioactive  substances1 the flattening 
is a very considerable fraction of the  
diameter of the moving body1 one-half or
 more1 and in the case of  the          
corpuscles of light1 if that theory were
 adopted1 this flattening  becomes equal
 to the diameter1 and their thickness is
 reduced to zero4    ,when we view      
,einstein's theories from the           
astronomical standpoint1  the earliest  
fact bearing on relativity that we need 
consider was  the discovery of          
aberration1 by ,bradley1 in #agbf1 as   
seen above4 ,in  #ahgb ,airy observed   

the star g ,draconis through a telescope
 filled  with water4 ,since the velocity
 of light is less in water than in  air1
 we should naturally expect to find the 
aberration appreciably  increased4 ,it  
was found1 on the other hand1 however1  
to be unaffected4    ,in #ahhg the      
results of the famous ,michelson-,morley
 experiment were  published4#g ,in this 
experiment the velocity of light was    
measured  in various directions with    
regard to the motion of the earth in its
  orbit4 ,if the ether were stationary1 
and the earth moving through it1        
different velocities should be obtained 
in different directions4 ,such  was not 
the case however1 and the experiment    
indicated that the ether  moved with the
 earth4 ,it thus flatly contradicted the
 conclusions  founded on aberration4    
,einstein's ,special ,theory of         
,relativity1 of #aije1 as we have seen1 
resolves this contradiction4 ,but as we 
shall presently see1 it is the  ,general
 ,theory1 of #aiae1 that leads to       

astronomical applications of  broad     
scope4 ,it indicates1 for instance1 that
 there is no essential  difference      
between gravitation and inertia4 ,this  
idea may be crudely  illustrated by our 
feelings of increased weight when an    
elevator  starts rapidly upwards4 ,a man
 while falling freely in space ceases   
to feel the pull of gravitation4    ,but
 we must not as yet conceive of the     
theory of relativity as a  universally  
accepted and unquestioned truth of      
science4 ,eddington is  its leading     
,english exponent1 and he is supported  
by such men as ,jeans1  ,larmor1 and    
,jeffreys4 ,on the other hand1 the      
theory has been severely  criticised by 
,lodge1 ,fowler1 ,silberstein1 and      
,sampson4 ,few ,american  scientists    
have expressed any opinions in print on 
the subject1  and the recent eclipse    
observations1 to which we shall refer   
later1  are to be repeated with more    
suitable instruments for verification   
in #aibb1 in the hope of obtaining more 

accurate and accordant results4#h    ,an
 appurtenance of the ,einstein theories 
which bears much the  same relation to  
them as does the ,lorentz-,fitzgerald   
contraction1  mentioned above1 is the   
idea1 first clearly stated by           
,minkowski1  that time is a kind of     
space--a fourth dimension4 ,this the    
reader  will doubtless find to be the   
most difficult portion of the theory  to
 picture in his own mind4 ,it is        
entirely unsupported by experiment  or  
observation1 necessarily so1 and is     
based wholly on mathematical  and       
philosophical conceptions4 ,our         
distinction between space and time      
seems to be that the direction in which 
we progress without effort  is time2 the
 other directions1 in which we have to  
make an exertion  to move ourselves1 or 
in which we are carried1 are space4 ,how
 many  dimensions empty space may have1 
we really have no means of knowing1     
because we can neither see nor feel it4 
,matter we know has three1  length1     

breadth1 and thickness1 also that it    
lies remote from us in  three           
corresponding directions4 ,these facts  
may have given us the  erroneous        
impression that space too has only three
 dimensions4 ,now  it is claimed that   
time is a fourth1 and that there are    
also others4    ,in order to illustrate 
this1 ,eddington asks us to imagine a   
movie film  taken of a man or of any    
moving object4 ,let the separate        
pictures be  cut apart and piled on one 
another4 ,this would form a sort of     
pictorial  history of the individual for
 a brief interval in his life1 in the   
form  of a cube4 ,if we attempt to pick 
it up1 it falls apart1 thus clearly     
showing the difference between time and 
space4 ,but suppose it now  all glued   
together in one solid cube1 so that it  
is no easier to cut  a section in one   
direction than in another4 ,that is     
,minkowski's idea  of space and time1   
and further1 that the direction in which
 we should  cut it depends merely on the

 velocity with which we are moving      
through  space4 ,i should cut it        
parallel to the films1 but a man on a   
rapidly  moving star1 in order to       
separate it into space and time1 would  
cut  it in an inclined direction4 ,that 
is a thing which may be true1 but it  is
 one which we believe no mortal man can 
clearly picture to himself4    ,on the  
other hand ,turner has recently made a  
very interesting  point1#i namely1 that 
the fourth dimension as actually treated
  by the mathematicians is not time     
itself1 but time multiplied  by a       
constant--the velocity of light4#aj     
,without affecting the  astronomical    
proofs of relativity at all1 this       
simplifies our  conceptions enormously4 
,in ordinary everyday life time and     
space  cannot be identical1 any more    
than a yard can be identical with a     
quart4 ,on what is known to physicists  
as the centimeter-gram-second  system1  
distance is represented by l1 mass by m1
 and time by  t4 ,velocity is then      

distance divided by time1 l_/t1 or as we
 say  in ,english units1 so many feet   
per second1 and the fourth dimension    
may be expressed as time multiplied by  
velocity1 t times l_/t   l4 ,that is to 
say1 it is simply distance1 just like   
the other three  dimensions4 ,to say    
that time is the fourth dimension from  
this point  of view1 appears to us just 
as ridiculous as it would be to attempt 
to measure the velocity of a train in   
quarts4 ,it is quite correct1  however1 
although unusual1 to speak of a given   
train as moving at a  speed of #aj      
quarts per square inch per second1      
l#c_/l#b t  l_/t4 ,this  would be       
equivalent to a velocity of #cc miles   
per hour4    ,if ,i wish to give a      
complete dimensional description of     
myself in my  four dimensions1 ,i must  
give my length1 my breadth1 and my      
thickness1  ever since ,i came into     
being1 and also the course ,i have      
traversed  through space since that     
time4 ,this latter distance will be     

expressed  in terms of a unit whose     
length is #ahf1jjj miles1 the distance  
traversed by light in one second4 ,the  
distance which ,i travel through  space 
annually is enormous1 and very complex  
as to direction4 ,it  involves not      
merely my own motions as ,i cross the   
room1 or take a  train or steamer1 but  
also those due to the rotation of the   
earth on  its axis1 its revolution round
 the sun1 and the motion of the latter  
through the heavens4 ,in general ,i     
travel1 or in other words increase  my  
length in the fourth dimension1 by over 
#d1jjj units a year4 ,the  fourth       
dimension accordingly1 if this view is  
accepted1 is simply a  distance like the
 other three1 and perfectly easy to     
understand4    ,we now come to the three
 actual tests by which the theory has   
been  tried4 ,the planets as is well    
known revolve about the sun in ellipses1
  with the sun in one of the foci4 ,that
 is to say1 the sun is not in  the      
center1 but a little on one side of it4 

,the end of the ellipse  where the      
planet comes nearest to the sun is      
called the perihelion1  and here the    
planet is moving most rapidly4 ,the     
other end is called  the aphelion1 and  
here the motion is slowest4 ,according  
to ,newton's  theory of gravitation1 if 
a spherical sun possesses a single      
planet  or companion1 its orbit will be 
permanently fixed in space unless       
perturbed by some other body4 ,if a     
second planet exist1 it will  cause the 
perihelion of the first slowly to       
advance4 ,according to  ,einstein the   
mass of a planet depends in part on its 
velocity4 ,it  will therefore be less at
 aphelion where it is moving slowly than
 at  perihelion where it is moving      
rapidly1 consequently in addition to    
the ,newtonian attraction we have       
another one which increases as we       
approach the sun4 ,the effect of this   
will be to cause the perihelion  of the 
orbit to advance1 whether there is a    
second planet or not4    ,among the     

larger planets ,mercury has the most    
eccentric orbit1 and it  also moves most
 rapidly1 so that it is particularly    
well adapted to  test the relativity    
theory4 ,the observed advance of its    
perihelion is  #egd'' per century1      
instead of the theoretical figure       
#ecb''1 due to the  other planets--a    
difference of #db''4aa ,this has long   
been a puzzling  discrepancy between    
observation and the law of gravitation4 
,prior to  ,einstein1 attempts were made
 to eliminate it by assuming a certain  
oblateness of the solar disk4 ,if the   
equatorial diameter exceeded the  polar 
by only #j''4e the whole advance would  
be accounted for1 but not  only has this
 ellipticity failed of detection1 but if
 it existed1  it should produce a very  
noticeable and inadmissible change in   
the  inclination of ,mercury's orbit1   
amounting to about #c'' per century1  as
 has been demonstrated by both ,herzer  
and ,newcomb4#ab    ,einstein from      
computations alone1 without introducing 

any new  constants or hypotheses        
whatever1 showed1 if the theory of      
relativity  be accepted1 that the sun   
should produce an acceleration of #dc'' 
per century1 thus entirely accounting   
for the observed discrepancy1  far      
within the limits of accuracy of the    
observations4 ,the only other  planet   
whose orbit has a large eccentricity1   
and that is suitable  for investigation1
 is the planet ,mars4 ,here the         
discrepancy between  observation and    
theory is very slight1 only #d''1 and a 
portion of  that may be due to the      
attraction of the asteroids4 ,this      
deviation  is so slight that it may well
 be due entirely to accidental errors   
of observation1 but however that may be1
 ,einstein's theory reduces  it to      
#b''4g4    ,this all seems very         
satisfactory and complete1 but the      
trouble  with it is that the coincidence
 for ,mercury is rather too good4 ,it   
is based on the assumption that the sun 
is a perfect sphere1 and  that the      

density of its surface is uniform from  
the equator to the  poles4 ,this would  
doubtless be true if the sun did not    
revolve on  its axis4 ,in point of fact 
it does revolve1 in a period in general 
of  about #bf days4 ,consequently an    
object on its equator must experience  a
 certain amount of centrifugal force4   
,therefore if its surface were  of      
uniform density the shape of the sun    
would be an oblate spheroid4    ,it can 
be readily shown that the theoretical   
excess of the equatorial  over the polar
 diameter1 due to the centrifugal force1
 should amount to  only #j''4jd1 an     
amount which could hardly be detected by
 observation1  and might readily be     
concealed by a slight excess of         
equatorial over  polar density4 ,any    
reasonable excess of density at the     
center would  diminish this result but  
slightly4 ,the molecular weight of the  
central  material#ac is probably about  
#b4 ,this computed equatorial excess is 
one-twelfth of the amount necessary to  

cause the observed advance1  and should 
therefore cause an advance of the       
perihelion of about #c''4e  per century1
 reducing the difference between the    
observed advance  and that caused by    
gravitation to #ch''4e4 ,according to   
,einstein the  advance due to relativity
 should be1 as we saw1 #dc''1 a         
discrepancy  of #d''4e per century1 or  
#aj per cent4 ,jeffreys has remarked    
that  any discrepancy such as #aj''     
would be fatal to a theory such as      
,einstein's1 which contains no arbitrary
 constituent capable of  adjustment to  
suit empirical facts4#ad ,it must be    
pointed out here  however1 that so far  
as known1 this small correction to the  
motion  of ,mercury's perihelion has not
 previously been suggested1 so that     
there has been no opportunity hitherto  
for its criticism by others4    ,it was 
due largely to the success with ,mercury
 that it was decided to  put the        
relativity theory to another test4      
,according to the ,newtonian  theory1 as

 stated by ,newton himself1 corpuscles  
as well as planets  have mass1 and must 
therefore be attracted by the sun4      
,according to  ,einstein1 owing to their
 high velocity1 this attraction must be 
twice  as great as it would be according
 to the theory of gravitation4 ,if the  
ray of light proceeding from a star were
 to pass nearly tangent to the  sun's   
limb it should be deflected #j''4hg     
according to ,newton4 ,according  to the
 theory of relativity it should be      
deflected #a''4ge4 ,stars  of course    
cannot usually be observed near the sun4
 ,it is therefore  necessary to take    
advantage of a total solar eclipse1 when
 the sun is  completely hidden by the   
moon1 in order to secure these          
observations4    ,two expeditions1 one  
to ,africa1 and one to ,south ,america1 
observed  successfully the total eclipse
 of ,may #bi1 #aiai4 ,the former was    
located on the ,island of ,principe in  
the ,gulf of ,guinea4 ,the latter  was  
located at ,sobral1 ,brazil4 ,their     

equipment and results are shown  in the 
following table1 where the successive   
columns give the location1  the aperture
 in inches of the telescopes employed1  
their focus in feet1  the number of     
plates secured1 the number of stars     
measured1 their  mean deduced deflection
 from their true positions by the       
attraction  of the sun1 and the         
deviations from the theoretical         
results4#ae ,in  the first and last line
 of the table shown herewith1 this      
,location   ,aperture   ,focus   ,plates
   ,stars   ,defl4    ,dev4             
,principe         #ac      #aa        #b
       #e   #a''4fj   -#j''4ae          
,sobral           #ac      #aa       #ai
      #ab   #j 4#ic    7#j 4#jf7        
11              #d      #ai        #h   
#g   #a 4#ih    #j 4#bc      deviation  
is taken from ,einstein's computed value
 of #a''4ge4 ,in the  second line the   
difference shown is from the value      
required by the  ,newtonian theory1     
#j''4hg4 ,the results obtained with this

 telescope  were rejected however1      
although they were much the most        
numerous1  because it was found that for
 some reason1 supposed to be the heating
  of the mirror by the sun before the   
eclipse1 the star images were  slightly 
out of focus1 and were therefore        
considered unreliable4 ,the  results    
with the two other telescopes were not  
very accordant1 but the  #d-;inch had   
the longer focus1 secured the greater   
number of plates1  and showed the       
greater number of stars4 ,the results   
obtained with  it therefore appear to   
have been the more reliable4 ,they      
differ from  ,einstein's prediction by  
#ac per cent4 ,in future expeditions to 
test  this question1 the mirror in front
 of the telescope will be eliminated4   
,we now come to the final test which has
 been applied to ,einstein's  theory4   
,einstein showed that in the intense    
gravitational field of the  sun1 the    
theory of relativity required that all  
of the spectrum lines  should be shifted

 slightly toward the red end4 ,the shift
 however is  exceedingly small1 and can 
only be detected and measured with the  
most powerful modern instruments4       
,moreover only certain lines can  be    
used1 because owing to varying pressure 
in the solar atmosphere1  which affects 
many lines1 as well as to rapid motion  
in the line of  sight1 which may affect 
all of them1 still larger displacements 
are  liable to occur4    ,according to  
the theory of relativity the            
displacement of the lines  should be    
#j4jjhj ,a4 ,st4 ,john at ,mt4 ,wilson  
found a displacement  for the cyanogen  
lines of only #j4jjah ,a4#af ,evershed  
at ,kodaikanal  found #j4jjfj at the    
north pole of the sun1 and #j4jjhj at   
the south  pole4 ,these latter values   
however were only for the stronger      
lines4 ,the  weaker lines give much     
smaller shifts1 as do those of calcium  
and  magnesium4#ag ,according to        
,einstein all lines should give nearly  
the  same shift1 an amount proportional 

to the wave length4 ,it therefore       
appears that we must conclude by saying 
that ,einstein's theory of  relativity  
has been partially1 but not completely1 
verified4        ,the reference numbers 
in the above text have nothing to do    
with the  numbers used in other parts of
 this volume to acknowledge the work  of
 the various contestants2 they refer to 
,dr4 ,pickering's sources1  as follows3 
#a ,journ4 ,brit4 ,astron4 ,assoc41     
#aiai1 #cj1 #gf4    #b ,comptes ,rendus1
 #afe1 #dbd1 and #afg1 #ga4    #c       
,monthly ,notices ,r4 ,a4 ,s41 #aiai1   
#hj1 #ajd4    #d ,monthly ,notices ,r4  
,a4 ,s41 #aiag1 #gg1 #cgi4    #e        
,astro-,physical ,journal1 #aiag1 #df1  
#bdi4 ,journ4 ,brit4 ,astron4 ,assoc41  
#aibj1 #cj1 #bgf4    #f ,monthly        
,notices1 ,r4 ,a4 ,s41 #aiag1 #gg1 #cgg4
    #g ,amer4 ,journ4 ,sci41 #cd1 #ccc4 
#h ,monthly ,notices1 ,r4 ,a4 ,s41      
#aibj1 #hj1 #fbh4    #i ,the            
,observatory #aibj1 ,april4 ,from an    
,oxford ,note ,book4    #aj ,monthly    

,notices1 ,r4 ,a4 ,s41 #aiag4 #gh1 #c   
,de ,sitter1 #aiai1 #hj1 #aba1  ,jeans1 
#hj1 #adf ,jeffreys4    #aa ,gravitation
 and the ,principle of ,relativity1     
,eddington4 ,royal  ,institution of     
,great ,britain1 #aiah4    #ab ,journ4  
,brit4 ,astron4 ,assoc41 #aibj1 #cj1    
#abe4    #ac ,the ,interior of a ,star1 
,eddington4 ,scientia1 #aiah1 #bc1 #ae4 
#ad ,monthly ,notices1 ,r4 ,a4 ,s41     
#aiai1 #hj1 #ach4    #ae ,monthly       
,notices4 ,r4 ,a4 ,s41 #aibj1 #hj1      
#dae4 ,journ4 ,brit4 ,astron4 ,assoc41  
#aiai1 #cj1 #df4    #af ,astro-,physical
 ,journ41 #aiag1 #df1 #bdi4    #ag      
,journ4 ,brit4 ,astron4 ,assoc41 #aibj1 
#cj1 #bgf4                    ,,xviii   
,,the ,,practical ,,significance ,,of   
,,relativity    ,the ,best ,discussion  
of the ,special ,theory ,among ,all the 
,competing  ,essays    ,,by ,,professor 
,,henry ,,norris ,,russell1 ,,princeton 
,,university      ,can a small child    
catch a baseball moving sixty miles an  
hour without  getting hurt8 ,we should  

probably answer ,no--but suppose that   
the boy  and his father were sitting    
side by side in an express train1 and   
the  ball was tossed lightly from one to
 the other4 ,then there would be  no    
trouble about it1 whether the train was 
standing still1 or going  at full speed4
 ,only the relative motion of ball and  
boy would count4    ,this every-day     
experience is a good illustration of the
 much discussed  ,principle of          
,relativity1 in its simplest form4 ,if  
there were no  jolting1 the motion of   
the train1 straight ahead at a uniform  
speed1  would have no effect at all upon
 the relative motions of objects  inside
 it1 nor on the forces required to      
produce or change these  motions4       
,indeed1 the motion of the earth in its 
orbit1 which is free  from all jar1 but 
a thousand times faster1 does not       
influence even  the most delicate       
apparatus4 ,we are quite unconscious of 
it1 and  would not know that the earth  
was moving1 if we could not see other   

bodies outside it4 ,this sort of        
relativity has been recognized for  more
 than two centuries and lies at the     
bottom of all our ordinary  dynamical   
reasoning1 upon which both science and  
engineering are based4    ,but there are
 other things in nature besides moving  
bodies1--above  all1 light1 which is    
intimately related to electricity and   
magnetism1  and can travel through empty
 space1 between the stars4 ,it moves at 
the enormous speed of #ahf1jjj miles per
 second1 and behaves exactly  like a    
series of vibrations or waves4 ,we      
naturally think of it as  travelling    
through some medium1 and call this      
thing1 which carries  the light1 the    
ether4    ,can we tell whether we are   
moving through this ether1 even though  
all  parts of our apparatus move        
together1 and at the same rate8 ,suppose
  that we have two mirrors1 ,m and ,n1  
at equal distances1 d1 from a  point ,o1
 but in directions at right angles to   
one another1 and send  out a flash of   

light from ,o4 ,if everything is at     
rest1 the reflected  flashes will       
evidently come back to ,o at the same   
instant1 and the  elapsed time will be  
#b;d_/c seconds if c is the velocity of 
light4    ,but suppose that ,o1 ,m1 and 
,n are fastened to a rigid frame work1  
and all moving in the direction ,o ,m1  
with velocity ,v4 ,the light  which goes
 from ,o toward ,m1 at the speed c1 will
 overtake it  with the difference of    
their speeds1 c-v1 taking d_/7c-v7      
seconds to  reach ,m4 ,on the way back1 
,o will be advancing to meet it1 and the
  return trip will occupy d_/7cv7       
seconds4 ,the elapsed time for the      
round trip comes out #b;cd_/7c#b-v#b7   
seconds1 which is longer than  when the 
system was at rest--the loss of time in 
the stern chase  exceeding the saving on
 the return4    ,the light which is     
reflected from ,n has a different       
history4 ,when  it starts1 ,o and ,n    
have certain positions in the ether1    
,o#a  and ,n#a4 ,by the time it reaches 

the mirror1 this is at ,n#b1 and  ,o is 
at ,o#b1 and when it returns1 it finds  
,o at ,o#c4 ,the  distances for the     
outward and inward journeys are now     
equal1 but  7as is obvious from the     
figure71 each of them is greater than d1
  or ,o#b,n#b1 and the time for the     
round trip will be correspondingly      
increased4 ,a simple calculation shows  
that it is #b;d_/sqrtc#b-v#b4    ,the   
increase above the time required when   
the system was at rest  is less in this 
case than the preceding4 ,hence1 if the 
apparatus  is moving through the ether1 
the flashes reflected from ,m and ,n    
will not return at the same instant4    
,for such velocities as are             
attainable--even the #ah miles per      
second of  the earth in its orbit--the  
difference is less than a               
hundred-millionth  of the elapsed time4 
,nevertheless1 ,michelson and ,morley   
tried to  detect it in their famous     
experiment4    ,a beam of light was     
allowed to fall obliquely upon a clear  

glass  mirror 7placed at ,o in the      
diagram7 which reflected part of it     
toward the mirror1 ,m1 and let the rest 
pass through to the mirror  ,n4 ,by     
reuniting the beams after their round   
trips1 it was possible  to tell whether 
one had gained upon the other by even a 
small fraction  of the time of vibration
 of a single light wave4 ,the apparatus 
was  so sensitive that the predicted    
difference1 though amounting to less    
than a millionth part of a billionth of 
a second1 could easily have  been       
measured2 but they actually found no    
difference at all--though  the earth is 
certainly in motion4    ,other optical  
experiments1 more intricate1 and even   
more delicate1  were attempted1 with the
 same object of detecting the motion of 
the  earth through the ether2 and they  
all failed4          ,,the ,,special    
,,theory ,,and ,,its ,,surprising       
,,consequences    ,it was upon these    
facts that ,einstein based his original1
 or special  theory of ,relativity4 ,he 

assumed boldly that the universe is so  
constituted that uniform straight-ahead 
motion of an observer and all  his      
apparatus will not produce any          
difference whatever in the result  of   
any physical process or experiment of   
any kind4 ,granting this1  it follows   
that if all objects in the visible      
universe were moving  uniformly together
 in any direction1 no matter how fast1  
we could  not find this out at all4 ,we 
cannot determine whether the universe1  
as a whole1 is at rest or in motion1 and
 may as well make one guess  as another4
 ,only the relative motions of its parts
 can be detected  or studied4    ,this  
seems simple and easy enough to         
understand4 ,but the consequences  which
 follow from it are extraordinary1 and  
at first acquaintance  seem almost      
absurd4    ,in the first place1 if an   
observer measures the velocity of light1
  he must always get the same result1 no
 matter how fast he and his  apparatus  
are moving1 or in what direction 7so    

long as the motion is  uniform and      
rectilinear74 ,this sounds harmless2 but
 let us go back to  the                 
,michelson-,morley experiment where the 
light came back in exactly  the same    
time from the two mirrors4 ,if the      
observer supposes himself  to be at     
rest1 he will say that the distances ,o 
,m and ,o ,n were  equal4 ,but if he    
fancies that the whole universe is      
moving in the  direction ,o ,m1 he will 
conclude that ,m is nearer to ,o than ,n
  is--for if they were equidistant1 the 
round-trip would take longer in  the    
first case1 as we have proved4 ,if once 
more he fancies that the  universe is   
moving in the direction ,o ,n1 he will  
conclude that ,n is  nearer to ,o than  
,m is4 ,his answer to the question which
 of the  two distances1 ,o ,m or ,o ,n1 
is the greater will therefore depend  on
 his assumption about the motion of the 
universe as a whole4    ,similar        
complications arise in the measurement  
of time4 ,suppose that  we have two     

observers1 ,a and ,b1 provided with     
clocks which run  with perfect          
uniformity1 and mirrors to reflect light
 signals to one  another4 ,at noon      
exactly by his clock1 ,a sends a flash  
of light  towards ,b4 ,b sees it come in
 at #ab3ja by his clock4 ,the flash     
reflected from ,b's mirror reaches ,a at
 #ab3jb by ,a's clock4 ,they            
communicate these observations to one   
another4    ,if ,a and ,b regard        
themselves as being at rest1 they will  
agree that  the light took as long to go
 out as it did to come back1 and        
therefore  that it reached ,b at just   
#ab3ja by ,a's clock1 and that the two  
clocks are synchronized4 ,but they may1 
if they please1 suppose that  they 7and 
the whole universe7 are moving in the   
direction from ,a  towards ,b1 with half
 the speed of light4 ,they will then say
 that  the light had a stern chase to   
reach ,b1 and took three times as  long 
to go out as to come back4 ,this means  
that it got to ,b at #a  #a_/b minutes  

past noon by ,a's clock1 and that ,b's  
clock is slow  compared with ,a's4 ,if  
they should assume that they were moving
 with  the same speed in the opposite   
direction1 they would conclude that     
,b's clock is half a minute fast4       
,hence their answer to the question     
whether two events at different  places 
happen at the same time1 or at different
 times1 will depend  on their assumption
 about the motion of the universe as a  
whole4    ,once more1 let us suppose    
that ,a and ,b1 with their clocks and   
mirrors1 are in relative motion1 with   
half the speed of light1 and pass  one  
another at noon by both clocks4 ,at     
#ab3jb by ,a's clock1 he sends a  flash 
of light1 which reaches ,b at #ab3jd by 
his clock1 is reflected1  and gets back 
to ,a's clock at #ab3jf4 ,they signal   
these results to  each other1 and sit   
down to work them out4 ,a thinks that he
 is at  rest1 and ,b moving4 ,he        
therefore concludes that the light had  
the  same distance to go out as to      

return to him and took two seconds each 
way1 reaching ,b at #ab3jd by ,a's      
clock1 and that the two clocks1  which  
agreed then1 as well as at noon1 are    
running at the same rate4    ,b1 on the 
contrary1 thinks that he is at rest and 
,a in motion4 ,he  then concludes that  
,a was much nearer when he sent out the 
flash  than when he got it back1 and    
that the light had three times as far to
  travel on the return journey4 ,this   
means that it was #ab3jc by ,a's  clock 
at the instant when the light reached ,b
 and ,b's clock read  #ab3jd4 ,hence    
,a's clock is running slow1 compared    
with ,b's4    ,hence the answer to the  
question whether two intervals of time1 
measured by observers who are in motion 
relative to one another1 are  of the    
same or of different durations1 depends 
upon their assumptions  about the motion
 of the universe as a whole4    ,now we 
must remember that one assumption about 
the motion of the  universe as a whole  
is exactly as good--or bad--as another4 

,no  possible experiment can distinguish
 between them4 ,hence on the  ,principle
 of ,relativity1 we have left no        
absolute measurement of  time or space4 
,whether two distances in different     
directions are to  be called equal or   
not--whether two events in different    
places are  to be called simultaneous or
 not--depends on our arbitrary choice of
  such an assumption1 or frame of       
reference4 ,all the various schemes  of 
measurement corresponding to these      
assumptions will1 when applied  to any  
imaginable experiment1 predict exactly  
the same phenomena4 ,but1  in certain   
important cases1 these predictions      
differ from those of the  old familiar  
theory1 and1 every time that such       
experiments have been  tried1 the result
 has agreed with the new theory1 and not
 with the old4    ,we are therefore     
driven to accept the theory of          
relativity1 strange  as it is1 as being 
more nearly true to nature than our     
older  ideas4 ,fortunately1 the         

difference between the results of the   
two  become important only when we      
assume that the whole visible universe  
is  moving together much faster than any
 of its parts are moving relatively  to 
one another4 ,unless we make such an    
unwarranted assumption1 the  differences
 are so small that it takes the most    
ingenious and precise  experiments to   
reveal them4          ,,the             
,,generalization    ,not content with   
all this1 ,einstein proceeded1 a few    
years ago1 to  develop a general theory 
of relativity1 which includes the       
effects  of gravitation4    ,to make    
this idea clear1 let us imagine two     
observers1 each1 with  his measuring    
instruments1 in a large and perfectly   
impervious box1  which forms his closed 
system4    ,the first observer1 with his
 box and its contents1 alone in space1  
is entirely at rest4    ,the second     
observer1 with his box and its contents1
 is1 it may be  imagined1 near the earth
 or the sun or some star1 and falling   

freely  under the influence of its      
gravitation4    ,this second box and its
 contents1 including the observer1 will 
then  fall under the gravitational      
force1 that is1 get up an               
ever-increasing  speed1 but at exactly  
the same rate1 so that there will be no 
tendency  for their relative positions  
to be altered4    ,according to         
,newton's principles1 this will make not
 the slightest  difference in the       
motions of the physical objects         
comprising the system  or their         
attractions on one another1 so that no  
dynamical experiment  can distinguish   
between the condition of the freely     
falling observer  in the second box and 
the observer at rest in the first4      
,but once more the question arises3     
,what could be done by an optical       
experiment8    ,einstein assumed that   
the principle of relativity still       
applied in  this case1 so that it would 
be impossible to distinguish between the
  conditions of the observers in the two

 boxes by any optical experiment4    ,it
 can easily be seen that it follows from
 this new generalized  relativity that  
light cannot travel in a straight line  
in a  gravitational field4    ,imagine  
that the first observer sets up three   
slits1 all in a straight  line4 ,a ray  
of light which passes through the first 
and second will  obviously pass exactly 
through the third4    ,suppose the      
observer in the freely falling system   
attempts the same  experiment1 having   
his slits ,p1 ,q1 ,r1 equally spaced1   
and placing  them at right angles to the
 direction in which he is falling4 ,when
  the light passes through ,p1 the slits
 will be in certain position  ,p#a ,q#a 
,r#a 7,figure74 ,by the time it reaches 
,q1 they will have  fallen to a lower   
level1 ,p#b ,q#b ,r#b1 and when it      
reaches ,r1  they will be still lower1  
,p#c ,q#c ,r#c4 ,the times which the    
light  takes to move from ,p to ,q and  
,q to ,r will be the same3 but1  since  
the system is falling ever faster and   

faster the distance ,r#b  ,r#c will be  
greater than ,q#a ,q#b4 ,hence1 if the  
light which has  passed through ,p and  
,q moves in a straight line1 it will    
strike  above ,r1 as is illustrated by  
the straight line in the figure4 ,but1  
on ,einstein's assumption1 the light    
must go through the third slit1  as it  
would do in the system at rest1 and must
 therefore move in a  curved line1 like 
the curved line in the figure1 and bend 
downward  in the direction of the       
gravitational force4          ,,the     
,,tests    ,calculation shows that the  
deviation of light by the moon or       
planets  would be too small to detect4  
,but for a ray which had passed near    
the sun1 the deflection comes out       
#a4g''1 which the modern astronomer     
regards as a large quantity1 easy to    
measure4 ,observations to test  this can
 be made only at a total eclipse1 when  
we can photograph stars  near the sun1  
on a nearly dark sky4 ,a very fine      
chance came in ,may1  #aiai1 and two    

,english expeditions were sent to       
,brazil and the ,african  coast4 ,these 
photographs were measured with extreme  
care1 and they  show that the stars     
actually appear to be shifted1 in almost
 exactly  the way predicted by          
,einstein's theory4    ,another         
consequence of general relativity is    
that ,newton's law of  gravitation needs
 a minute correction4 ,this is so small 
that there is  but a single case in     
which it can be tested4 ,on ,newton's   
theory1 the  line joining the sun to the
 nearest point upon a planet's orbit    
7its  perihelion7 should remain fixed in
 direction1 7barring certain effects  of
 the attraction of the other planets1   
which can be allowed for74 ,on          
,einstein's theory it should move slowly
 forward4 ,it has been known for  years 
that the perihelion of ,mercury was     
actually moving forward1 and  all       
explanations had failed4 ,but           
,einstein's theory not only predicts    
the direction of the motion1 but exactly

 the observed amount4    ,einstein also 
predicts that the lines of any element  
in the solar  spectrum should be        
slightly shifted towards the red1 as    
compared  with those produced in our    
laboratories4 ,different observers have 
investigated this1 and so far they      
disagree4 ,the trouble is that  there   
are several other influences which may  
shift the lines1 such  as pressure in   
the sun's atmosphere1 motion of currents
 on the sun's  surface1 etc41 and it is 
very hard to disentangle this ,gordian  
knot4 ,at  present1 the results of these
 observations can neither be counted    
for or against the theory1 while those  
in the other two cases are  decisively  
favorable4    ,the mathematical         
expression of this general relativity is
 intricate  and difficult4              
,mathematicians--who are used to        
conceptions which  are unfamiliar1 if   
not incomprehensible1 to most of        
us--find that  these expressions may be 
described 7to the trained student7 in   

terms  of space of four dimensions and  
of the non-,euclidean geometry4 ,we     
therefore hear such phrases as time as a
 sort of fourth dimension1  curvature of
 space and others4 ,but these are simply
 attempts--not  altogether              
successful--to put mathematical         
relationships into ordinary  language1  
instead of algebraic equations4    ,more
 important to the general reader are the
 physical bearings of  the new theory1  
and these are far easier to understand4 
,various assumptions which we may make  
about the motion of the universe  as a  
whole1 though they do not influence the 
observed facts of nature1  will lead us 
to different ways of interpreting our   
observations as  measurements of space  
and time4    ,theoretically1 one of     
these assumptions is as good as any     
other4 ,hence  we no longer believe in  
absolute space and time4 ,this is of    
great  interest philosophically4        
,practically1 it is unimportant1 for1   
unless our choice of an assumption is   

very wild1 our conclusions and          
measurements will agree substantially   
with those which are already  familiar4 
,finally1 the general relativity shows  
that gravitation and  electro-magnetic  
phenomena--7including light7 do not form
 two  independent sides of nature1 as we
 once supposed1 but influence one       
another 7though slightly7 and are parts 
of one greater whole4                   
,,xix    ,,einstein',s ,,theory ,,of    
,,relativity    ,a ,simple ,explanation 
of ,his ,postulates and ,their          
,consequences    ,,by ,t4 ,,royds1      
,,kodaikanal ,,observatory1 ,,india     
,einstein's theory of relativity seeks  
to represent to us the world as  it     
really is instead of the world of       
appearances which may be deceiving  us4 
,when ,i was in town last week to buy #e
 yards of calico ,i watched  the draper 
very carefully as he measured the cloth 
to make sure ,i was  not cheated4 ,yet  
experiment can demonstrate1 and         
,einstein's theory  can explain1 that   

the draper's yardstick became longer or 
shorter  according to the direction in  
which it was held4 ,the length of the   
yardstick did not appear to me to change
 simply because everything else  in the 
same direction1 the store1 the draper1  
the cloth1 the retina  of my eye1       
changed length in the same ratio4       
,einstein's theory points  out not only 
this1 but every case where appearances  
are deceptive1  and tries to show us the
 world of reality4    ,einstein's theory
 is based on the principle of relativity
 and  before we try to follow his       
reasoning we must spend a little time   
in understanding what he means by       
relativity and in grasping how  the idea
 arises4 ,suppose ,i wish to define my  
motion as ,i travel  along in an        
automobile4 ,i may be moving at the rate
 of #be miles an  hour relative to      
objects fixed on the roadside1 but      
relative to a  fellow-passenger ,i am   
not moving at all2 relative to the sun  
,i am  moving with a speed of #ah #a_/b 

miles per second in an elliptical orbit1
  and again relative to the stars ,i am 
moving in the direction of the  star    
,vega at a speed of #ab miles per       
second4 ,thus motion can only be        
defined relative to some object or point
 of reference4 ,now this is  not        
satisfactory to the exact scientist4    
,scientists are not content  with       
knowing1 for example1 that the          
temperature of boiling water is  #ajj   
,c4 relative to the temperature of      
freezing2 they have set out  to         
determine absolute temperatures and have
 found that water boils at  #cgc ,c4    
above absolute zero4 ,why should ,i not1
 therefore1 determine  the absolute     
motion of the automobile1 not its motion
 relative to  the road1 earth1 sun or   
stars1 but relative to absolute rest8   
,michelson and ,morley set out in their 
famous experiment to measure the        
absolute velocity of their laboratory1  
which was1 of course1 fixed  on the     
earth4 ,the experiment consisted of     

timing two rays of light  over two equal
 tracks at right angles to each other4  
,when one track  was situated in the    
direction of the earth's motion they    
expected to  get the same result as when
 two scullers of equal prowess are      
racing  in a river1 one up and down the 
stream and the other across and back2   
the winner will be the sculler rowing   
across the stream1 as working  out an   
example will convince4 ,even if the     
earth had been stationary  at the time  
of one experiment1 the earth's motion   
round the sun would  have been reversed 
#f months later and would then have     
given double the  effect4 ,they found1  
however1 that the two rays of light     
arrived always  an exact dead heat4 ,all
 experimenters who have tried since have
 arrived  at the same result and found  
it impossible to detect absolute motion4
    ,the principle of relativity has its
 foundation in fact on these  failures  
to detect absolute motion4 ,this        
principle states that the  only motion  

we can ever know about is relative      
motion4 ,if we devise  an experiment    
which ought to reveal absolute motion1  
nature will  enter into a conspiracy to 
defeat us4 ,in the ,michelson and       
,morley  experiment the conspiracy was  
that the track in the direction of the  
earth's absolute motion should contract 
its length by just so much  as would    
allow the ray of light along it to      
arrive up to time4    ,we see1          
therefore1 that according to the        
principle of relativity  motion must    
always remain a relative term1 in much  
the same way as  vertical and           
horizontal1 right and left1 are relative
 terms having  only meaning when        
referred to some observer4 ,we do not   
expect to  find an absolute vertical and
 are wise enough not to attempt it2  in 
seeking to find absolute motion         
physicists were not so wise and  only   
found themselves baffled4    ,the       
principle that all motion is relative   
now requires to be worked  out to all   

its consequences1 as has been done by   
,einstein1 and we  have his theory of   
relativity4 ,einstein conceives a world 
of four  dimensions built up of the     
three dimensions of space1 namely up and
  down1 backwards and forwards1 right   
and left1 with time as the fourth       
dimension4 ,this is an unusual          
conception to most of us1 so let us     
simplify it into something which we can 
more easily picture but which  will     
still allow us to grasp ,einstein's     
ideas4 ,let us confine ourselves  for   
the present to events which happen on   
this sheet of paper1 i4e41  to space of 
two dimensions only and take time as our
 third dimension  at right angles to the
 plane of the paper4 ,we have thus built
 up a  three dimensional world of       
space-time which is every bit as useful 
to us as a four dimensional             
representation so long as we only need  
study objects moving over the sheet of  
paper4    ,suppose a fly is crawling    
over this sheet of paper and let us make

  a movie record of it4 ,if we cut up   
the strip of movie film into the        
individual pictures and cement them     
together one above another in  their    
proper order1 we shall build up a solid 
block of film which  will be a model of 
our simplified world of space-time and  
in which  there will be a series of dots
 representing the motion of the fly     
over the paper4 ,just as ,i can state   
the exact position of an object  in my  
room by defining its height above the   
floor1 its distance  from the north wall
 and its distance from the east wall1 so
 we can  reduce the positions of the    
dots to figures for use in calculations 
by measuring their distances from the   
three faces intersecting in the  lines  
,,ox1 ,,oy1 and ,,ot1 where ,,oxaytbcd  
represents the block of  film4 ,the     
mathematician would call the three lines
 ,,ox1 ,,oy1 ,,ot  the coordinate axes4 
,measuring all the dots in this way we  
shall  obtain the motion of the fly     
relative to the coordinate axes ,,ox1   

,,oy1 ,,ot4 ,if we add a block          
,,otdyefgh of plain film we can use     
,,ex1 ,,eh1 ,,ef as coordinate axes and 
again obtain the motion of  the fly     
relative to these new axes2 or we can   
add block after block  so as to keep the
 axes moving4 ,we can conceive of other 
changes of  axes4 ,the operator making  
the movie record might have taken the   
fly  for the hero of the piece and moved
 the camera about so as to keep the  fly
 more or less central in the picture2 or
 he might1 by turning the  handle first 
fast and then slow and by moving the    
camera1 have made  the fly appear to be 
doing stunts4 ,moving the camera would  
change the  axes of x and y1 and turning
 the handle at different speeds would   
change the axis of time4 ,again1 we     
might change the axes by pushing  the   
block out of shape or by distorting it  
into a state of strain4  ,whatever      
change of axes we make1 any dot in the  
block of film will  signify a           
coincidence of the fly with a certain   

point of the paper  at a certain time1  
and the series of dots will1 in every   
case1 be a  representation of the motion
 of the fly4 ,maybe the representation  
will be a distorted one1 but who is to  
say which is the absolutely  undistorted
 representation8 ,the principle of      
relativity which we laid  down before   
says that no one set of coordinates will
 give the absolute  motion of the fly1  
so that one set is as good as another4  
,the principle  that all motion is      
relative means1 therefore1 that no      
matter how we  change our coordinates of
 space-time1 the laws of motion which we
  deduce must be the same for all       
changes4    ,to use an analogy1 the     
sculptured head of ,shakespeare on my   
table may  appear to have hollow cheeks 
when ,i admit light from the east window
  only1 or to have sunken eyes with     
light from the skylight in the roof1    
but the true shape of the head remains  
the same in all lights4    ,hence1 if   
with reference to two consecutive dots  

in our block of  film a mathematical    
quantity can be found which will not    
change no  matter how we changes our    
axes of coordinates1 that quantity must 
be an expression of the true law of     
motion of the fly between the  two      
points of the paper and the two times   
represented by these two  dots4         
,einstein has worked out such a quantity
 remaining constant for  all changes of 
coordinates of the four dimensional     
world of space-time4    ,in passing we  
may notice a feature of ,einstein's     
world of space-time  which we shall     
doubtless find it difficult to conceive1
 namely1  that there is no essential    
difference between a time and a distance
  in space4 ,since one set of           
coordinates is as good as another1 we   
can transform time into space and space 
into time according as we  choose our   
axes4 ,for example if we change ,,ox1   
,,ot1 the axes of x  and time in ,fig4  
#b1 into ,,ox'1 ,,ot' by a simple       
rotation1 the new  time represented by  

,,ot' consists partly of ,,oa in the old
 time  and partly of ,,ob in the old x  
direction4 ,referring to our block  of  
movie film again1 it means that although
 ,i might separate the  block into space
 and time by slicing it into the        
original pictures1  ,i can just as      
readily slice it in any direction ,i    
choose and still  get individual        
pictures representing the motion of the 
fly but with1  of course1 new time and  
space4 ,so whilst ,i may be believe that
 a liner  has travelled #c1jjj miles in 
#d days1 an observer on a star who knows
  nothing of my particular axes in      
space-time may say1 with equal truth1   
that it went #b1jjj miles in #g days4   
,thus1 time and space are not two       
separate identities in ,einstein's view2
 there only exists a world of  four     
dimensions which we can split up into   
time and space as we choose4    ,let us 
see now how ,einstein explains          
gravitation4 ,when a body is not  acted 
on by any forces 7except gravitation7   

the quantity which remains  constant for
 all changes of coordinates implies that
 the body will  follow that path in the 
space of an outside observer which takes
 the  least time4 ,it is an observed    
fact that one body attracts another by  
gravitation2 that is1 the path of one   
body is bent from its course by  the    
presence of another4 ,now we can bend   
the path of the fly in our  block of    
film by straining the block in some way4
 ,suppose1 therefore1  that ,i strain   
the world so as to bend the path of a   
body exactly as  the gravitation due to 
some other body bends it2 i4e41 by a    
change  of coordinates ,i have obtained 
the same effect as that produced  by    
gravitation4 ,einstein's theory1        
therefore1 explains gravitation  as a   
distortion of the world of space-time   
due to the presence of  matter4 ,suppose
 first that a body is moving with no    
other bodies near2  according to        
,einstein it will take the path in space
 which requires the  least time1 i4e41 a

 straight line as agrees with our       
experience4 ,if now  the world be       
strained by the presence of another body
 or by a change  of coordinates it will 
still pursue the path of least time1 but
 this  path is now distorted from the   
straight line1 just as in a similar way 
the path on a globe requiring the least 
time to travel follows a great  circle4 
,so1 on ,einstein's view of gravitation1
 the earth moves in an  elliptical path 
around the sun not because a force is   
acting on it1  but because the world of 
space-time is so distorted by the       
presence of  the sun that the path of   
least time through space is the         
elliptical  path observed4 ,there is1   
therefore1 no need to introduce any idea
 of  force of gravitation4 ,einstein's  
theory explains gravitation only  in the
 sense that he has explained it away as 
a force of nature and  makes it a       
property of space-time1 namely1 a       
distortion not different  from an       
appropriate change of coordinates4 ,he  

does not1 however1  explain how or why a
 body can distort space-time4 ,it is    
noteworthy  that whilst the law of      
gravitation and the law of uniform      
motion in  a straight line when no force
 is acting were separate and independent
  laws under ,newton1 ,einstein finds   
one explanation for both under the      
principle of relativity4 #ab            
,,xx    ,,einstein',s ,,theory ,,of     
,,gravitation    ,the ,discussion of the
 ,general ,theory and ,its ,most        
,important  ,application1 from the      
,essay by    ,,prof4 ,w4 ,f4 ,g4        
,,swann1 ,,university ,,of ,,minnesota1 
,,minneapolis      ,newton's great      
discovery regarding the motion of the   
planets consisted  in his showing that  
these could all be summed up in the     
following  statement3 consider any      
planet in its relation to all particles 
in the  universe4 ,write down1 for the  
planet1 in the line joining it to any   
particle1 an acceleration proportional  
to the mass of the particle and  to the 

inverse square of its distance from the 
planet4 ,then calculate  the planet's   
resultant acceleration by combining all 
the accelerations  thus obtained4    ,we
 have here purposely avoided the use of 
the word force1 for  ,newton's law is   
complete as a practical statement of    
fact without it2  and this word adds    
nothing to the law by way of enhancing  
its power  in actual use4 ,nevertheless1
 the fact that the acceleration is made 
up as it were of non-interfering        
contributions from each particle  in the
 line joining it to the planet strongly 
suggests to the mind  something of the  
nature of an elastic pull for which the 
particle is  responsible1 and to which  
the planet's departure from a           
straight-line  motion is due4 ,the mind 
likes to think of the elastic2 ever     
since the  time of ,newton people have  
sought to devise some mechanism by which
  these pulls might be visualized as    
responsible for the phenomena in  the   
same way as one pictures an elastic     

thread as controlling the  motion of a  
stone which swings around at its end4   
,this search has been always without    
success2 and now ,einstein has  found a 
rather different law which fits the     
facts better than ,newton's  law4 ,it is
 of such a type that it does not lend   
itself conveniently  to expression in   
terms of force2 the mind would gain     
nothing by  trying to picture such      
forces as are necessary4 ,it compensates
 for  this1 however1 in being capable of
 visualization in terms of what is      
ultimately a much simpler concept4      
,in order to appreciate the fundamental 
ideas involved1 suppose for  a moment   
that gravitation could be annihilated1  
completely1 and  suppose ,i find myself 
upon this earth in empty space4 ,you    
shall be  seated at some point in space 
and shall watch my doings4 ,if ,i am in 
the condition of mind of the people of  
the reign of ,king ,henry ,,viii1  ,i   
shall believe that the earth does not   
rotate4 ,if ,i let go a stone1  there   

being no gravity1 ,i shall find that it 
flies away from me with  an             
acceleration4 ,you will know1 however1  
that the stone really moves  in a       
straight line with constant velocity1   
and that the apparent  acceleration     
which ,i perceive is due to the earth's 
rotation4 ,if ,i  have argued that      
acceleration is due to force1 ,i shall  
say that the  earth repels the stone1   
and shall try to find the law governing 
the  variation of this force with       
distance4 ,i may go farther1 and try to 
imagine some reason for the force1 some 
pushing action transmitted from  the    
earth to the stone through a surrounding
 medium2 and1 you will  pity me for all 
this wasted labor1 and particularly for 
my attempt  to find a mechanism to      
account for the force1 since you know   
that if  ,i would only accept your      
measurements all would appear so simple4
    ,let us probe this matter a little  
farther1 however1 from the  stand-point 
of myself4 ,i must believe in the       

reality of the force1  since ,i have to 
be tied to my chair to prevent my       
departure from  the earth4 ,i might     
wonder how this field of force would    
affect the  propagation of light1       
chemical action and so forth4 ,for1 even
 though ,i  had discovered that1 by     
using your measures1 ,i could transform 
away the  apparent effects of my field  
of force as far as concerned its power  
to hurl stones about1 ,i could still    
regard this as a mathematical  accident1
 and believe that the force was really  
there4 ,although ,i  might suspect that 
the same transformation of view-point   
that would  annul the field's effect as 
regards the stones would also annul its 
effect as regards light1 etc41 ,i should
 not be sure of this1 as you  would be2 
and my conscience would hardly allow me 
to do more than  look upon the          
assumption of complete equivalence      
between the apparent  field and a change
 in the system of measurement as a      
hypothesis4 ,i  should be strongly      

tempted to make the hypothesis1 however4
    ,now the question raised by         
,einstein is whether the force of       
gravity1  which we experience as a very 
real thing1 may be put upon a footing   
which is in some way analogous to that  
of the obviously fictitious  centrifugal
 force cited above3 whether gravitation 
may be regarded as  a figment of our    
imagination engendered by the way in    
which we measure  things4 ,he found that
 it could be so regarded4 ,he went still
 farther1  and in his ,principle of     
,equivalence1 he postulated that the    
apparent  effects of gravitation in all 
phenomena could be attributed to the    
same  change in the system of our       
measurements that would account for the 
ordinary phenomena of gravitation4 ,on  
the basis of this hypothesis  he was    
able to deduce for subsequent           
experimental verification1  the effects 
of gravitation on light4 ,he did not    
limit himself to  such simple changes in
 our measurements as were sufficient to 

serve  the purpose of the problem of    
centrifugal force cited above2 but1     
emboldened by the assumptions1 in the   
older theory of relativity1  of change  
in standards of length and time on      
account of motion1  he went even farther
 than this1 and considered the          
possibility of  change of our measures  
due to mere proximity to matter4    ,his
 problem amounted to an attempt to find 
some way in which it was  possible to   
conceive our scales and clocks as       
altered1 relatively to  some more       
fundamental set1 so as to allow of the  
planetary motions  being uniform and    
rectilinear with respect to these       
fundamental  measures1 although they    
appear as they do to us4 ,if we allow   
our  imaginations perfect freedom as to 
how the scales may be altered1 we  shall
 not balk at assuming alterations       
varying in any way we please1  ,einstein
 does1 however1 introduce restrictions  
for reasons which we  will now discern4 
,if we imagine our whole universe1 with 

its observers1 planetary  orbits1       
instruments1 and everything else1       
embedded in a jelly1 and  then distort  
the jelly and contents in any way1 the  
numbers at which  our planetary orbits  
7or rather their telescopic images7     
intersect  our scales will be unaltered4
 ,moreover1 we could vary1 in any       
manner1  the times at which all objects 
7including the clock hands7 occupied    
their distorted positions1 and the hand 
of some clock near the point  where the 
planetary image crossed the scale would 
record for this  occurrence the same    
dial reading as before4 ,an inhabitant  
of this  distorted universe would be    
absolutely unconscious of the change4   
,now  the ,general ,theory of           
,relativity which expresses itself in   
slightly  varied forms1 amounts to      
satisfying a certain philosophical      
craving  of the mind1 by asserting that 
the laws of nature which control our    
universe ought to be such that another  
universe like the above1 whose          

inhabitants would be unconscious of     
their change1 would also satisfy  these 
laws1 not merely from the standpoint of 
its own inhabitants1  but also from the 
standpoint of our measurements4 ,in     
other words1  this second universe ought
 to appear possible to us as well as to 
its inhabitants4    ,einstein decides to
 make his theory conform to this        
philosophical  desire1 and this greatly 
limits the modifications of clocks and  
scales which he permits himself for the 
purpose of representing  gravitation4   
,further1 if we express the alterations 
of the measures  as functions of        
proximity to matter1 velocity and so    
forth1 our  expressions for these       
alterations will include1 as a          
particular case1  that where matter is  
absent1 although the scales and observer
 may still  remain4 ,our alteration of  
the scales and clocks with velocity must
 thus  revert1 for this case1 to that   
corresponding to the older theory of    
relativity1 in order to avoid predicting

 that two observers1 in uniform  motion 
relative to each other in empty space1  
will measure different  values for the  
velocity of light4 ,in this way1 the    
velocity of light  comes to play a part 
in expressing the alterations of the    
measures4    ,even with these           
restrictions1 ,einstein was able to do  
the equivalent  of finding an alteration
 of scales and clocks in the presence of
  matter which would account for our    
finding that the planetary motions  take
 place very nearly in accordance with   
,newton's law4 ,the new law  has        
accounted with surprising accuracy for  
certain astronomical  irregularities for
 which ,newton's law failed to account1 
and has  predicted at least one         
previously unknown phenomenon which was 
immediately verified4    ,in conclusion1
 it may be of interest to state how the 
new law describes  the motion of a      
particle in the vicinity of a body like 
the earth4 ,the  law amounts to stating 
that1 if we measure a short distance1   

radially  as regards the earth's center1
 we must allow for the peculiarity of   
our units by dividing by                
sqrt#a-;frac#bm,gc#br    where r is the 
distance from the earth's center1 m the 
mass  of the earth1 c the velocity of   
light1 and ,g the ,newtonian            
gravitational constant4 ,tangential     
measurements require no correction1  but
 intervals of time as measured by our   
clocks must be multiplied1  for each    
particular place1 by the above factor4  
,then1 in terms of  the corrected       
measures so obtained1 the particle will 
be found to  describe a straight line   
with constant velocity although1 in     
terms  of our actual measures1 it       
appears to fall with an acceleration4   
,,xxi    ,,the ,,equivalence            
,,hypothesis    ,the ,discussion of     
,this1 ,with ,its ,difficulties and the 
,manner in ,which  ,einstein ,has       
,resolved ,them1 from the ,essay by     
,,prof4 ,e4 ,n4 ,,da ,c4 ,,andrade1     
,,ordnance ,,college1 ,,woolwich1       

,,england      ,having shown that1 of   
several systems all moving with         
reference to one  another with uniform  
motion1 no one is entitled to any       
preference over  the others1 and having 
deduced the laws for such systems1      
,einstein  was confronted with a        
difficulty which had long been felt4 ,a 
body  rotating1 which is a special case 
of an accelerated body1 can be          
distinguished from one at rest1 without 
looking outside it1 by the  existence of
 the so-called centrifugal forces4      
,this circumstance1 which gives certain 
bodies an absolute or  preferential     
motion1 is unpalatable to the           
relativist2 he would  like there to be  
no difference as regards forces #ac     
between the  case when the earth rotates
 with reference to outside bodies 7the  
stars7 considered as fixed1 and the case
 when the earth is considered  fixed and
 all outside bodies rotate around it4   
,this point cannot be  investigated by  
direct experiment2 we can spin a top but

 we cannot  keep a top at rest and spin 
the world round it1 to see if the forces
  are same4    ,in considering the      
problem of how to devise laws which     
should make all  rotations relative1    
,einstein conceived the brilliant yet   
simple idea  that gravitation could be  
brought into the scheme as an           
acceleration  effect1 since both        
ordinary accelerational forces and      
gravitational  forces are proportional  
to the same thing1 the mass of a body4  
,the  impossibility of separating the   
two kinds of effect can be easily  seen 
by considering the starting of an       
elevator4 ,when the elevator  is quickly
 accelerated upwards we feel a downward 
pull1 just as if  the gravitational pull
 had been increased1 and if the         
acceleration  continued to be uniform1  
bodies tested with a spring balance     
would  all weigh more in the elevator   
than they did on firm ground4 ,in a     
similar way the whole of the            
gravitational pull may be considered to 

be an accelerational effect1 the        
difficulty being to devise laws of      
motion which will give the effects that 
we find by actual observation4    ,but  
it is obvious that we cannot1 by        
ordinary mechanics1 consider the  earth 
as being accelerated in all directions1 
which we should have to  do1 apparently1
 to account for the fact that the       
gravitational pull  is always toward the
 center4 ,it is obvious that we cannot  
explain  gravitation by assuming that   
the earth's surface is continually      
moving  outward with an accelerated     
velocity4#bbg ,so ,einstein found that1 
as  long as we treat the problem by     
,euclid's geometry1 we cannot reach  a  
satisfactory solution4 ,but he found    
that to the four-dimensional  space made
 up of the three ordinary dimensions of 
space1 together with  the time-dimension
 which we have already mentioned in     
discussing the  special theory1 may be  
attributed a peculiar geometry1 the     
nature of  which departs more and more  

from ,euclidean geometry as we approach 
a  gravitational body1 and the net      
result of which is to make possible  the
 universal correspondence of gravitation
 and acceleration4    ,this modification
 of the geometry of space is often      
spoken of as the  curvature of space1 an
 expression which is puzzling1          
especially as  the space which is curved
 is four-dimensional time-space4 ,but we
  can get an idea of what is meant by   
considering figures1 triangles  say1    
drawn on the surface1 of a sphere4      
,these triangles1 although  drawn on a  
surface1 will not have the same         
properties as triangles  drawn on flat  
paper--their three angles will not      
together equal right  angles4 ,they will
 be non-,euclidean4 ,this is only a     
rough analogy1 but  we can see that the 
curvature of the surface causes a       
departure from  ,euclidean geometry for 
plane figures1 and consequently the     
departure  from ,euclidean laws extended
 to four dimensions may be referred to  

as caused by curvature of space4    ,it 
is difficult to imagine a lump of matter
 affecting the geometry of  the space   
round it4 ,once more we must use a rough
 illustration4 ,imagine  a very hot     
body1 and that1 knowing nothing of its  
properties1 we  have to measure up the  
space round it with metal               
measuring-rods4 ,the  nearer we are to  
the body1 the longer the rods will      
become1 owing to  the expansion of the  
metal4 ,when we measure out a square1   
one side  of which is nearer the body   
than the opposite side1 its angles will 
not be right angles4 ,if we knew nothing
 of the laws of heat we should  say that
 the body had made the space round it   
non-,euclidean4    ,einstein found1     
then1 that by taking the properties of  
space1 as given  by measurement1 to be  
modified in the neighborhood of masses  
of matter1  he could devise general laws
 according to which gravitational       
effects  would be produced1 and there   
would be no absolute rotation4 ,all     

forces will be the same whether a body  
rotates with everything  outside it     
fixed1 or the body is fixed1 and        
everything rotates  round it4 ,all      
motion is then relative1 and the theory 
is one of  general relativity4 ,the     
velocity of light is1 however1 no longer
  constant1 and its path is not a       
straight line1 if it is passing near    
gravitating matter4 ,this does not      
contradict the special theory1  which   
did not allow for gravitation4 ,rather1 
the special theory is  a particular case
 to which the generalised theory reduces
 when there  is no matter about1 just as
 the ,newtonian dynamics is a special   
case  of the special theory1 which we   
obtain when all velocities are small    
compared to that of light4              
,,xxii    ,,the ,,general ,,theory      
,fragments of ,particular ,merit on     
,this ,phase of the ,subject    ,,by    
,,various ,,contributors      ,when     
,dorothy was carried by the cyclone from
 her home in ,kansas to the  land of    

,oz1 together with her uncle's house and
 her little dog ,toto1  she neglected to
 lower the trap door over the hole in   
the floor which  formerly led to the    
cyclone cellar and ,toto stepped        
through4 ,dorothy  rushed to the opening
 expecting to see him dashed onto the   
rocks below  but found him floating just
 below the floor4 ,she drew him back    
into  the room and closed the trap4     
,the author of the chronicle of         
,dorothy's adventures explains that the 
same force which held up the house held 
up ,toto but this explanation  is not   
necessary4 ,dorothy was now floating    
through space and house  and dog were   
subject to the same forces of           
gravitation which gave  them identical  
motions4 ,dorothy must have pushed the  
dog down onto the  floor and in doing so
 must herself have floated to the       
ceiling whence  she might have pushed   
herself back to the floor4 ,in fact     
gravitation  was apparently suspended   
and ,dorothy was in a position to have  

tried  certain experiments which        
,einstein has never tried because he was
  never in ,dorothy's unique            
position4#ahh        ,the ,principle of 
,equivalence1 of which ,einstein's      
suspended cage  experiment is the usual 
illustration1 and upon which the        
generalized  theory of relativity is    
built1 is thus stated by ,prof4         
,eddington3 ,a  gravitational field of  
force is precisely equivalent to an     
artificial  field of force1 so that in  
any small region it is impossible1 by   
any  conceivable experiment1 to         
distinguish between them4 ,in other     
words1  force is purely relative4       
,this may be otherwise stated by going  
back to our idea of a  four-dimensional 
world1 the points of which represent the
 positions  and times of events4 ,if we 
mark in such a space-time the successive
  positions of an object we get a line1 
or curve1 which represents the  whole   
history of the object1 inasmuch as it   
shows us the position  of the object at 

every time4 ,the reader may imagine that
 all events  happen in one plane1 so    
that only two perpendicular dimensions  
are needed to fix positions in space1   
with a third perpendicular  dimension   
for time4 ,he may then conceive1 if he  
may not picture1  an analogous process  
for four-dimensional space-time4 ,these 
lines1  tracks of objects through       
space-time1 were called by ,minkowski   
world-lines4 ,we may now say that all   
the events we observe are the           
intersections of world-lines4 ,the      
temperature at noon was #gj4 ,this      
means that if ,i plot the world-line of 
the top of the mercury column  and the  
world-line of a certain mark on the     
glass they intersect in  a certain point
 of space-time4 ,all that we know are   
intersections of  these world-lines4    
,suppose now we have a large number of  
them drawn  in our four-dimensional     
world1 satisfying all known             
intersections1  and let us suppose the  
whole embedded in a jelly4 ,we may      

distort  this jelly in any way1 changing
 our coordinates as we please1 but we   
shall neither destroy nor create        
intersections of world-lines4 ,it  may  
be proved that a change from one system 
of reference1 to which  observations are
 referred1 to any other system1 moving  
in any way  with respect to the first   
system1 may be pictured as a distortion 
of  the four-dimensional jelly4 ,the    
laws of nature1 therefore1 being laws   
that describe intersections1 must be    
expressible in a form independent  of   
the reference system chosen4    ,from   
these postulates1 ,einstein was able to 
show such a formulation  possible4 ,his 
law may be stated very simply3--,all    
bodies move through  space-time in the  
straightest possible tracks4    ,the    
fact that an easy non-mathematical      
explanation can not be given1  of how   
this law is reached1 or of just why the 
straightest track of  ,mercury through  
space-time will give us an ellipse in   
space after we  have split space-time up

 into space and time1 is no valid       
objection to  the theory4 ,newton's law 
that bodies attract with a force        
proportional  to their masses and       
inversely proportional to the square of 
the  distance is simple1 but no one has 
ever given an easy non-mathematical     
proof of how that law requires the path 
of ,mercury to be an ellipse1  with the 
sun at a focus1 instead of some other   
curve4#ahb        ,one of the grave     
difficulties we have in gaining a       
satisfactory  comprehension of          
,einstein's conceptions1 is that they do
 not readily  relate themselves to our  
modes of geometrical thought4 ,within   
limits  we may choose our own geometry1 
but it may be at the cost of unwieldy   
complication4 ,if we think with ,newton 
in terms of ,euclidean geometry  and    
consider the earth as revolving around  
the sun1 the motions of  our solar      
system can be stated in comparatively   
simple terms4 ,if1  on the other hand1  
we should persist in stating them1 as   

,ptolemy  would have done1 from the     
earth as a relatively stationary center1
  our formulas will become complicated  
beyond ready comprehension4 ,for  this  
reason it is much simpler in applying   
the theory of relativity1  and in       
considering and describing what actually
 happens in the physical  universe1 to  
use geometrical conceptions to which the
 actual conditions  can be easily       
related4 ,we find such an instrument in 
non-,euclidean  geometry1 wherein space 
will appear as though it were projected 
from a slightly concave mirror4 ,it is  
in this sense that some speak  of space 
as curved4 ,the analogy is so suggestive
 it tempts one to  linger over it4      
,unless there were material objects     
within the range  of the mirror1 its    
conformation would be immaterial2 the   
thought of  the space which the mirror1 
as it were1 circumscribes1 is dependent 
upon the presence of such material      
objects4 ,the lines of light and of  all
 other movement will not be quite       

straight from the view-point  of        
,euclidean geometry4 ,a line drawn in a 
universe of such a nature  must         
inevitably return upon itself4 ,nothing 
therefore1 can ever pass  out of this   
unlimitedly great but yet finite cosmos4
 ,but even now1  since our imaginary    
mirror is only very slightly concave1 it
 follows  that for limited regions like 
the earth or even the solar system1 our 
conception of geometry may well be      
rectilinear and ,euclidean4 ,newton's   
law of gravitation will be quite        
accurate with only a theoretical        
modification drawn from the theory of   
relativity4#hb        ,the way in which 
a curvature of space might appear to us 
as a force  is made plainer by an       
example4 ,suppose that in a certain room
 a  marble dropped anywhere on the floor
 always rolled to the center of  the    
room2 suppose the same thing happened to
 a baseball1 a billiard  ball1 and a    
tennis ball4 ,these results could be    
explained in two  ways2 we might assume 

that a mysterious force of attraction   
existed  at the center of the floor1    
which affected all kinds of balls alike2
  or we might assume that the floor was 
curved4 ,we naturally prefer the  latter
 explanation4 ,but when we find that in 
the neighborhood of a  large material   
body all other bodies move toward it in 
exactly the  same manner1 regardless of 
their nature or their condition1 we are 
accustomed to postulate a mysterious    
attractive force 7gravitation72         
,einstein1 on the contrary1 adopts the  
other alternative1 that the  space      
around the body is curved4#bbc          
,in the ordinary analytical geometry1   
the position and motion of  all the     
points considered is referred to a rigid
 body or frame of  reference4 ,this     
usually consists of an imaginary room of
 suitable  size4 ,the position of any   
point is then given by three numbers1   
i4e41  its distances from one side wall 
and from the back wall and its height   
above the floor4 ,these three numbers   

can only give one point1 every  other   
point having at least one number        
different4 ,in four-dimensional         
geometry a fourth wall may be vaguely   
imagined as perpendicular to  all three 
walls1 and a fourth number added1 giving
 the distance of  the point from this   
wall also4 ,since rigid bodies do not   
exist in  gravitational fields the frame
 of reference must be non-rigid4 ,the   
frame of reference in the ,gaussian     
system need not be rigid1 it can  be of 
any shape and moving in any manner1 in  
fact a kind of jelly4 ,a  point or event
 in the four dimensioned world is still 
given by four  numbers but these numbers
 do not represent distances from        
anywhere2  all that is necessary is that
 no two events shall have exactly the   
same four numbers to represent them1 and
 that two events which are  very close  
together shall be represented by numbers
 which differ only  slightly from one   
another4 ,this system assumes so little 
that it will  be seen to be very wide in

 its scope2 although to the ordinary    
mind1  what is gained in scope seems    
more than that lost in concreteness4    
,this  does not concern the             
mathematician1 however1 and by using    
this system  he gains his object1       
proving that the general laws of nature 
remain  the same when expressed in any  
,gaussian coordinate system             
whatever4#bbj        ,einstein          
enunciates a general principle that it  
is possible to find  a transformation of
 coordinate axes which is exactly       
equivalent  to any force1 and in        
particular one which is equivalent to   
the  force of gravitation4 ,that is by  
concentrating our attention on  the     
transformation which is a purely        
mathematical operation we can  afford to
 neglect the force completely4 ,to get a
 better idea of this  principle of      
equivalence as it is called1 let us     
consider a relatively  simple example   
7which actually has nothing to do with  
gravitation1  but which will serve to   

make our notions clearer47 ,a person on 
the  earth unconsciously refers all his 
experiences1 i4e41 the motions  of the  
objects around him to a set of axes     
fixed in the earth on  which he stands4 
,however1 we know that the earth is     
rotating about  its axis1 and his axes  
of reference are also rotating with     
respect to  the space about him4 ,from  
the point of view of general relativity 
it  is exactly because we do refer      
motions on the surface of the earth  to 
axes rotating with the earth that we    
experience the so-called  centrifugal   
force of the earth's rotation1 with     
which everyone is  familiar4 ,if we     
could find it convenient to transform   
from moving axes  to fixed axes1 the    
force would vanish1 since it is exactly 
equivalent  to the transformation from  
one set of axes to the other4 ,however1 
we find it unnatural to refer daily     
experiences to axes that are not  placed
 where we happen to be1 and so we prefer
 to take the force and  rotating axes   

instead of no force and fixed axes4#bgb 
,we seem to have a direct experience of 
force in our muscular  sensations4 ,by  
pushing or pulling we can set bodies in 
motion4 ,it  is natural to assume1 that 
something similar occurs1 when ,nature  
set  bodies in motion4 ,but is this not 
a relic of animism8 ,the savage and  the
 ancients peopled all the woods and     
skies with ,gods and demons1 who        
carries on the activities of nature by  
their own bodily efforts4 ,today  we    
have dispossessed the demons1 but the   
ghost of a muscular pull  still holds   
the planets in place4#ada        ,the   
general theory is an extension of the   
special theory which enables  the law of
 gravitation to be deduced4 ,not in     
,newton's form1 it is true1  but in a   
better form1 that is1 one that accounts 
for two important  facts otherwise not  
explained4 ,but it is a far more general
 theory  that indicated above4 ,it is a 
complete study of the relations between 
laws expressed by means of any four     

coordinates 7of which three space  and  
one time is a special case71 and the    
same laws expressed in the  four        
coordinates of a system having any      
motion whatever with respect  to the    
first system4 ,by restricting this      
general study in accordance  with       
certain postulates about the nature of  
the universe we live in1  we arrive at a
 number of conclusions which fit more   
closely with  observed facts that the   
conclusions drawn from ,newton's        
theory4#bba                    ,,notes  
#a ,it will be noted that ,mr4 ,bolton  
pronounces the geometry of  space to be 
,euclidean in the absence of            
gravitational fields1 not  that of      
space-time4 ,this is in accord with what
 was pointed out on  page               
#afa4--;,editor4    #b ,the author here 
comes perilously close to ascribing to  
this  contraction the sort of physical  
reality which it does not  possess4 ,see
 page #if4--;,editor4    #c ,dr4 ,davis 
went rather fully into the algebra of   

the  ,michelson-,morley experiment4 ,but
 ,dr4 ,russell has covered the same     
ground in a form somewhat more          
advantageous from the typographical     
viewpoint1 and the point is not one     
which it is profitable to discuss       
twice2 so we eliminate this part of ,dr4
 ,davis' text4--,editor4    #d ,this    
statement is objectionable1 as explained
 in ,chapter  ,,iv4--,editor4    #e ,a4 
,einstein3 ,die ,grundlage der          
allgemeinen  ,relativittstheorie4 ,ann4 
d4 ,physik4 #d1 vol4 #di1 page #ggf4    
#f ,at this point we have again used the
 blue pencil on ,dr4 ,davis'  text1 his 
discussion of the three observational   
tests of the ,general  ,theory adding   
nothing to ,dr4 ,pickering's4--,the     
,editor4    #g ,commander ,mc,hardy uses
 the term event in a sense somewhat     
different from that seen in a majority  
of the essays4 ,he reserves  for the    
four-dimensional element--the instant of
 time at a point in  space--the name    
point-event2 and the term event he      

applies to a  collection of these       
forming1 together1 an observable whole4 
,an actual  physical happening1 like a  
railroad wreck or a laboratory          
experiment1  it will be realized is of  
the latter sort1 occupying an           
appreciable  region of space rather than
 a single point1 and an appreciable     
interval  of time rather than a single  
second4 ,to the element1 the point-event
  of ,commander ,mc,hardy's essay1 this 
bears the same relation that the        
geometer's solid bears to his point4    
,this comment is in no sense  to be     
taken as criticism of ,commander        
,mc,hardy's terminology1 which  rather  
appeals to us2 we make it merely to     
guard against confusion  in the reader's
 mind4--,editor4    #h ,this paragraph  
is the result of an editorial revision  
of the author's text1 designed to retain
 the substance of his  presentation1    
while tying up what he has to say more  
definitely with  the preceding essays1  
and eliminating the distinction between 

finite  and infinitesimal intervals1    
which we believe to be out of place in  
an essay of this character4 ,we will not
 apologize to our mathematical  readers 
for having used finite and differential 
notation in the same  equation1 in      
violation of mathematical               
convention4--,editor4    #i ,although   
gravitational force in a small region   
can be imitated  or annulled by         
accelerating motion1 there remains the  
disturbing  influence of gravitational  
matter already referred to and expressed
 in  the fabric curvature4 ,it is this  
that defines how unique tracks run1  or 
rather1 how bodies progress4--,author4  
#aj ,not all gravitational fields may be
 transformed away by a  proper choice of
 coordinates4 ,if this were so1 the     
space1 whose  nature is independent of  
any choice of coordinates1 would always 
be ,euclidean4--,author4    #aa ,thus   
when it it said that a body contracts or
 that a clock runs  slow when it is put 
in motion no actual physical change is  

implied4 ,the  judgment of different    
observers--one at rest with respect to  
the body  and one not--are              
different4--,author4    #ab ,the balance
 of ,dr4 ,royds' essay is given to a    
discussion of  the phenomena of         
,mercury's perihelial advance1 the      
deflection of  light under the          
gravitational field of the sun1 and the 
shift in  spectral lines1 in connection 
with which alone ,einstein's theory     
makes predictions which are sufficiently
 at variance with those  of ,newtonian  
science to be of value in checking up   
the theory  observationally4 ,in the    
interest of space conservation and in   
the  presence of ,dr4 ,pickering's very 
complete discussion of these matters  we
 omit ,dr4 ,royds' statement4--,editor4 
#ac ,there is1 on any view1 no          
difference as regards observation of    
position only4--,author4