,,sidelights ,,on ,,relativity  ,by     
,alb]t ,e9/e9  ,cont5ts  ,,e!r ,,& ,,!  
,,!ory ,,( ,,relativity  ,an ,address   
deliv]$ on ,may #e?1 #aibj1 9 !         
,univ]sity ( ,leyd5  ,,geometry ,,&     
,,exp]i5ce  ,an exp&$ =m ( an ,address  
to ! ,prussian ,academy ( ,sci5ces 9    
,b]l9 on ,janu>y #bg?1 #aiba4      ,,e!r
 ,,& ,,! ,,!ory ,,( ,,relativity  ,an   
,address deliv]$ on ,may #e?1 #aibj1 9 !
 ,univ]sity ( ,leyd5    ,h[ does x come 
ab\t t alongside ( ! idea ( pond]able   
matt]1 :i* is d]iv$ by ab/raction f     
ev]yday life1 ! physici/s set ! idea ( !
 exi/5ce ( ano!r k9d ( matt]1 ! e!r8 ,! 
explanation is probably to be s\ght 9 ^?
 ph5om5a :i* h giv5 rise to ! !ory (    
action at a di/ance1 & 9 ! prop]ties (  
light :i* h l$ to ! undulatory !ory4    
,let u devote a little :ile to !        
consid]ation ( ^! two subjects4         
,outside ( physics we "k no?+ ( action  
at a di/ance4 ,:5 we try to connect     
cause & effect 9 ! exp]i5ces :i* natural
 objects af=d u1 x seems at fir/ z if   

!re 7 no o!r mutual actions ?an ^? (    
imm$iate contact1 e4g4 ! communication (
 motion by impact1 pu% & pull1 heat+ or 
induc+ combu/ion by means ( a flame1    
etc4 ,x is true t ev5 9 ev]yday exp]i5ce
 weight1 :i* is 9 a s5se action at a    
di/ance1 plays a v important "p4 ,b s9ce
 9 daily exp]i5ce ! weight ( bodies     
meets u z some?+ con/ant1 some?+ n l9k$ 
to any cause :i* is v>iable 9 "t or     
place1 we d n 9 ev]yday life speculate z
 to ! cause ( gravity1 & !re=e d n      
become consci\s ( its "* z action at a  
di/ance4 ,x 0 ,newton's !ory (          
gravitation t fir/ assign$ a cause =    
gravity by int]pret+ x z action at a    
di/ance1 proce$+ f masses4 ,newton's    
!ory is probably ! greate/ /ride ev]    
made 9 ! ef=t t[>ds ! causal nexus (    
natural ph5om5a4 ,& yet ?is !ory evok$ a
 lively s5se ( discom=t among ,newton's 
contempor>ies1 because x seem$ to be 9  
conflict ) ! pr9ciple spr++ f ! re/ (   
exp]i5ce1 t !re c be reciprocal action  
only ?r\gh contact1 & n ?r\gh imm$iate  

action at a di/ance4 ,x is only )       
reluctance t man's desire = k endures a 
dualism ( ?is k9d4 ,h[ 0 unity to be    
pres]v$ 9 8 compreh5sion ( ! =ces (     
nature8 ,ei!r by try+ to look ^u contact
 =ces z be+ !mselves di/ant =ces :i*    
admitt$ly are obs]vable only at a v     
small di/ance--& ?is 0 ! road :i*       
,newton's foll[]s1 :o 7 entirely "u !   
spell ( 8 doctr9e1 mo/ly pref]r$ to     
take2 or by assum+ t ! ,newtonian action
 at a di/ance is only app>5tly imm$iate 
action at a di/ance1 b 9 tru? is convey$
 by a m$ium p]meat+ space1 :e!r by      
movem5ts or by ela/ic de=mation ( ?is   
m$ium4 ,?us ! endeav\r t[>d a unifi$    
view ( ! nature ( =ces leads to !       
hypo!sis ( an e!r4 ,?is hypo!sis1 to be 
sure1 did n at fir/ br+ ) x any advance 
9 ! !ory ( gravitation or 9 physics     
g5]ally1 s t x became cu/om>y to treat  
,newton's law ( =ce z an axiom n fur!r  
r$ucible4 ,b ! e!r hypo!sis 0 b\nd      
always to play "s "p 9 physical sci5ce1 
ev5 if at fir/ only a lat5t "p4  ,:5 9 !

 fir/ half ( ! n9ete5? c5tury ! f>-rea*+
 simil>ity 0 reveal$ :i* subsi/s betwe5 
! prop]ties ( light & ^? ( ela/ic waves 
9 pond]able bodies1 ! e!r hypo!sis f\nd 
fre% support4 ,x appe>$ beyond "q t     
light mu/ be int]pret$ z a vibratory    
process 9 an ela/ic1 in]t m$ium fill+ up
 univ]sal space4 ,x also seem$ to be a  
necess>y consequ5ce ( ! fact t light is 
capable ( pol>isation t ?is m$ium1 !    
e!r1 mu/ be ( ! nature ( a solid body1  
because transv]se waves are n possible 9
 a fluid1 b only 9 a solid4 ,?us !      
physici/s 7 b\nd to arrive at ! !ory ( !
 quasi-rigid lum9if]\s e!r1 ! p>ts ( :i*
 c c>ry out no movem5ts relatively to "o
 ano!r except ! small movem5ts (        
de=mation :i* correspond to light-waves4
  ,?is !ory--also call$ ! !ory ( !      
/ation>y lum9if]\s e!r--moreov] f\nd a  
/rong support 9 an exp]im5t :i* is also 
( fundam5tal importance 9 ! special !ory
 ( relativity1 ! exp]im5t ( ,fizeau1 f  
:i* "o 0 oblig$ to inf] t ! lum9if]\s   
e!r does n take "p 9 ! movem5ts (       

bodies4 ,! ph5om5on ( ab]ration also    
fav\r$ ! !ory ( ! quasi-rigid e!r4  ,!  
developm5t ( ! !ory ( electricity along 
! pa? op5$ up by ,maxwell & ,lor5tz gave
 ! developm5t ( our ideas conc]n+ ! e!r 
q a peculi> & unexpect$ turn4 ,=        
,maxwell himself ! e!r inde$ /ill _h    
prop]ties :i* 7 purely me*anical1 al?\gh
 ( a mu* m complicat$ k9d ?an !         
me*anical prop]ties ( tangible solid    
bodies4 ,b nei!r ,maxwell nor 8 foll[]s 
succe$$ 9 elaborat+ a me*anical model = 
! e!r :i* might furni% a satisfactory   
me*anical int]pretation ( ,maxwell's    
laws ( ! electro-magnetic field4 ,! laws
 7 cle> & simple1 ! me*anical           
int]pretations clumsy & contradictory4  
,almo/ imp]ceptibly ! !oretical         
physici/s adapt$ !mselves to a situation
 :i*1 f ! /&po9t ( _! me*anical         
programme1 0 v depress+4 ,!y 7          
p>ticul>ly influ5c$ by !                
electro-dynamical inve/igations (       
,he9ri* ,h]tz4 ,= :]eas !y previ\sly _h 
requir$ ( a conclusive !ory t x %\ld    

cont5t itself ) ! fundam5tal concepts   
:i* belong exclusively to me*anics 7e4g4
 d5sities1 velocities1 de=mations1      
/resses7 !y gradually accu/om$ !mselves 
to admitt+ electric & magnetic =ce z    
fundam5tal concepts side by side ) ^? ( 
me*anics1 )\t requir+ a me*anical       
int]pretation = !m4 ,?us ! purely       
me*anical view ( nature 0 gradually     
ab&on$4 ,b ?is *ange l$ to a fundam5tal 
dualism :i* 9 ! long-run 0              
insupportable4 ,a way ( escape 0 n[     
s\ght 9 ! rev]se direction1 by r$uc+ !  
pr9ciples ( me*anics to ^? (            
electricity1 & ?is especially z         
confid5ce 9 ! /rict validity ( !        
equations ( ,newton's me*anics 0 %ak5 by
 ! exp]im5ts ) beta-rays & rapid ka?ode 
rays4  ,?is dualism /ill confronts u 9  
unext5uat$ =m 9 ! !ory ( ,h]tz1 :]e     
matt] appe>s n only z ! be>] (          
velocities1 k9etic en]gy1 & me*anical   
pressures1 b also z ! be>] (            
electromagnetic fields4 ,s9ce su* fields
 also occur 9 vacuo--i4e4 9 free e!r--! 

e!r also appe>s z be>] ( electromagnetic
 fields4 ,! e!r appe>s indi/+ui%able 9  
its functions f ord9>y matt]4 ,)9 matt] 
x takes "p 9 ! motion ( matt] & 9 empty 
space x has ev]y:]e a velocity2 s t !   
e!r has a def9itely assign$ velocity    
?r\gh\t ! :ole ( space4 ,!re is no      
fundam5tal diff]5ce betwe5 ,h]tz's e!r &
 pond]able matt] 7:i* 9 "p subsi/s 9 !  
e!r74  ,! ,h]tz !ory suff]$ n only f !  
defect ( ascrib+ to matt] & e!r1 on ! "o
 h& me*anical /ates1 & on ! o!r h&      
electrical /ates1 :i* d n /& 9 any      
conceivable relation to ea* o!r2 x 0    
also at v>iance ) ! result ( ,fizeau's  
important exp]im5t on ! velocity ( !    
propagation ( light 9 mov+ fluids1 & )  
o!r e/abli%$ exp]im5tal results4  ,su* 0
 ! /ate ( ?+s :5 ,h4 ,a4 ,lor5tz ent]$  
^u ! sc5e4 ,he br\ght !ory into h>mony )
 exp]i5ce by means ( a wond]ful         
simplification ( !oretical pr9ciples4   
,he a*iev$ ?is1 ! mo/ important advance 
9 ! !ory ( electricity s9ce ,maxwell1 by
 tak+ f e!r its me*anical1 & f matt] its

 electromagnetic qualities4 ,z 9 empty  
space1 s too 9 ! int]ior ( mat]ial      
bodies1 ! e!r1 & n matt] view$          
atomi/ically1 0 exclusively ! seat (    
electromagnetic fields4 ,accord+ to     
,lor5tz ! elem5t>y p>ticles ( matt]     
alone are capable ( c>ry+ out movem5ts2 
_! electromagnetic activity is entirely 
conf9$ to ! c>ry+ ( electric *>ges4 ,?us
 ,lor5tz succe$$ 9 r$uc+ all            
electromagnetic happ5+s to ,maxwell's   
equations = free space4  ,z to !        
me*anical nature ( ! ,lor5tzian e!r1 x  
may be said ( x1 9 a some:at playful _s1
 t immobility is ! only me*anical       
prop]ty ( :i* x has n be5 depriv$ by ,h4
 ,a4 ,lor5tz4 ,x may be add$ t ! :ole   
*ange 9 ! conception ( ! e!r :i* !      
special !ory ( relativity br\ght ab\t1  
consi/$ 9 tak+ away f ! e!r its la/     
me*anical quality1 namely1 its          
immobility4 ,h[ ?is is to be und]/ood w 
=?) be exp\nd$4  ,! space-"t !ory & !   
k9ematics ( ! special !ory ( relativity 
7 modell$ on ! ,maxwell-,lor5tz !ory ( !

 electromagnetic field4 ,?is !ory !re=e 
satisfies ! conditions ( ! special !ory 
( relativity1 b :5 view$ f ! latt] x    
acquires a novel aspect4 ,= if ,k be a  
sy/em ( co-ord9ates relatively to :i* ! 
,lor5tzian e!r is at re/1 !             
,maxwell-,lor5tz equations are valid    
prim>ily ) ref]5ce to ,k4 ,b by !       
special !ory ( relativity ! same        
equations )\t any *ange ( mean+ also    
hold 9 relation to any new sy/em (      
co-ord9ates ,k' :i* is mov+ 9 uni=m     
translation relatively to ,k4 ,n[ comes 
! anxi\s "q3--,:y mu/ ,i 9 ! !ory       
di/+ui% ! ,k sy/em above all ,k' sy/ems1
 :i* are physically equival5t to x 9 all
 respects1 by assum+ t ! e!r is at re/  
relatively to ! ,k sy/em8 ,= !          
!oretician su* an asymmetry 9 !         
!oretical /ructure1 ) no correspond+    
asymmetry 9 ! sy/em ( exp]i5ce1 is      
intol]able4 ,if we assume ! e!r to be at
 re/ relatively to ,k1 b 9 motion       
relatively to ,k'1 ! physical equival5ce
 ( ,k & ,k' seems to me f ! logical     

/&po9t1 n inde$ d[nright incorrect1 b   
nev]!less inacceptable4  ,! next        
position :i* x 0 possible to take up 9  
face ( ?is /ate ( ?+s appe>$ to be !    
foll[+4 ,! e!r does n exi/ at all4 ,!   
electromagnetic fields are n /ates ( a  
m$ium1 & are n b\nd d[n to any be>]1 b  
!y are indep5d5t realities :i* are n    
r$ucible to any?+ else1 exactly l !     
atoms ( pond]able matt]4 ,?is conception
 sugge/s itself ! m readily z1 accord+  
to ,lor5tz's !ory1 electromagnetic      
radiation1 l pond]able matt]1 br+s      
impulse & en]gy ) x1 & z1 accord+ to !  
special !ory ( relativity1 bo? matt] &  
radiation are b special =ms ( di/ribut$ 
en]gy1 pond]able mass los+ its isolation
 & appe>+ z a special =m ( en]gy4  ,m   
c>eful reflection tea*es u1 h[ev]1 t !  
special !ory ( relativity does n compel 
u to d5y e!r4 ,we may assume ! exi/5ce (
 an e!r2 only we mu/ give up ascrib+ a  
def9ite /ate ( motion to x1 i4e4 we mu/ 
by ab/raction take f x ! la/ me*anical  
*>act]i/ic :i* ,lor5tz _h /ill left x4  

,we %all see lat] t ?is po9t ( view1 !  
conceivability ( :i* ,i %all at once    
endeav\r to make m intelligible by a    
some:at halt+ comp>ison1 is ju/ifi$ by !
 results ( ! g5]al !ory ( relativity4   
,?9k ( waves on ! surface ( wat]4 ,"h we
 c describe two entirely diff]5t ?+s4   
,ei!r we may obs]ve h[ ! undulatory     
surface =m+ ! b\nd>y betwe5 wat] & air  
alt]s 9 ! c\rse ( "t2 or else--) ! help 
( small floats1 = in/ance--we c obs]ve  
h[ ! position ( ! sep>ate p>ticles (    
wat] alt]s 9 ! c\rse ( "t4 ,if ! exi/5ce
 ( su* floats = track+ ! motion ( !     
p>ticles ( a fluid 7 a fundam5tal       
impossibility 9 physics--if1 9 fact1    
no?+ else :atev] 7 obs]vable ?an ! %ape 
( ! space occupi$ by ! wat] z x v>ies 9 
"t1 we %\ld h no gr\nd = ! assumption t 
wat] consi/s ( movable p>ticles4 ,b all 
! same we c\ld *>act]ise x z a m$ium4   
,we h some?+ l ?is 9 ! electromagnetic  
field4 ,= we may picture ! field to     
ourselves z consi/+ ( l9es ( =ce4  ,if  
we wi% to int]pret ^! l9es ( =ce to     

ourselves z some?+ mat]ial 9 ! ord9>y   
s5se1 we are tempt$ to int]pret !       
dynamic processes z motions ( ^! l9es ( 
=ce1 su* t ea* sep>ate l9e ( =ce is     
track$ ?r\gh ! c\rse ( "t4 ,x is well   
kn[n1 h[ev]1 t ?is way ( reg>d+ !       
electromagnetic field leads to          
contradictions4  ,g5]alis+ we mu/ say   
?is3--,!re may be suppos$ to be ext5d$  
physical objects to :i* ! idea ( motion 
_c be appli$4 ,!y may n be ?\ght ( z    
consi/+ ( p>ticles :i* all[ !mselves to 
be sep>ately track$ ?r\gh "t4 ,9        
,m9k[ski's idiom ?is is express$ z      
foll[s3--,n e ext5d$ con=mation 9 !     
f\r-dim5sional _w c be reg>d$ z compos$ 
( _w-?reads4 ,! special !ory (          
relativity =bids u to assume ! e!r to   
consi/ ( p>ticles obs]vable ?r\gh "t1 b 
! hypo!sis ( e!r 9 itself is n 9        
conflict ) ! special !ory ( relativity4 
,only we mu/ be on our gu>d aga9/       
ascrib+ a /ate ( motion to ! e!r4       
,c]ta9ly1 f ! /&po9t ( ! special !ory ( 
relativity1 ! e!r hypo!sis appe>s at    

fir/ to be an empty hypo!sis4 ,9 !      
equations ( ! electromagnetic field !re 
occur1 9 addition to ! d5sities ( !     
electric *>ge1 only ! int5sities ( !    
field4 ,! c>e] ( electromagnetic        
processes 9 vacuo appe>s to be          
completely det]m9$ by ^! equations1     
un9flu5c$ by o!r physical quantities4 ,!
 electromagnetic fields appe> z         
ultimate1 irr$ucible realities1 & at    
fir/ x seems sup]flu\s to po/ulate a    
homog5e\s1 isotropic e!r-m$ium1 & to    
envisage electromagnetic fields z /ates 
( ?is m$ium4  ,b on ! o!r h& !re is a   
weighty argum5t to be adduc$ 9 fav\r ( !
 e!r hypo!sis4 ,to d5y ! e!r is         
ultimately to assume t empty space has  
no physical qualities :atev]4 ,!        
fundam5tal facts ( me*anics d n h>monize
 ) ?is view4 ,= ! me*anical behavi\r ( a
 corporeal sy/em hov]+ freely 9 empty   
space dep5ds n only on relative         
positions 7di/ances7 & relative         
velocities1 b also on its /ate (        
rotation1 :i* physically may be tak5 z a

 *>act]i/ic n app]ta9+ to ! sy/em 9     
itself4 ,9 ord] to be able to look ^u ! 
rotation ( ! sy/em1 at lea/ =mally1 z   
some?+ real1 ,newton objectivises space4
  ,s9ce he classes 8 absolute space     
toge!r ) real ?+s1 = him rotation       
relative to an absolute space is also   
some?+ real4 ,newton might no less well 
h call$ 8 absolute space ,e!r2 :at is   
ess5tial is m]ely t besides obs]vable   
objects1 ano!r ?+1 :i* is n p]ceptible1 
mu/ be look$ ^u z real1 to enable       
accel]ation or rotation to be look$ ^u z
 some?+ real4  ,x is true t ,ma* tri$ to
 avoid hav+ to accept z real some?+ :i* 
is n obs]vable by endeav\r+ to sub/itute
 9 me*anics a mean accel]ation ) ref]5ce
 to ! totality ( ! masses 9 ! univ]se 9 
place ( an accel]ation ) ref]5ce to     
absolute space4 ,b in]tial resi/ance    
oppos$ to relative accel]ation ( di/ant 
masses presupposes action at a di/ance2 
& z ! mod]n physici/ does n believe t he
 may accept ?is action at a di/ance1 he 
comes back once m1 if he foll[s ,ma*1 to

 ! e!r1 :i* has to s]ve z m$ium = !     
effects ( in]tia4 ,b ?is conception ( ! 
e!r to :i* we are l$ by ,ma*'s way (    
?9k+ diff]s ess5tially f ! e!r z        
conceiv$ by ,newton1 by ,fresnel1 & by  
,lor5tz4 ,ma*'s e!r n only conditions ! 
behavi\r ( in]t masses1 b is also       
condition$ 9 its /ate by !m4  ,ma*'s    
idea f9ds its full developm5t 9 ! e!r ( 
! g5]al !ory ( relativity4 ,accord+ to  
?is !ory ! metrical qualities ( !       
cont9uum ( space-"t diff] 9 ! environm5t
 ( diff]5t po9ts ( space-"t1 & are p>tly
 condition$ by ! matt] exi/+ outside ( !
 t]ritory "u consid]ation4  ,?is        
space-"t v>iability ( ! reciprocal      
relations ( ! /&>ds ( space & "t1 or1   
p]haps1 ! recognition ( ! fact t empty  
space 9 its physical relation is nei!r  
homog5e\s nor isotropic1 compell+ u to  
describe its /ate by t5 functions 7!    
gravitation pot5tials g7mn771 has1 ,i   
?9k1 f9ally dispos$ ( ! view t space is 
physically empty4 ,b !re) ! conception (
 ! e!r has aga9 acquir$ an intelligible 

cont5t1 al?\gh ?is cont5t diff]s widely 
f t ( ! e!r ( ! me*anical undulatory    
!ory ( light4  ,! e!r ( ! g5]al !ory (  
relativity is a m$ium :i* is itself     
devoid ( all me*anical & k9ematical     
qualities1 b helps to det]m9e me*anical 
7& electromagnetic7 ev5ts4  ,:at is     
fundam5tally new 9 ! e!r ( ! g5]al !ory 
( relativity z oppos$ to ! e!r ( ,lor5tz
 consi/s 9 ?is1 t ! /ate ( ! =m] is at e
 place det]m9$ by connections ) ! matt] 
& ! /ate ( ! e!r 9 neighb\r+ places1 :i*
 are am5able to law 9 ! =m ( diff]5tial 
equations2 :]eas ! /ate ( ! ,lor5tzian  
e!r 9 ! abs5ce ( electromagnetic fields 
is condition$ by no?+ outside itself1 & 
is ev]y:]e ! same4 ,! e!r ( ! g5]al !ory
 ( relativity is transmut$ conceptually 
into ! e!r ( ,lor5tz if we sub/itute    
con/ants = ! functions ( space :i*      
describe ! =m]1 disreg>d+ ! causes :i*  
condition its /ate4 ,?us we may also    
say1 ,i ?9k1 t ! e!r ( ! g5]al !ory (   
relativity is ! outcome ( ! ,lor5tzian  
e!r1 ?r\gh relativation4  ,z to ! "p :i*

 ! new e!r is to play 9 ! physics ( !   
future we are n yet cle>4 ,we "k t x    
det]m9es ! metrical relations 9 !       
space-"t cont9uum1 e4g4 ! configurative 
possibilities ( solid bodies z well z ! 
gravitational fields2 b we d n "k :e!r x
 has an ess5tial %>e 9 ! /ructure ( !   
electrical elem5t>y p>ticles con/itut+  
matt]4 ,nor d we "k :e!r x is only 9 !  
proximity ( pond]able masses t its      
/ructure diff]s ess5tially f t ( !      
,lor5tzian e!r2 :e!r ! geometry ( spaces
 ( cosmic ext5t is approximately        
,euclidean4 ,b we c ass]t by reason ( ! 
relativi/ic equations ( gravitation t   
!re mu/ be a dep>ture f ,euclidean      
relations1 ) spaces ( cosmic ord] (     
magnitude1 if !re exi/s a positive mean 
d5sity1 no matt] h[ small1 ( ! matt] 9 !
 univ]se4 ,9 ?is case ! univ]se mu/ (   
necessity be spatially unb\nd$ & ( f9ite
 magnitude1 its magnitude be+ det]m9$ by
 ! value ( t mean d5sity4  ,if we       
consid] ! gravitational field & !       
electromagnetic field f ! /&-po9t ( !   

e!r hypo!sis1 we f9d a rem>kable        
diff]5ce betwe5 ! two4 ,!re c be no     
space nor any "p ( space )\t            
gravitational pot5tials2 = ^! conf] ^u  
space its metrical qualities1 )\t :i* x 
_c be imag9$ at all4 ,! exi/5ce ( !     
gravitational field is insep>ably b\nd  
up ) ! exi/5ce ( space4 ,on ! o!r h& a  
"p ( space may v well be imag9$ )\t an  
electromagnetic field2 ?us 9 contra/ ) !
 gravitational field1 ! electromagnetic 
field seems to be only second>ily l9k$  
to ! e!r1 ! =mal nature ( !             
electromagnetic field be+ z yet 9 no way
 det]m9$ by t ( gravitational e!r4 ,f ! 
pres5t /ate ( !ory x looks z if !       
electromagnetic field1 z oppos$ to !    
gravitational field1 re/s ^u an entirely
 new =mal motif1 z ?\gh nature might j z
 well h end[$ ! gravitational e!r )     
fields ( q ano!r type1 = example1 )     
fields ( a scal> pot5tial1 in/ead (     
fields ( ! electromagnetic type4  ,s9ce 
accord+ to our pres5t conceptions !     
elem5t>y p>ticles ( matt] are also1 9 _!

 ess5ce1 no?+ else ?an cond5sations ( ! 
electromagnetic field1 our pres5t view (
 ! univ]se pres5ts two realities :i* are
 completely sep>at$ f ea* o!r           
conceptually1 al?\gh connect$ causally1 
namely1 gravitational e!r &             
electromagnetic field1 or--z !y might   
also be call$--space & matt]4  ,( c\rse 
x w\ld be a great advance if we c\ld    
succe$ 9 compreh5d+ ! gravitational     
field & ! electromagnetic field toge!r z
 "o unifi$ con=mation4 ,!n = ! fir/ "t !
 epo* ( !oretical physics f\nd$ by      
,f>aday & ,maxwell w\ld rea* a          
satisfactory conclusion4 ,! contra/     
betwe5 e!r & matt] w\ld fade away1 &1   
?r\gh ! g5]al !ory ( relativity1 ! :ole 
( physics w\ld become a complete sy/em (
 ?\ght1 l geometry1 k9ematics1 & ! !ory 
( gravitation4  ,an exce$+ly +5i\s      
attempt 9 ?is direction has be5 made by 
! ma!matician ,h4 ,weyl2 b ,i d n       
believe t 8 !ory w hold its gr\nd 9     
relation to reality4 ,fur!r1 9          
contemplat+ ! imm$iate future (         

!oretical physics we ought n            
unconditionally to reject ! possibility 
t ! facts compris$ 9 ! quantum !ory may 
set b\nds to ! field !ory beyond :i* x  
_c pass4  ,recapitulat+1 we may say t   
accord+ to ! g5]al !ory ( relativity    
space is end[$ ) physical qualities2 9  
?is s5se1 !re=e1 !re exi/s an e!r4      
,accord+ to ! g5]al !ory ( relativity   
space )\t e!r is un?9kable2 = 9 su*     
space !re n only w\ld be no propagation 
( light1 b also no possibility ( exi/5ce
 = /&>ds ( space & "t 7measur+-rods &   
clocks71 nor !re=e any space-"t int]vals
 9 ! physical s5se4 ,b ?is e!r may n be 
?\ght ( z end[$ ) ! quality *>act]i/ic (
 pond]able m$ia1 z consi/+ ( p>ts :i*   
may be track$ ?r\gh "t4 ,! idea ( motion
 may n be appli$ to x4     ,,geometry   
,,& ,,exp]i5ce  ,an exp&$ =m ( an       
,address to ! ,prussian ,academy (      
,sci5ces 9 ,b]l9 on ,janu>y #bg?1 #aiba4
    ,"o reason :y ma!matics enjoys      
special e/eem1 above all o!r sci5ces1 is
 t its laws are absolutely c]ta9 &      

indisputable1 :ile ^? ( all o!r sci5ces 
are to "s ext5t debatable & 9 con/ant   
dang] ( be+ ov]?r[n by newly discov]$   
facts4 ,9 spite ( ?is1 ! inve/igator 9  
ano!r dep>tm5t ( sci5ce w\ld n ne$ to   
envy ! ma!matician if ! laws ( ma!matics
 ref]r$ to objects ( our m]e imag9ation1
 & n to objects ( reality4 ,= x _c      
occasion surprise t diff]5t p]sons %\ld 
arrive at ! same logical conclusions :5 
!y h already agre$ ^u ! fundam5tal laws 
7axioms71 z well z ! me?ods by :i* o!r  
laws are to be d$uc$ !refrom4 ,b !re is 
ano!r reason = ! high repute (          
ma!matics1 9 t x is ma!matics :i* af=ds 
! exact natural sci5ces a c]ta9 measure 
( security1 to :i* )\t ma!matics !y c\ld
 n atta94  ,at ?is po9t an enigma       
pres5ts itself :i* 9 all ages has       
agitat$ inquir+ m9ds4 ,h[ c x be t      
ma!matics1 be+ aft] all a product (     
human ?\ght :i* is indep5d5t ( exp]i5ce1
 is s admirably appropriate to ! objects
 ( reality8 ,is human reason1 !n1 )\t   
exp]i5ce1 m]ely by tak+ ?\ght1 able to  

fa?om ! prop]ties ( real ?+s4  ,9 my    
op9ion ! answ] to ?is "q is1 briefly1   
?is3--,z f> z ! laws ( ma!matics ref] to
 reality1 !y are n c]ta92 & z f> z !y   
are c]ta91 !y d n ref] to reality4 ,x   
seems to me t complete cle>ness z to ?is
 /ate ( ?+s fir/ became common prop]ty  
?r\gh t new dep>ture 9 ma!matics :i* is 
kn[n by ! "n ( ma!matical logic or      
,axiomatics4 ,! progress a*iev$ by      
axiomatics consi/s 9 its hav+ neatly    
sep>at$ ! logical-=mal f its objective  
or intuitive cont5t2 accord+ to         
axiomatics ! logical-=mal alone =ms !   
subject-matt] ( ma!matics1 :i* is n     
conc]n$ ) ! intuitive or o!r cont5t     
associat$ ) ! logical-=mal4  ,let u = a 
mom5t consid] f ?is po9t ( view any     
axiom ( geometry1 = in/ance1 !          
foll[+3--,?r\gh two po9ts 9 space !re   
always passes "o & only "o /raight l9e4 
,h[ is ?is axiom to be int]pret$ 9 !    
old] s5se & 9 ! m mod]n s5se8  ,! old]  
int]pretation3--,e "o kn[s :at a /raight
 l9e is1 & :at a po9t is4 ,:e!r ?is k   

spr+s f an ability ( ! human m9d or f   
exp]i5ce1 f "s collaboration ( ! two or 
f "s o!r s\rce1 is n = ! ma!matician to 
decide4 ,he leaves ! "q to ! philosoph]4
 ,be+ bas$ ^u ?is k1 :i* prec$es all    
ma!matics1 ! axiom /at$ above is1 l all 
o!r axioms1 self-evid5t1 t is1 x is !   
expression ( a "p ( ?is a priori k4  ,! 
m mod]n int]pretation3--,geometry treats
 ( entities :i* are d5ot$ by ! words    
/raight l9e1 po9t1 etc4 ,^! entities d n
 take = grant$ any k or intuition       
:atev]1 b !y presuppose only ! validity 
( ! axioms1 su* z ! "o /at$ above1 :i*  
are to be tak5 9 a purely =mal s5se1    
i4e4 z void ( all cont5t ( intuition or 
exp]i5ce4 ,^! axioms are free creations 
( ! human m9d4 ,all o!r propositions (  
geometry are logical inf]5ces f ! axioms
 7:i* are to be tak5 9 ! nom9ali/ic s5se
 only74 ,! matt] ( :i* geometry treats  
is fir/ def9$ by ! axioms4 ,s*lick 9 8  
book on epi/emology has !re=e *>act]is$ 
axioms v aptly z implicit def9itions4   
,?is view ( axioms1 advocat$ by mod]n   

axiomatics1 purges ma!matics ( all      
extrane\s elem5ts1 & ?us dispels ! my/ic
 obscurity :i* =m]ly surr\nd$ !         
pr9ciples ( ma!matics4  ,b a pres5tation
 ( its pr9ciples ?us cl>ifi$ makes x    
also evid5t t ma!matics z su* _c        
pr$icate any?+ ab\t p]ceptual objects or
 real objects4 ,9 axiomatic geometry !  
words po9t1 /raight l9e1 etc41 /& only =
 empty conceptual s*emata4 ,t :i* gives 
!m sub/ance is n relevant to ma!matics4 
,yet on ! o!r h& x is c]ta9 t ma!matics 
g5]ally1 & p>ticul>ly geometry1 owes its
 exi/5ce to ! ne$ :i* 0 felt ( le>n+    
some?+ ab\t ! relations ( real ?+s to "o
 ano!r4 ,! v ^w geometry1 :i*1 ( c\rse1 
means e>?-measur+1 proves ?is4 ,=       
e>?-measur+ has to d ) ! possibilities (
 ! disposition ( c]ta9 natural objects )
 respect to "o ano!r1 namely1 ) p>ts ( !
 e>?1 measur+-l9es1 measur+-w&s1 etc4 ,x
 is cle> t ! sy/em ( concepts (         
axiomatic geometry alone _c make any    
ass]tions z to ! relations ( real       
objects ( ?is k9d1 :i* we w call        

practically-rigid bodies4 ,to be able to
 make su* ass]tions1 geometry mu/ be    
/ripp$ ( its m]ely logical-=mal "* by ! 
co-ord9ation ( real objects ( exp]i5ce )
 ! empty conceptual frame-"w ( axiomatic
 geometry4 ,to accompli% ?is1 we ne$    
only add ! proposition3--,solid bodies  
are relat$1 ) respect to _! possible    
dispositions1 z are bodies 9 ,euclidean 
geometry ( ?ree dim5sions4 ,!n !        
propositions ( ,euclid conta9           
affirmations z to ! relations (         
practically-rigid bodies4  ,geometry ?us
 complet$ is evid5tly a natural sci5ce2 
we may 9 fact reg>d x z ! mo/ anci5t    
bran* ( physics4 ,its affirmations re/  
ess5tially on induction f exp]i5ce1 b n 
on logical inf]5ces only4 ,we w call ?is
 complet$ geometry practical geometry1 &
 %all di/+ui% x 9 :at foll[s f purely   
axiomatic geometry4 ,! "q :e!r !        
practical geometry ( ! univ]se is       
,euclidean or n has a cle> mean+1 & its 
answ] c only be furni%$ by exp]i5ce4    
,all l9e> measurem5t 9 physics is       

practical geometry 9 ?is s5se1 s too is 
geodetic & a/ronomical l9e> measurem5t1 
if we call to our help ! law ( exp]i5ce 
t light is propagat$ 9 a /raight l9e1 & 
inde$ 9 a /raight l9e 9 ! s5se (        
practical geometry4  ,i atta* special   
importance to ! view ( geometry :i* ,i h
 j set =?1 because )\t x ,i %\ld h be5  
unable to =mulate ! !ory ( relativity4  
,)\t x ! foll[+ reflection w\ld h be5   
impossible3--,9 a sy/em ( ref]5ce rotat+
 relatively to an in]t sy/em1 ! laws (  
disposition ( rigid bodies d n          
correspond to ! rules ( ,euclidean      
geometry on acc\nt ( ! ,lor5tz          
contraction2 ?us if we admit non-in]t   
sy/ems we mu/ ab&on ,euclidean geometry4
 ,! decisive /ep 9 ! transition to g5]al
 co-v>iant equations w\ld c]ta9ly n h   
be5 tak5 if ! above int]pretation _h n  
s]v$ z a /epp+-/one4 ,if we d5y !       
relation betwe5 ! body ( axiomatic      
,euclidean geometry & !                 
practically-rigid body ( reality1 we    
readily arrive at ! foll[+ view1 :i* 0  

ent]ta9$ by t acute & pr(\nd ?9k]1 ,h4  
,po9c>e3--,euclidean geometry is        
di/+ui%$ above all o!r imag9able        
axiomatic geometries by its simplicity4 
,n[ s9ce axiomatic geometry by itself   
conta9s no ass]tions z to ! reality :i* 
c be exp]i5c$1 b c d s only 9 comb9ation
 ) physical laws1 x %\ld be possible &  
reasonable--:atev] may be ! nature (    
reality--to reta9 ,euclidean geometry4  
,= if contradictions betwe5 !ory &      
exp]i5ce manife/ !mselves1 we %\ld r    
decide to *ange physical laws ?an to    
*ange axiomatic ,euclidean geometry4 ,if
 we d5y ! relation betwe5 !             
practically-rigid body & geometry1 we   
%all inde$ n easily free ourselves f !  
conv5tion t ,euclidean geometry is to be
 reta9$ z ! simple/4 ,:y is ! equival5ce
 ( ! practically-rigid body & ! body (  
geometry--:i* sugge/s itself s          
readily--d5i$ by ,po9c>e & o!r          
inve/igators8 ,simply because "u clos]  
inspection ! real solid bodies 9 nature 
are n rigid1 because _! geometrical     

behavi\r1 t is1 _! possibilities (      
relative disposition1 dep5d ^u          
temp]ature1 ext]nal =ces1 etc4 ,?us !   
orig9al1 imm$iate relation betwe5       
geometry & physical reality appe>s      
de/roy$1 & we feel impell$ t[>d ! foll[+
 m g5]al view1 :i* *>act]izes ,po9c>e's 
/&po9t4 ,geometry 7,g7 pr$icates no?+   
ab\t ! relations ( real ?+s1 b only     
geometry toge!r ) ! purport 7,p7 (      
physical laws c d s4 ,us+ symbols1 we   
may say t only ! sum ( 7,g7  7,p7 is    
subject to ! control ( exp]i5ce4 ,?us   
7,g7 may be *os5 arbitr>ily1 & also p>ts
 ( 7,p72 all ^! laws are conv5tions4    
,all t is necess>y to avoid             
contradictions is to *oose ! rema9d] (  
7,p7 s t 7,g7 & ! :ole ( 7,p7 are toge!r
 9 accord ) exp]i5ce4 ,envisag$ 9 ?is   
way1 axiomatic geometry & ! "p ( natural
 law :i* has be5 giv5 a conv5tional     
/atus appe> z epi/emologically          
equival5t4  ,sub specie aet]ni ,po9c>e1 
9 my op9ion1 is "r4 ,! idea ( !         
measur+-rod & ! idea ( ! clock          

co-ord9at$ ) x 9 ! !ory ( relativity d n
 f9d _! exact correspond5ce 9 ! real _w4
 ,x is also cle> t ! solid body & !     
clock d n 9 ! conceptual edifice (      
physics play ! "p ( irr$ucible elem5ts1 
b t ( composite /ructures1 :i* may n    
play any indep5d5t "p 9 !oretical       
physics4 ,b x is my conviction t 9 !    
pres5t /age ( developm5t ( !oretical    
physics ^! ideas mu/ /ill be employ$ z  
indep5d5t ideas2 = we are /ill f> f     
possess+ su* c]ta9 k ( !oretical        
pr9ciples z to be able to give exact    
!oretical con/ructions ( solid bodies & 
clocks4  ,fur!r1 z to ! objection t !re 
are no really rigid bodies 9 nature1 & t
 !re=e ! prop]ties pr$icat$ ( rigid     
bodies d n apply to physical            
reality1--?is objection is by no means s
 radical z might appe> f a ha/y         
exam9ation4 ,= x is n a difficult task  
to det]m9e ! physical /ate ( a          
measur+-rod s accurately t its behavi\r 
relatively to o!r measur+-bodies %all be
 suffici5tly free f ambiguity to all[ x 

to be sub/itut$ = ! rigid body4 ,x is to
 measur+-bodies ( ?is k9d t /atem5ts z  
to rigid bodies mu/ be ref]r$4  ,all    
practical geometry is bas$ ^u a pr9ciple
 :i* is accessible to exp]i5ce1 & :i* we
 w n[ try to realise4 ,we w call t :i*  
is enclos$ betwe5 two b\nd>ies1 m>k$ ^u 
a practically-rigid body1 a tract4 ,we  
imag9e two practically-rigid bodies1 ea*
 ) a tract m>k$ out on x4 ,^! two tracts
 are said to be equal to "o ano!r if !  
b\nd>ies ( ! "o tract c be br\ght to    
co9cide p]man5tly ) ! b\nd>ies ( ! o!r4 
,we n[ assume t3  ,if two tracts are    
f\nd to be equal once & any:]e1 !y are  
equal always & ev]y:]e4  ,n only !      
practical geometry ( ,euclid1 b also its
 ne>e/ g5]alisation1 ! practical        
geometry ( ,riemann1 & !re) ! g5]al !ory
 ( relativity1 re/ ^u ?is assumption4 ,(
 ! exp]im5tal reasons :i* w>rant ?is    
assumption ,i w m5tion only "o4 ,!      
ph5om5on ( ! propagation ( light 9 empty
 space assigns a tract1 namely1 !       
appropriate pa? ( light1 to ea* int]val 

( local "t1 & conv]sely4 ,!nce x foll[s 
t ! above assumption = tracts mu/ also  
hold good = int]vals ( clock-"t 9 ! !ory
 ( relativity4 ,consequ5tly x may be    
=mulat$ z foll[s3--,if two ideal clocks 
are go+ at ! same rate at any "t & at   
any place 7be+ !n 9 imm$iate proximity  
to ea* o!r71 !y w always g at ! same    
rate1 no matt] :]e & :5 !y are aga9     
comp>$ ) ea* o!r at "o place4--,if ?is  
law 7 n valid = real clocks1 ! prop]    
frequ5cies = ! sep>ate atoms ( ! same   
*emical elem5t w\ld n be 9 su* exact    
agreem5t z exp]i5ce demon/rates4 ,!     
exi/5ce ( %>p spectral l9es is a conv9c+
 exp]im5tal pro( ( ! above-m5tion$      
pr9ciple ( practical geometry4 ,?is is !
 ultimate f\ndation 9 fact :i* enables u
 to speak ) mean+ ( ! m5suration1 9     
,riemann's s5se ( ! ^w1 ( !             
f\r-dim5sional cont9uum ( space-"t4  ,! 
"q :e!r ! /ructure ( ?is cont9uum is    
,euclidean1 or 9 accordance ) ,riemann's
 g5]al s*eme1 or o!rwise1 is1 accord+ to
 ! view :i* is "h be+ advocat$1 prop]ly 

speak+ a physical "q :i* mu/ be answ]$  
by exp]i5ce1 & n a "q ( a m]e conv5tion 
to be select$ on practical gr\nds4      
,riemann's geometry w be ! "r ?+ if !   
laws ( disposition ( practically-rigid  
bodies are trans=mable into ^? ( !      
bodies ( ,euclid's geometry ) an        
exactitude :i* increases 9 proportion z 
! dim5sions ( ! "p ( space-"t "u        
consid]ation are dim9i%$4  ,x is true t 
?is propos$ physical int]pretation (    
geometry breaks d[n :5 appli$ imm$iately
 to spaces ( sub-molecul> ord] (        
magnitude4 ,b nev]!less1 ev5 9 que/ions 
z to ! con/itution ( elem5t>y p>ticles1 
x reta9s "p ( its importance4 ,= ev5 :5 
x is a "q ( describ+ ! electrical       
elem5t>y p>ticles con/itut+ matt]1 !    
attempt may /ill be made to ascribe     
physical importance to ^? ideas ( fields
 :i* h be5 physically def9$ = ! purpose 
( describ+ ! geometrical behavi\r (     
bodies :i* are l>ge z comp>$ ) !        
molecule4 ,success alone c decide z to !
 ju/ification ( su* an attempt1 :i*     

po/ulates physical reality = !          
fundam5tal pr9ciples ( ,riemann's       
geometry outside ( ! doma9 ( _! physical
 def9itions4 ,x might possibly turn out 
t ?is extrapolation has no bett] w>rant 
?an ! extrapolation ( ! idea (          
temp]ature to p>ts ( a body ( molecul>  
ord] ( magnitude4  ,x appe>s less       
problematical to ext5d ! ideas (        
practical geometry to spaces ( cosmic   
ord] ( magnitude4 ,x might1 ( c\rse1 be 
object$ t a con/ruction compos$ ( solid 
rods dep>ts m & m f ideal rigidity 9    
proportion z its spatial ext5t becomes  
great]4 ,b x w h>dly be possible1 ,i    
?9k1 to assign fundam5tal significance  
to ?is objection4 ,!re=e ! "q :e!r !    
univ]se is spatially f9ite or n seems to
 me decid$ly a pregnant "q 9 ! s5se (   
practical geometry4 ,i d n ev5 consid] x
 impossible t ?is "q w be answ]$ be=e   
long by a/ronomy4 ,let u call to m9d :at
 ! g5]al !ory ( relativity tea*es 9 ?is 
respect4 ,x (f]s two possibilities3--   
#a4 ,! univ]se is spatially inf9ite4    

,?is c be s only if ! av]age spatial    
d5sity ( ! matt] 9 univ]sal space1      
conc5trat$ 9 ! />s1 vani%es1 i4e4 if !  
ratio ( ! total mass ( ! />s to !       
magnitude ( ! space ?r\gh :i* !y are    
scatt]$ approximates indef9itely to !   
value z]o :5 ! spaces tak5 into         
consid]ation are con/antly great] &     
great]4  #b4 ,! univ]se is spatially    
f9ite4 ,?is mu/ be s1 if !re is a mean  
d5sity ( ! pond]able matt] 9 univ]sal   
space diff]+ f z]o4 ,! small] t mean    
d5sity1 ! great] is ! volume ( univ]sal 
space4  ,i mu/ n fail to m5tion t a     
!oretical argum5t c be adduc$ 9 fav\r ( 
! hypo!sis ( a f9ite univ]se4 ,! g5]al  
!ory ( relativity tea*es t ! in]tia ( a 
giv5 body is great] z !re are m         
pond]able masses 9 proximity to x2 ?us x
 seems v natural to r$uce ! total effect
 ( in]tia ( a body to action & reaction 
betwe5 x & ! o!r bodies 9 ! univ]se1 z  
inde$1 ev] s9ce ,newton's "t1 gravity   
has be5 completely r$uc$ to action &    
reaction betwe5 bodies4  ,f ! equations 

( ! g5]al !ory ( relativity x c be d$uc$
 t ?is total r$uction ( in]tia to       
reciprocal action betwe5 masses--z      
requir$ by ,e4 ,ma*1 = example--is      
possible only if ! univ]se is spatially 
f9ite4  ,on _m physici/s & a/ronom]s ?is
 argum5t makes no impression4 ,exp]i5ce 
alone c f9ally decide :i* ( ! two       
possibilities is realis$ 9 nature4 ,h[ c
 exp]i5ce furni% an answ]8 ,at fir/ x   
might seem possible to det]m9e ! mean   
d5sity ( matt] by obs]vation ( t "p ( ! 
univ]se :i* is accessible to our        
p]ception4 ,?is hope is illusory4 ,!    
di/ribution ( ! visible />s is extremely
 irregul>1 s t we on no acc\nt may      
v5ture to set d[n ! mean d5sity (       
/>-matt] 9 ! univ]se z equal1 let u say1
 to ! mean d5sity 9 ! ,milky ,way4 ,9   
any case1 h[ev] great ! space exam9$ may
 be1 we c\ld n feel conv9c$ t !re 7 no m
 />s beyond t space4 ,s x seems         
impossible to e/imate ! mean d5sity4 ,b 
!re is ano!r road1 :i* seems to me m    
practicable1 al?\gh x also pres5ts great

 difficulties4 ,= if we inquire into !  
deviations %[n by ! consequ5ces ( !     
g5]al !ory ( relativity :i* are         
accessible to exp]i5ce1 :5 ^! are comp>$
 ) ! consequ5ces ( ! ,newtonian !ory1 we
 fir/ ( all f9d a deviation :i* %[s     
itself 9 close proximity to gravitat+   
mass1 & has be5 confirm$ 9 ! case ( !   
planet ,m]cury4 ,b if ! univ]se is      
spatially f9ite !re is a second         
deviation f ! ,newtonian !ory1 :i*1 9 ! 
language ( ! ,newtonian !ory1 may be    
express$ ?us3--,! gravitational field is
 9 its nature su* z if x 7 produc$1 n   
only by ! pond]able masses1 b also by a 
mass-d5sity ( negative sign1 di/ribut$  
uni=mly ?r\gh\t space4 ,s9ce ?is        
factiti\s mass-d5sity w\ld h to be      
enorm\sly small1 x c\ld make its pres5ce
 felt only 9 gravitat+ sy/ems ( v great 
ext5t4  ,assum+ t we "k1 let u say1 !   
/ati/ical di/ribution ( ! />s 9 ! ,milky
 ,way1 z well z _! masses1 !n by        
,newton's law we c calculate !          
gravitational field & ! mean velocities 

:i* ! />s mu/ h1 s t ! ,milky ,way %\ld 
n collapse "u ! mutual attraction ( its 
/>s1 b %\ld ma9ta9 its actual ext5t4 ,n[
 if ! actual velocities ( ! />s1 :i* c1 
( c\rse1 be measur$1 7 small] ?an !     
calculat$ velocities1 we %\ld h a pro( t
 ! actual attractions at great di/ances 
are small] ?an by ,newton's law4 ,f su* 
a deviation x c\ld be prov$ indirectly t
 ! univ]se is f9ite4 ,x w\ld ev5 be     
possible to e/imate its spatial         
magnitude4  ,c we picture to ourselves a
 ?ree-dim5sional univ]se :i* is f9ite1  
yet unb\nd$8  ,! usual answ] to ?is "q  
is ,no1 b t is n ! "r answ]4 ,! purpose 
( ! foll[+ rem>ks is to %[ t ! answ]    
%\ld be ,yes4 ,i want to %[ t )\t any   
extraord9>y difficulty we c illu/rate ! 
!ory ( a f9ite univ]se by means ( a     
m5tal image to :i*1 ) "s practice1 we   
%all soon gr[ accu/om$4  ,fir/ ( all1 an
 obs]vation ( epi/emological nature4 ,a 
geometrical-physical !ory z su* is      
incapable ( be+ directly pictur$1 be+   
m]ely a sy/em ( concepts4 ,b ^! concepts

 s]ve ! purpose ( br++ a multiplicity ( 
real or imag9>y s5sory exp]i5ces into   
connection 9 ! m9d4 ,to visualise a     
!ory1 or br+ x home to "o's m9d1 !re=e  
means to give a repres5tation to t      
abundance ( exp]i5ces = :i* ! !ory      
supplies ! s*ematic arrangem5t4 ,9 !    
pres5t case we h to ask ourselves h[ we 
c repres5t t relation ( solid bodies )  
respect to _! reciprocal disposition    
7contact7 :i* corresponds to ! !ory ( a 
f9ite univ]se4 ,!re is really no?+ new 9
 :at ,i h to say ab\t ?is2 b innum]able 
que/ions address$ to me prove t !       
requirem5ts ( ^? :o ?ir/ = k ( ^! matt]s
 h n yet be5 completely satisfi$4  ,s1 w
 ! initiat$ please p>don me1 if "p ( :at
 ,i %all br+ =w>d has long be5 kn[n8    
,:at d we wi% to express :5 we say t our
 space is inf9ite8 ,no?+ m ?an t we     
might lay any numb] :atev] ( bodies (   
equal sizes side by side )\t ev] fill+  
space4 ,suppose t we are provid$ ) a    
great _m wood5 cubes all ( ! same size4 
,9 accordance ) ,euclidean geometry we c

 place !m above1 beside1 & beh9d "o     
ano!r s z to fill a "p ( space ( any    
dim5sions2 b ?is con/ruction w\ld nev]  
be f9i%$2 we c\ld g on add+ m & m cubes 
)\t ev] f9d+ t !re 0 no m room4 ,t is   
:at we wi% to express :5 we say t space 
is inf9ite4 ,x w\ld be bett] to say t   
space is inf9ite 9 relation to          
practically-rigid bodies1 assum+ t !    
laws ( disposition = ^! bodies are giv5 
by ,euclidean geometry4  ,ano!r example 
( an inf9ite cont9uum is ! plane4 ,on a 
plane surface we may lay squ>es (       
c>dbo>d s t ea* side ( any squ>e has !  
side ( ano!r squ>e adjac5t to x4 ,!     
con/ruction is nev] f9i%$2 we c always g
 on lay+ squ>es--if _! laws (           
disposition correspond to ^? ( plane    
figures ( ,euclidean geometry4 ,! plane 
is !re=e inf9ite 9 relation to ! c>dbo>d
 squ>es4 ,accord+ly we say t ! plane is 
an inf9ite cont9uum ( two dim5sions1 &  
space an inf9ite cont9uum ( ?ree        
dim5sions4 ,:at is "h meant by ! numb] (
 dim5sions1 ,i ?9k ,i may assume to be  

kn[n4  ,n[ we take an example ( a       
two-dim5sional cont9uum :i* is f9ite1 b 
unb\nd$4 ,we imag9e ! surface ( a l>ge  
globe & a quantity ( small pap] discs1  
all ( ! same size4 ,we place "o ( !     
discs any:]e on ! surface ( ! globe4 ,if
 we move ! disc ab\t1 any:]e we l1 on ! 
surface ( ! globe1 we d n come ^u a     
limit or b\nd>y any:]e on ! j\rney4     
,!re=e we say t ! sph]ical surface ( !  
globe is an unb\nd$ cont9uum4 ,moreov]1 
! sph]ical surface is a f9ite cont9uum4 
,= if we /ick ! pap] discs on ! globe1 s
 t no disc ov]laps ano!r1 ! surface ( ! 
globe w f9ally become s full t !re is no
 room = ano!r disc4 ,?is simply means t 
! sph]ical surface ( ! globe is f9ite 9 
relation to ! pap] discs4 ,fur!r1 !     
sph]ical surface is a non-,euclidean    
cont9uum ( two dim5sions1 t is to say1 !
 laws ( disposition = ! rigid figures   
ly+ 9 x d n agree ) ^? ( ! ,euclidean   
plane4 ,?is c be %[n 9 ! foll[+ way4    
,place a pap] disc on ! sph]ical        
surface1 & ar\nd x 9 a circle place six 

m discs1 ea* ( :i* is to be surr\nd$ 9  
turn by six discs1 & s on4 ,if ?is      
con/ruction is made on a plane surface1 
we h an un9t]rupt$ disposition 9 :i* !re
 are six discs t\*+ e disc except ^? :i*
 lie on ! outside4  ,figure #a3 ,discs  
maximally pack$ on a plane  ,on !       
sph]ical surface ! con/ruction also     
seems to promise success at ! outset1 & 
! small] ! radius ( ! disc 9 proportion 
to t ( ! sph]e1 ! m promis+ x seems4 ,b 
z ! con/ruction progresses x becomes m &
 m pat5t t ! disposition ( ! discs 9 !  
mann] indicat$1 )\t int]ruption1 is n   
possible1 z x %\ld be possible by       
,euclidean geometry ( ! ! plane surface4
 ,9 ?is way creatures :i* _c leave !    
sph]ical surface1 & _c ev5 peep out f ! 
sph]ical surface into ?ree-dim5sional   
space1 might discov]1 m]ely by exp]im5t+
 ) discs1 t _! two-dim5sional space is n
 ,euclidean1 b sph]ical space4  ,f !    
late/ results ( ! !ory ( relativity x is
 probable t our ?ree-dim5sional space is
 also approximately sph]ical1 t is1 t ! 

laws ( disposition ( rigid bodies 9 x   
are n giv5 by ,euclidean geometry1 b    
approximately by sph]ical geometry1 if  
only we consid] p>ts ( space :i* are    
suffici5tly great4 ,n[ ?is is ! place   
:]e ! read]'s imag9ation boggles4       
,nobody c imag9e ?is ?+1 he cries       
indignantly4 ,x c be said1 b _c be      
?\ght4 ,i c repres5t to myself a        
sph]ical surface well en\gh1 b no?+     
analog\s to x 9 ?ree dim5sions4  ,figure
 #b3 ,a circle project$ f a sph]e onto a
 plane  ,we mu/ try to surm\nt ?is b>ri]
 9 ! m9d1 & ! pati5t read] w see t x is 
by no means a p>ticul>ly difficult task4
 ,= ?is purpose we w fir/ give our      
att5tion once m to ! geometry (         
two-dim5sional sph]ical surfaces4 ,9 !  
adjo9+ figure let ,k be ! sph]ical      
surface1 t\*$ at ,s by a plane1 ,e1 :i*1
 = facility ( pres5tation1 is %[n 9 !   
draw+ z a b\nd$ surface4 ,let ,l be a   
disc on ! sph]ical surface4  ,n[ let u  
imag9e t at ! po9t ,n ( ! sph]ical      
surface1 diametrically opposite to ,s1  

!re is a lum9\s po9t1 ?r[+ a %ad[ ,l' ( 
! disc ,l ^u ! plane ,e4 ,e po9t on !   
sph]e has its %ad[ on ! plane4 ,if !    
disc on ! sph]e ,k is mov$1 its %ad[ ,l'
 on ! plane ,e also moves4 ,:5 ! disc ,l
 is at ,s1 x almo/ exactly co9cides )   
its %ad[4  ,if x moves on ! sph]ical    
surface away f ,s upw>ds1 ! disc %ad[   
,l' on ! plane also moves away f ,s on !
 plane outw>ds1 gr[+ bigg] & bigg]4 ,z !
 disc ,l approa*es ! lum9\s po9t ,n1 !  
%ad[ moves (f to inf9ity1 & becomes     
inf9itely great4  ,n[ we put ! "q1 ,:at 
are ! laws ( disposition ( ! disc-%ad[s 
,l' on ! plane ,e8 ,evid5tly !y are     
exactly ! same z ! laws ( disposition ( 
! discs ,l on ! sph]ical surface4 ,= to 
ea* orig9al figure on ,k !re is a       
correspond+ %ad[ figure on ,e4 ,if two  
discs on ,k are t\*+1 _! %ad[s on ,e    
also t\*4 ,! %ad[-geometry on ! plane   
agrees ) ! ! disc-geometry on ! sph]e4  
,if we call ! disc-%ad[s rigid figures1 
!n sph]ical geometry holds good on !    
plane ,e ) respect to ^! rigid figures4 

,moreov]1 ! plane is f9ite ) respect to 
! disc-%ad[s1 s9ce only a f9ite numb] ( 
! %ad[s c f9d room on ! plane4  ,at ?is 
po9t somebody w say1 ,t is nons5se4 ,!  
disc-%ad[s are n rigid figures4 ,we h   
only to move a two-foot rule ab\t on !  
plane ,e to conv9ce ourselves t ! %ad[s 
con/antly increase 9 size z !y move away
 f ,s on ! plane t[>ds inf9ity4 ,b :at  
if ! two-foot rule 7 to behave on !     
plane ,e 9 ! same way z ! disc-%ad[s    
,l'8 ,x w\ld !n be impossible to %[ t ! 
%ad[s increase 9 size z !y move away f  
,s2 su* an ass]tion w\ld !n no long] h  
any mean+ :atev]4 ,9 fact ! only        
objective ass]tion t c be made ab\t !   
disc-%ad[s is j ?is1 t !y are relat$ 9  
exactly ! same way z are ! rigid discs  
on ! sph]ical surface 9 ! s5se (        
,euclidean geometry4  ,we mu/ c>efully  
be> 9 m9d t our /atem5t z to ! gr[? ( ! 
disc-%ad[s1 z !y move away f ,s t[>ds   
inf9ity1 has 9 itself no objective      
mean+1 z long z we are unable to employ 
,euclidean rigid bodies :i* c be mov$   

ab\t on ! plane ,e = ! purpose ( comp>+ 
! size ( ! disc-%ad[s4 ,9 respect ( !   
laws ( disposition ( ! %ad[s ,l'1 ! po9t
 ,s has no special privileges on ! plane
 any m ?an on ! sph]ical surface4  ,!   
repres5tation giv5 above ( sph]ical     
geometry on ! plane is important = u1   
because x readily all[s itself to be    
transf]r$ to ! ?ree-dim5sional case4    
,let u imag9e a po9t ,s ( our space1 & a
 great numb] ( small sph]es1 ,l'1 :i* c 
all be br\ght to co9cide ) "o ano!r4 ,b 
^! sph]es are n to be rigid 9 ! s5se (  
,euclidean geometry2 _! radius is to    
increase 79 ! s5se ( ,euclidean         
geometry7 :5 !y are mov$ away f ,s t[>ds
 inf9ity1 & ?is increase is to take     
place 9 exact accordance ) ! same law z 
applies to ! increase ( ! radii ( !     
disc-%ad[s ,l' on ! plane4  ,aft] hav+  
ga9$ a vivid m5tal image ( ! geometrical
 behavi\r ( our ,l' sph]es1 let u assume
 t 9 our space !re are no rigid bodies  
at all 9 ! s5se ( ,euclidean geometry1 b
 only bodies hav+ ! behavi\r ( our ,l'  

sph]es4 ,!n we %all h a vivid           
repres5tation ( ?ree-dim5sional sph]ical
 space1 or1 r ( ?ree-dim5sional sph]ical
 geometry4 ,"h our sph]es mu/ be call$  
rigid sph]es4 ,_! increase 9 size z !y  
dep>t f ,s is n to be detect$ by measur+
 ) measur+-rods1 any m ?an 9 ! case ( ! 
disc-%ad[s on ,e1 because ! /&>ds (     
measurem5t behave 9 ! same way z !      
sph]es4 ,space is homog5e\s1 t is to    
say1 ! same sph]ical configurations are 
possible 9 ! environm5t ( all po9ts499  
,our space is f9ite1 because1 9         
consequ5ce ( ! gr[? ( ! sph]es1 only a  
f9ite numb] ( !m c f9d room 9 space4  99
 ,?is is intelligible )\t calculation--b
 only = ! two-dim5sional case--if we    
rev]t once m to ! case ( ! disc on !    
surface ( ! sph]e4  ,9 ?is way1 by us+ z
 /epp+-/ones ! practice 9 ?9k+ &        
visualisation :i* ,euclidean geometry   
gives u1 we h acquir$ a m5tal picture ( 
sph]ical geometry4 ,we may )\t          
difficulty imp>t m dep? & vig\r to ^!   
ideas by c>ry+ out special imag9>y      

con/ructions4 ,nor w\ld x be difficult  
to repres5t ! case ( :at is call$       
elliptical geometry 9 an analog\s mann]4
 ,my only aim to-day has be5 to %[ t !  
human faculty ( visualisation is by no  
means b\nd to capitulate to             
non-,euclidean geometry4