Skip to main content

Full text of "The fourth dimension"

See other formats 


SATURDAY     2    APR  1955 


SATURDAY     2 


1955 


THE    FOURTH    DIMENSION 


SOME  OPINIONS  OF  THE   PRESS 

"  Mr.  C.  H.  Hinton  discusses  the  subject  of  the  higher  dimensionality  of 
space,  his  aim  being  to  avoid  mathematical  subtleties  and  technicalities,  and 
thus  enable  his  argument  to  be  followed  by  readers  who  are  not  sufficiently 
conversant  with  mathematics  to  follow  these  processes  of  reasoning." — 
NOTTS  GUABDIAN. 

"  The  fourth  dimension  is  a  subject  which  has  /tad  a  great  fascination  for 
many  teachers,  and  though  one  cannot  pretend  to  have  quite  grasped 
Mr.  Hinton's  conceptions  and  arguments,  yet  it  must  be  admitted  that  he 
reveals  the  elusive  idea  in  quite  a  fascinating  light.  Quite  apart  from  the 
main  thesis  of  the  book  many  chapters  are  of  great  independent  interest. 
Altogether  an  interesting,  clever  and  ingenious  book." — DUNDEE  COURIER. 

"  The  book  will  well  repay  the  study  of  men  who  like  to  exercise  their  wits 
upon  the  problems  of  abstract  thought." — SCOTSMAN. 

"Professor  Hinton  has  done  well  to  attempt  a  treatise  of  moderate  size, 
which  shall  at  once  be  clear  in  method  and  free  from  technicalities  of  the 
schools." — PALL  MALL  GAZETTE. 

"  A  very  interesting  book  he  has  made  of  it." — PUBLISHERS'  CIRCULAR. 
"Mr.  Hinton  tries  to  explain  the  theory  of  the  fourth  dimension  so  that 
the  ordinary  reasoning  mind  can  get  a  grasp  of  what  metaphysical 
mathematicians  mean  by  it.  If  he  is  not  altogether  successful  it  is  not  from 
want  of  clearness  on  his  part,  but  because  the  whole  theory  comes  as  such  an 
absolute  shock  to  all  one's  preconceived  ideas." — BRISTOL  TIMES. 

"  Mr.  Hinton's  enthusiasm  is  only  the  result  of  an  exhaustive  study,  which 
has  enabled  him  to  set  his  subject  before  the  reader  with  far  more  than  the 
amount  of  lucidity  to  which  it  is  accustomed." — PALL  MALL  GAZETTE. 

"  The  book  throughout  is  a  very  solid  piece  of  reasoning  in  tlie  domain  of 
higher  mathematics." — GLASGOW  HERALD. 

"  Those  who  wish  to  grasp  the  meaning  of  this  somewhat  difficult  subject 
would  do  well  to  read  The  Fourth  Dimension.  No  mathematical  knowledge 
is  demanded  of  the  reader,  and  any  one,  who  is  not  afraid  of  a  little  hard 
thinking,  should  be  able  to  follow  the  argument." — LIGHT. 

''  A  splendidly  clear  re-statement  of  the  old  problem  of  the  fourth  dimension. 
All  who  are  interested  in  this  subject  will  find  the  work  not  only  fascinating, 
but  lucid,  it  being  written  in  a  style  easily  understandable.  The  illustrations 
make  still  more  clear  the  letterpress,  and  the  whole  is  most  admirably  adapted 
to  the  requirements  of  the  novice  or  the  student." — Two  WORLDS. 

"  Those  in  search  of  mental  gymnastics  ivill  find  abundance  of  exercise  in 
Mr.  C.  Ht  Hinton's  Fourth  Dimension." — WESTMINSTER  REVIEW. 


FIRST  EDITION,  April  1904;  SECOND  EDITION,  May  1906; 
THIRD  EDITION,  January  1912. 


THE 


FOURTH  DIMENSION 


BY 


C,    HOWARD    HINTON,    M.A. 

AUTHOR      OF      "  SCIENTIFIC     ROMANCES  " 
"A  NEW   ERA   OF  THOUGHT,"   ETC.,   ETC. 


LONDON 

GEORGE    ALLEN    &    CO.,    LTD. 

EUSKIN     HOUSE 
44,   45   RATHBONE   PLACE 

1912 


PREFACE    TO    FIRST    EDITION 

I  HAVE  endeavoured  to  present  the  subject  of  the  higher 
dimensionality  of  space  in  a  clear  manner,  devoid  of 
mathematical  subtleties  and  technicalities.  In  order  to 
engage  the  interest  of  the  reader,  I  have  in  the  earlier 
chapters  dwelt  on  the  perspective  the  hypothesis  of  a 
fourth  dimension  opens,  and  have  treated  of  the  many 
connections  there  are  between  this  hypothesis  and  the 
ordinary  topics  of  our  thoughts. 

A  lack  of  mathematical  knowledge  will  prove  of  no 
disadvantage  to  the  reader,  for  I  have  used  no  mathe- 
matical processes  of  reasoning.  I  have  taken  the  view 
that  the  space  which  we  ordinarily  think  of,  the  space 
of  real  things  (which  I  would  call  permeable  matter), 
is  different  from  the  space  treated  of  by  mathematics. 
Mathematics  will  tell  us  a  great  deal  about  space,  just 
as  the  atomic  theory  will  tell  us  a  great  deal  about  the 
chemical  combinations  of  bodies.  But  after  all,  a  theory 
is  not  precisely  equivalent  to  the  subject  with  regard 
to  which  it  is  held.  There  is  an  opening,  therefore,  from 
the  side  of  our  ordinary  space  perceptions  for  a  simple, 
altogether  rational,  mechanical,  and  observational  way 


VI  PBEFACE 

of  treating  this   subject   of  higher  space,    and  of  this 
opportunity  I  have  availed  myself. 

The  details  introduced  in  the  earlier  chapters,  especially 
in  Chapters  VIII.,  IX.,  X.,  may  perhaps  be  found 
wearisome.  They  are  of  no  essential  importance  in  the 
main  line  of  argument,  and  if  left  till  Chapters  XI. 
and  XII.  have  been  read,  will  be  found  to  afford 
interesting  and  obvious  illustrations  of  the  properties 
discussed  in  the  later  chapters. 

My  thanks  are  due  to  the  friends  who  have  assisted 
me  in  designing  and  preparing  the  modifications  of 
my  previous  models,  and  in  no  small  degree  to  the 
publisher  of  this  volume,  Mr.  Sonnenschein,  to  whose 
unique  appreciation  of  the  line  of  thought  of  this,  as 
of  my  former  essays,  their  publication  is  owing.  By 
the  provision  of  a  coloured  plate,  in  addition  to  the  other 
illustrations,  he  has  added  greatly  to  the  convenience 
of  the  reader. 

C.  HOWARD  HINTON. 


CONTENTS 


CHAP. 


PAGE 


I.       FOUR-DIMENSIONAL    SPACE  .  ,  1 

II.     THE  ANALOGY  OF  A  PLANE  WORLD  ...         6 

III.  THE    SIGNIFICANCE     OF   A     FOUR-DIMENSIONAL 

EXISTENCE     .  .15 

IV.  THE  FIRST  CHAPTER  IN  THE  HISTORY  OF  FOUR 

SPACE    .....  .23 

V.     THE    SECOND    CHAPTER    IN    THK    HISTORY    OF 

FOUR  SPACE 41 

Lobatchewsky,  Bolyai,  and  Gauss 
Metageometry 

VI.  THE  HIGHER  WORLD        .         .        »  61 

VII.  THE  EVIDENCES  FOR  A  FOURTH  DIMENSION        .  76 

VIII.  THE  USE  OF  FOUR  DIMENSIONS  IN  THOUGHT    .  85 

IX.  APPLICATION  TO  KANT'S  THEORY  OF  EXPERIENCE  107 

X.       A    FoUR-DlMEXSIONAL    FlGURE  .  .122 

XI.     NOMENCLATURE  AND  ANALOGIES  136 


Vlll  CONTENTS 

CHAP.  PAGE 

XII.     THE  SIMPLEST  FOUR-DIMENSIONAL  SOLID  .         .  157 

XTII.     REMARKS  ON  THE  FIGURES        .        .         .  178 

XIV.     A    RECAPITULATION    AND    EXTENSION    OF    THE 

PHYSICAL  ARGUMENT 203 

APPENDIX   I.— THE  MODELS 231 

II. — A  LANGUAGE  OF  SPACE  248 


THE     FOURTH     DIMENSION 


CHAPTER    I 

POUR  DIMENSIONAL    SPACE 

THERE  is  nothing  more  indefinite,  and  at  the  same  time 
more  real,  than  that  which  we  indicate  when  we  speak 
of  the  "  higher."  In  our  social  life  we  see  it  evidenced 
in  a  greater  complexity  of  relations.  But  this  com- 
plexity is  not  all.  There  is,  at  the  same  time,  a  contact 
with,  an  apprehension  of,  something  more  fundamental, 
more  real. 

With  the  greater  development  of  man  there  comes 
a  consciousness  of  something  more  than  all  the  forms 
in  which  it  shows  itself.  There  is  a  readiness  to  give 
up  all  the  visible  and  tangible  for  the  sake  of  those 
principles  and  values  of  which  the  visible  and  tangible 
are  the  representation.  The  physical  life  of  civilised 
man  and  of  a  mere  savage  are  practically  the  same,  but 
the  civilised  man  has  discovered  a  depth  in  his  existence, 
which  makes  him  feel  that  that  which  appears  all  to 
the  savage  is  a  mere  externality  and  appurtenage  to  his 
true  being. 

Now,  this  higher — how  shall  we  apprehend  it  ?  It  is 
generally  embraced  by  our  religious  faculties,  by  our 
idealising  tendency.  But  the  higher  existence  has  two 
sides.  It  has  a  being  as  well  as  qualities.  And  in  trying 

1 


2  THE   FOURTH   DIMENSION 

to  realise  it  through  our  emotions  we  are  always  taking  the 
subjective  view.  Our  attention  is  always  fixed  on  what  we 
feel,  what  we  think.  Is  there  any  way  of  apprehending 
the  higher  after  the  purely  objective  method  of  a  natural 
science  ?  I  think  that  there  is. 

Plato,  in  a  wonderful  allegory,  speaks  of  some  men 
living  in  such  a  condition  that  they  were  practically 
reduced  to  be  the  denizens  of  a  shadow  world.  They 
were  chained,  and  perceived  but  the  shadows  of  them- 
selves and  all  real  objects  projected  on  a  wall,  towards 
which  their  faces  were  turned.  All  movements  to  them 
were  but  movements  on  the  surface,  all  shapes  but  the 
shapes  of  outlines  with  no  substantiality. 

Plato  uses  this  illustration  to  portray  the  relation 
between  true  being  and  the  illusions  .of  the  sense  world. 
He  says  that  just  as  a  man  liberated  from  his  chains 
could  learn  and  discover  that  the  world  was  solid  and 
real,  and  could  go  back  and  tell  his  bound  companions  of 
this  greater  higher  reality,  so  the  philosopher  who  has 
been  liberated,  who  has  gone  into  the  thought  of  the 
ideal  world,  into  the  world  of  ideas  greater  and  more 
real  than  the  things  of  sense,  can  come  and  tell  his  fellow 
men  of  that  which  is  more  true  than  the  visible  sun — 
more  noble  than  Athens,  the  visible  state. 

Now,  I  take  Plato's  suggestion  ;  but  literally,  not 
metaphorically.  He  imagines  a  world  which  is  lower 
than  this  world,  in  that  shadow  figures  and  shadow 
motions  are  its  constituents  ;  and  to  it  he  contrasts  the  real 
world.  As  the  real  world  is  to  this  shadow  world,  so  is  the 
higher  world  to  our  world.  I  accept  his  analogy.  As  our 
world  in  three  dimensions  is  to  a  shadow  or  plane  world, 
so  is  the  higher  world  to  our  three-dimensional  world. 
That  is,  the  higher  world  is  four-dimensional ;  the  higher 
being  is,  so  far  as  its  existence  is  concerned  apart  from  its 
qualities,  to  be  sought  through  the  conception  of  an  actual 


FOUR-UIMENSIONAL    SPACE  3 

existence  spatially  higher  than  that  which  we  realise  with 
our  senses. 

Here  you  will  observe  I  necessarily  leave  out  all  that 
gives  its  charm  and  interest  to  Plato's  writings.  All 
those  conceptions  of  the  beautiful  and  good  which  live 
immortally  in  his  pages. 

All  that  I  keep  from  his  great  storehouse  of  wealth  is 
this  one  thing  simply — a  world  spatially  higher  than  this 
world,  a  world  which  can  only  be  approached  through  the 
stocks  and  stones  of  it,  a  world  which  must  be  appre- 
hended laboriously,  patiently,  through  the  material  things 
of  it,  the  shapes,  the  movements,  the  figures  of  it. 

We  must  learn  to  realise  the  shapes  of  objects  in 
this  world  of  the  higher  man ;  we  must  become  familiar 
with  the  movements  that  objects  make  in  his  world,  so 
that  we  can  learn  something  about  his  daily  experience, 
his  thoughts  of  material  objects,  his  machinery. 

The  means  for  the  prosecution  of  this  enquiry  are  given 
in  the  conception  of  space  itself. 

It  often  happens  that  that  which  we  consider  to  be 
unique  and  unrelated  gives  us,  within  itself,  those  relations 
by  means  of  which  we  are  able  to  see  it  as  related  to 
others,  determining  and  determined  by  them. 

Thus,  on  the  earth  is  given  that  phenomenon  of  weight 
by  means  of  which  Newton  brought  the  earth  into  its 
true  relation  to  the  sun  and  other  planets.  Our  terrestrial 
globe  was  determined  in  regard  to  other  bodies  of  the 
solar  system  by  means  of  a  relation  which  subsisted  on 
the  earth  itself. 

And  so  space  itself  bears  within  it  relations  of  which 
we  can  determine  it  as  related  to  other  space.  For  within 
space  are  given  the  conceptions  of  point  and  line,  line  and 
plane,  which  really  involve  the  relation  of  space  to  a 
higher  space. 

Where  one  segment  of  a  straight  line  leaves  off  and 


4  THE   FOURTH    DIMENSION 

another  begins  is  a  point,  and  the  straight  line  itself  can 
be  generated  by  the  motion  of  the  point. 

One  portion  of  a  plane  is  bounded  from  another  by  a 
straight  line,  and  the  plane  itself  can  be  generated  by 
the  straight  line  moving  in  a  direction  not  contained 
in  itself. 

Again,  two  portions  of  solid  space  are  limited  with 
regard  to  each  other  by  a  plane ;  and  the  plane,  moving 
in  a  direction  not  contained  in  itself,  can  generate  solid 
space. 

Thus,  going  on,  we  may  say  that  space  is  that  which 
limits  two  portions  of  higher  space  from  each  other,  and 
that  our  space  will  generate  the  higher  space  by  moving 
in  a  direction  not  contained  in  itself. 

Another  indication  of  the  nature  of  four-dimensional 
space  can  be  gained  by  considering  the  problem  of  the 
arrangement  of  objects. 

If  I  have  a  number  of  swords  of  varying  degrees  of 
brightness,  I  can  represent  them  in  respect  of  this  quality 
by  points  arranged  along  a  straight  line. 

If  I  place  a  sword  at  A,  fig.  1,  and  regard  it  as  having 
a  certain  brightness,  then  the  other  swords 

can  be   arranged   in   a   series   along  the 

line,   as    at   A,    B,    c,    etc.,   according    to 
their  degrees  of  brightness. 

If  now  I  take  account  of  another  quality,  say  length, 
thev  can  be  arranged  in  a  plane.      Starting  from  A,  B,  c,  I 
can  find    points  to   represent   different 
E     degrees  of  length  along   such  lines   as 
I  AF,  BD,  CE,  drawn  from  A  and  B  and  C. 

— ' '       Points  on  these  lines  represent  different 

*  *• 2i  degrees  of  length  with  the  same  degree  of 

brightness.  Thus  the  whole  plane  is  occupied  by  points 
representing  all  conceivable  varieties  of  brightness  and 
length. 


FOUK-DIMKNSIONAL    SPACE  5 

Bringing  in  a  third  quality,  say  sharpness,  I  can  draw, 
as  in  fig.  3,  any  number  of  upright 
lines.  Let  distances  along  these 
upright  lines  represent  degrees  of 
sharpness,  thus  the  points  F  and  G 
will  represent  swords  of  certain 
definite  degrees  of  the  three  qualities 
mentioned,  and  the  whole  of  space  will  serve  to  represent 
all  conceivable  degrees  of  these  three  qualities. 

If  now  I  bring  in  a  fourth  quality,  such  as  weight,  and 
try  to  find  a  means  of  representing  it  as  I  did  the  other 
three  qualities,  I  find  a  difficulty.  Every  point  in  space  is 
taken  up  by  some  conceivable  combination  of  the  three 
qualities  already  taken. 

To  represent  four  qualities  in  the  same  way  as  that  in 
which  I  have  represented  three,  I  should  need  another 
dimension  of  space. 

Thus  we  may  indicate  the  nature  of  four-dimensional 
space  by  saying  that  it  is  a  kind  of  space  which  would 
give  positions  representative  of  four  qualities,  as  three- 
dimensional  space  gives  positions  representative  of  three 
qualities. 


CHAPTER   II 
THE    ANALOGY    OF    A    PLANE    WORLD 

AT  the  risk  of  some  prolixity  I  will  go  fully  into  the 
experience  of  a  hypothetical  creature  confined  to  motion 
on  a  plane  surface.  By  so  doing  I  shall  obtain  an  analogy 
which  will  serve  in  our  subsequent  enquiries,  because  the 
change  in  our  conception,  which  we  make  in  passing  from 
the  shapes  and  motions  in  two  dimensions  to  those  in 
three,  affords  a  pattern  by  which  we  can  pass  on  still 
further  to  the  conception  of  an  existence  in  four-dimensional 
space. 

A  piece  of  paper  on  a  smooth  table  affords  a  ready 
image  of  a  two-dimensional  existence.  If  we  suppose  the 
being  represented  by  the  piece  of  paper  to  have  no 
knowledge  of  the  thickness  by  which  he  projects  above  the 
surface  of  the  table,  it  is  obvious  that  he  can  have  no 
knowledge  of  objects  of  a  similar  desciiption,  except  by 
the  contact  with  their  edges.  His  body  and  the  objects 
in  his  world  have  a  thickness  of  which  however,  he  has  no 
consciousness.  Since  the  direction  stretching  up  from 
the  table  is  unknown  to  him  he  will  think  of  the  objects 
of  his  world  as  extending  in  two  dimensions  only.  Figures 
are  to  him  completely  bounded  by  their  lines,  just  as  solid 
objects  are  to  us  by  their  surfaces.  He  cannot  conceive 
of  approaching  the  centre  of  a  circle,  except  by  breaking 
through  the  circumference,  for  the  circumference  encloses 
the  centre  in  the  directions  in  which  motion  is  possible  to 


THE    ANALOGY    OF   A    PLANE    WOELD  7 

him.  The  plane  surface  over  which  he  slips  and  with 
which  he  is  always  in  contact  will  be  unknown  to  him ; 
there  are  no  differences  by  which  he  can  recognise  its 
existence. 

But  for  the  purposes  of  our  analogy  this  representation 
is  deficient. 

A  being  as  thus  described  has  nothing  about  him  to 
push  off  from,  the  surface  over  which  he  slips  affords  no 
means  by  which  he  can  move  in  one  direction  rather  than 
another.  Placed  on  a  surface  over  which  he  slips  freely, 
he  is  in  a  condition  analogous  to  that  in  which  we  should 
be  if  we  were  suspended  free  in  space.  There  is  nothing 
which  he  can  push  off  from  in  any  direction  known  to  him. 

Let  us  therefore  modify  our  representation.  Le't  us 
suppose  a  vertical  plane  against  which  particles  of  thin 
matter  slip,  never  leaving  the  surface.  Let  these  particles 
possess  an  attractive  force  and  cohere  together  into  a  disk ; 
this  disk  will  represent  the  globe  of  a  plane  being.  He 
must  be  conceived  as  existing  on  the  rim. 

Let  1  represent  this  vertical  disk  of  flat  matter  and  2 
the  plane  being  on  it,  standing  upon  its 
rim  as  we  stand  on  the  surface  of  our  earth. 
The  direction  of  the  attractive  force  of  his 
matter  will  give  the  creature  a  knowledge 
of  up  and  down,  determining  for  him  one 
direction  in  his  plane  space.  Also,  since 
Fig.  4.  he  can  move  along  the  surface  of  his  earth, 
he  will  have  the  sense  of  a  direction  parallel  to  its  surface, 
which  we  may  call  forwards  and  backwards. 

He  will  have  no  sense  of  right  and  left — that  is,  of  the 
direction  which  we  recognise  as  extending  out  from  the 
plane  to  our  right  and  left. 

The  distinction  of  right  and  left  is  the  one  that  we 
must  suppose  to  be  absent,  in  order  to  project  ourselves, 
into  the  condition  of  a  plane  being. 


8  THE    FOURTH    DIMENSION 

Let  the  reader  imagine  himself,  as  he  looks  along  the 
plane,  fig.  4,  to  become  more  and  more  identified  with 
the  thin  body  on  it,  till  he  finally  looks  along  parallel  to 
the  surface  of  the  plane  earth,  and  up  and  down,  losing 
the  sense  of  the  direction  which  stretches  right  and  left. 
This  direction  will  be  an  unknown  dimension  to  him. 

Our  space  conceptions  are  so  intimately  connected  with 
those  which  we  derive  from  the  existence  of  gravitation 
that  it  is  difficult  to  realise  the  condition  of  a  plane  being, 
without  picturing  him  as  in  material  surroundings  with 
a  definite  direction  of  up  and  down.  Hence  the  necessity 
of  our  somewhat  elaborate  scheme  of  representation,  which, 
when  its  import  has  been  grasped,  can  be  dispensed  with 
for  the  simpler  one  of  a  thin  object  slipping  over  a 
smooth  surface,  which  lies  in  front  of  us. 

It  is  obvious  that  we  must  suppose  some  means  by 
which  the  plane  being  is  kept  in  contact  with  the  surface 
on  which  he  slips.  The  simplest  supposition  to  make  is 
that  there  is  a  transverse  gravity,  which  keeps  him  to  the 
plane.  This  gravity  must  be  thought  of  as  different  to 
the  attraction  exercised  by  his  matter,  and  as  unperceived 
by  him. 

At  this  stage  of  our  enquiry  I  do  not  wish  to  enter 
into  the  question  of  how  a  plane  being  could  arrive  at 
a  knowledge  of  the  third  dimension,  but  simply  to  in- 
vestigate his  plane  consciousness. 

It  is  obvious  that  the  existence  of  a  plane  being  must 
be  very  limited.  A  straight  line  standing  up  from  the 
surface  of  his  earth  affords  a  bar  to  his  progress.  An 
object  like  a  wheel  which  rotates  round  an  axis  would 
be  unknown  to  him,  for  there  is  no  conceivable  way  in 
which  he  can  get  to  the  centre  without  going  through 
the  circumference.  He  would  have  spinning  disks,  but 
could  not  get  to  the  centre  of  them.  The  plane  being 
can  represent  the  motion  from  any  one  point  of  his  space 


THE    ANALOGY    OF   A    PLANE    WORLD 


(J 


to  any  other,  by  means  of  two  straight  lines  drawn  at 
right  angles  to  each  other. 

Let  AX  and  AY  be  two  such  axes.     He  can  accomplish 
the  translation  from  A  to  B  by  going  along  AX  to  C,  and 
then  from  c  along  CB  parallel  to  AY. 

The  same  result  can  of  course  be  obtained 
by  moving  to  D  along  AY  and  then  parallel 
to  AX  from  D  to  B,  or  of  course  by  any 
diagonal  movement  compounded  by  these 
axial  movements. 

By  means  of  movements  parallel  to 
these  two  axes  he  can  proceed  (except  for 


C 


rig.  5. 


material  obstacles)  from  any  one  point  of  his   space   to 
any  other. 

If  now  we  suppose  a  third  line  drawn 
out  from  A  at  right  angles  to  the  plane 
it  is  evident  that  no  motion  in  either 
of  the  two  dimensions  he  knows  will 
carry  him  in  the  least  degree  in  the 

^"^^ Z    direction  represented  by  A  z. 

Fig.  6.  The   lines  AZ   and  AX  determine  a 

plane.  If  he  could  be  taken  off  his  plane,  and  trans- 
ferred to  the  plane  AXZ,  he  would  be  in  a  world  exactly 
like  his  own.  From  every  line  in  his 
world  there  goes  off  a  space  world  exactly 
like  his  own. 

From  every  point  in  his  world  a  line  can 
be  drawn  parallel  to  AZ  in  the  direction 
unknown  to  him.  If  we  suppose  the  square 
in  fig.  7  to  be  a  geometrical  square  from 
every  point  of  it,  inside  as  well  as  on  the 
contour,  a  straight  line  can  be  drawn  parallel 
to  AZ.  The  assemblage  of  these  lines  constitute  a  solid 
figure,  of  which  the  square  in  the  plane  is  the  base.  If 
we  consider  the  square  to  represent  an  object  in  the  plane 


Fig.  7. 


10  THE   FOURTH   DIMENSION 

being's  world  then  we  must  attribute  to  it  a  very  small 
thickness,  for  every  real  thing  must  possess  all  three 
dimensions.  This  thickness  he  does  not  preceive,  but 
thinks  of  this  real  object  as  a  geometrical  square.  He 
thinks  of  it  as  possessing  area  only,  and  no  degree  of 
solidity.  The  edges  which  project  from  the  plane  to  a 
very  small  extent  he  thinks  of  as  having  merely  length 
and  no  breadth — as  being,  in  fact,  geometrical  lines. 

With  the  first  step  in  the  apprehension  of  a  third 
dimension  there  would  come  to  a  plane  being  the  con- 
viction that  he  had  previously  formed  a  wrong  conception 
of  the  nature  of  his  material  objects.  He  had  conceived 
them  as  geometrical  figures  of  two  dimensions  only. 
If  a  third  dimension  exists,  such  figures  are  incapable 
of  real  existence.  Thus  he  would  admit  that  all  his  real 
objects  had  a  certain,  though  very  small  thickness  in  the 
unknown  dimension,  and  that  the  conditions  of  his 
existence  demanded  the  supposition  of  an  extended  sheet 
of  matter,  from  contact  with  which  in  their  motion  his 
objects  never  diverge. 

Analogous  conceptions  must  be  formed  by  us  on  the 
supposition  of  a  four-dimensional  existence.  We  must 
suppose  a  direction  in  which  we  can  never  point  extending 
from  every  point  of  our  space.  We  must  draw  a  dis- 
tinction between  a  geometrical  cube  and  a  cube  of  real 
matter.  The  cube  of  real  matter  we  must  suppose  to 
have  an  extension  in  an  unknown  direction,  real,  but  so 
small  as  to  be  imperceptible  by  us.  From  every  point 
of  a  cube,  interior  as  well  as  exterior,  we  must  imagine 
that  it  is  possible  to  draw  a  line  in  the  unknown  direction. 
The  assemblage  of  these  lines  would  constitute  a  higher 
.'olid.  The  lines  going  off  in  the  unknown  direction  from 
the  face  of  a  cube  would  constitute  a  cube  starting  from 
that  face.  Of  this  cube  all  that  we  should  see  in  our 
space  would  be  the  face 


THE    ANALOGY    OF   A    PLANE    WORLD  11 

Again,  just  as  the  plane  being  can  represent  any 
motion  in  his  space  by  two  axes,  so  we  can  represent  any 
motion  in  our  three-dimensional  space  by  means  of  three 
axes.  There  is  no  point  in  our  space  to  which  we  cannot 
move  by  some  combination  of  movements  on  the  directions 
marked  out  by  these  axes. 

On  the  assumption  of  a  fourth  dimension  we  have 
to  suppose  a  fourth  axis,  which  we  will  call  AW.  It  must 
be  supposed  to  be  at  right  angles  to  each  and  every 
one  of  the  three  axes  AX,  AY,  AZ.  Just  as  the  two  axes, 
AX,  AZ,  determine  a  plane  which  is  similar  to  the  original 
plane  on  which  we  supposed  the  plane  being  to  exist,  but 
which  runs  off  from  it,  and  only  meets  it  in  a  line ;  so  in 
our  space  if  we  take  any  three  axes  such  as  AX,  AY,  and 
AW,  they  determine  a  space  like  our  space  world.  This 
space  runs  off  from  our  space,  and  if  we  were  transferred 
to  it  we  should  find  ourselves  in  a  space  exactly  similar  to 
our  own. 

We  must  give  up  any  attempt  to  picture  this  space  in 
its  relation  to  ours,  just  as  a  plane  being  would  have  to 
give  up  any  attempt  to  picture  a  plane  at  right  angles 
to  his  plane. 

Such  a  space  and  ours  run  in  different  directions  from 
the  plane  of  AX  and  AY.  They  meet  in  this  plane  but 
have  nothing  else  in  common,  just  as  the  plane  space 
of  AX  and  AY  and  that  of  AX  and  AZ  run  in  different 
directions  and  have  but  the  line  AX  in  common. 

Omitting  all  discussion  of  the  manner  on  which  a  plane 
being  might  be  conceived  to  form  a  theory  of  a  three- 
dimensional  existence,  let  us  examine  how,  with  the  means 
at  his  disposal,  he  could  represent  the  properties  of  three- 
dimensional  objects. 

There  are  two  ways  in  which  the  plane  being  can  think 
of  one  of  our  solid  bodies.  He  can  think  of  the  cube, 
fig.  8,  as  composed  of  a  number  of  sections  parallel  to 


12 


THE    FOURTH    DIMENSION 


Fig.  8. 


his   plane,   each   lying   in  the   third  dimension  a  little 

further  off  from  his  plane  than 
the  preceding  one.  These  sec- 
tions he  can  represent  as  a 
series  of  plane  figures  lying  in 
his  plane,  but  in  so  representing 
them  he  destroys  the  coherence 
of  them  in  the  higher  figure. 
The  set  of  squares,  A,  B,  c,  D, 
represents  the  section  parallel 
to  the  plane  of  the  cube  shown  in  figure,  but  they  are 
not  in  their  proper  relative  positions. 

The  plane  being  can  trace  out  a  movement  in  the  third 
dimension  by  assuming  discontinuous  leaps  from  one 
section  to  another.  Thus,  a  motion  along  the  edge  of 
the  cube  from  left  to  right  would  be  represented  in  the 
set  of  sections  in  the  plane  as  the  succession  of  the 
corners  of  the  sections  A,  B,  c,  D.  A  point  moving  from 
A  through  BCD  in  our  space  must  be  represented  in  the 
plane  as  appearing  in  A,  then  in  B,  and  so  on,  without 
passing  through  the  intervening  plane  space. 

In  these  sections  the  plane  being  leaves  out,  of  course, 
the  extension  in  the  third  dimension  ;  the  distance  between 
any  two  sections  is  not  represented.     In  order  to  realise 
this  distance  the  conception  of  motion  can  be  employed. 
Let  fig.  9  represent  a  cube  passing  transverse  to  the 
plane.     It  will  appear  to  the  plane  being  as  a 
square   object,   but   the   matter   of   which   this 
^|     object  is  composed  will  be  continually  altering. 

{. I     One  material  particle  takes  the  place  of  another, 

but  it  does  not  come  from  anywhere  or  go 
anywhere  in  the  space  which  the  plane  being 
knows. 

The  analogous  manner  of  representing  a  higher  solid  in 
our  case,  is  to  conceive  it  as  composed  of  a  number  of 


TUE  ANALOGY  OF  A  PLANE  WORLD 


13 


sections,  each  lying  a  little  farther  off  in  the  unknown 

direction  than  the  preceding. 

We  can  represent  these  sections  as  a  number  of  solids. 

Thus  the  cubes  A,  B,  c,  D, 
may  be  considered  as 
the  sections  at  different 


B 


C 

Fig.  1U 


O  intervals  in  the  unknown 

dimension  of  a  higher 
cube.  Arranged  thus  their  coherence  in  the  higher  figure 
is  destroyed,  they  are  mere  representations. 

A  motion  in  the  fourth  dimension  from  A  through  B,  c, 
etc.,  would  be  continuous,  but  we  can  only  represent  it  as 
the  occupation  of  the  positions  A,  B,  c,  etc.,  in  succession. 
We  can  exhibit  the  results  of  the  motion  at  different 
stages,  but  no  more. 

In  this  representation  we  have  left  out  the  distance 
between  one  section  and  another ;  we  have  considered  the 
higher  body  merely  as  a  series  of  sections,  and  so  left  out 
its  contents.  The  only  way  to  exhibit  its  contents  is  to 
call  in  the  aid  of  the  conception  of  motion. 

If  a  higher  cube  passes  transverse  to  our  space,  it  will 
appear  as  a  cube  isolated  in  space,  the  part 
that  has  not  come  into  our  space  and  the  part 
that  has  passed  through  will  not  be  visible. 
The  gradual  passing  through  our  space  would 
appear  as  the  change  of  the  matter  of  the  cube 


Fig.  11. 


before  us.  One  material  particle  in  it  is  succeeded  by 
another,  neither  coming  nor  going  in  any  direction  we  can 
point  to.  In  this  manner,  by  the  duration  of  the  figure, 
we  can  exhibit  the  higher  dimensionality  of  it ;  a  cube  of 
our  matter,  under  the  circumstances  supposed,  namely, 
that  it  has  a  motion  transverse  to  our  space,  would  instantly 
disappear.  A  higher  cube  would  last  till  it  had  passed 
transverse  to  our  space  by  its  whole  distance  of  extension 
in  the  fourth  dimension. 


14 


THE    FOURTH    DIMENSION 


As  the  plane  being  can  think  of  the  cube  as  consisting 
of  sections,  each  like  a  figure  he  knows,  extending  away 
from  his  plane,  so  we  can  think  of  a  higher  solid  as  com- 
posed of  sections,  each  like  a  solid  which  we  know,  but 
extending  away  from  our  space. 

Thus,  taking  a  higher  cube,  we  can  look  on  it  as 
starting  from  a  cube  in  our  space  and  extending  in  the 
unknown  dimension. 

Take  the  face  A  and  conceive  it  to  exist  as  simply  a 


Fig.  12. 

face,  a  square  with  no  thickness.  From  this  face  the 
cube  in  our  space  extends  by  the  occupation  of  space 
which  we  can  see. 

But  from  this  face  there  extends  equally  a  cube  in  the 
unknown  dimension.  We  can  think  of  the  higher  cube, 
then,  by  taking  the  set  of  sections  A,  B,  c,  D,  etc.,  and 
considering  that  from  each  of  them  there  runs  a  cube. 
These  cubes  have  nothing  in  common  with  each  other, 
and  of  each  of  them  in  its  actual  position  all  that  we  can 
have  in  our  space  is  an  isolated  square.  It  is  obvious  that 
we  can  take  our  series  of  sections  in  any  manner  we 
please.  We  can  take  them  parallel,  for  instance,  to  any 
one  of  the  three  isolated  faces  shown  in  the  figure. 
Corresponding  to  the  three  series  of  sections  at  right 
angles  to  each  other,  which  we  can  make  of  the  cube 
in  space,  we  must  conceive  of  the  higher  cube,  as  com- 
posed of  cubes  starting  from  squares  parallel  to  the  faces 
of  the  cube,  and  of  these  cubes  all  that  exist  in  our  space 
are  the  isolated  squares  from  which  they  start. 


CHAPTER   III 

THE   SIGNIFICANCE   OF  A  FOUR- 
DIMENSIONAL  EXISTENCE 

HAVING  now  obtained  the  conception  of  a  four-dimensional 
space,  and  having  formed  the  analogy  which,  without 
any  further  geometrical  difficulties,  enables  us  to  enquire 
into  its  properties,  I  will  refer  the  reader,  whose  interest 
is  principally  in  the  mechanical  aspect,  to  Chapters  VI. 
and  VII.  In  the  present  chapter  I  will  deal  with  the 
general  significance  of  the  enquiry,  and  in  the  next 
with  the  historical  origin  of  the  idea. 

First,  with  regard  to  the  question  of  whether  there 
is  any  evidence  that  we  are  really  in  four-dimensional 
space,  I  will  go  back  to  the  analogy  of  the  plane  world. 

A  being  in  a  plane  world  could  not  have  any  ex- 
perience of  three-dimensional  shapes,  but  he  could  have 
an  experience  of  three-dimensional  movements. 

We  have  seen  that  his  matter  must  be  supposed  to 
have  an  extension,  though  a  very  small  one,  in  the  third 
dimension.  And  thus,  in  the  small  particles  of  his 
matter,  three-dimensional  movements  may  well  be  con- 
ceived to  take  place.  Of  these  movements  he  would  only 
perceive  the  resultants.  Since  all  movements  of  an 
observable  size  in  the  plane  world  are  two-dimensional, 
he  would  only  perceive  the  resultants  in  two  dimensions 
of  the  small  three-dimensional  movements.  Thus,  there 
would  be  phenomena  which  he  could  not  explain  by  his 


15 


16  THE   FOURTH   DIMENSION 

theory  of  mechanics — motions  would  take  place  which 
he  could  not  explain  by  his  theory  of  motion.  Hence, 
to  determine  if  we  are  in  a  four-dimensional  world,  we 
must  examine  the  phenomena  of  motion  in  our  space. 
If  movements  occur  which  are  not  explicable  on  the  sup- 
positions of  our  three-dimensional  mechanics,  we  should 
have  an  indication  of  a  possible  four-dimensional  motion, 
and  if,  moreover,  it  could  be  shown  that  such  movements 
would  be  a  consequence  of  a  four-dimensional  motion  in 
the  minute  particles  of  bodies  or  of  the  el  her,  we  should 
have  a  strong  presumption  in  favour  of  the  reality  of 
the  fourth  dimension. 

By  proceeding  in  the  direction  of  finer  r.nd  finer  sub- 
division, we  come  to  forms  of  matter  possessing  properties 
different  to  those  of  the  larger  masses.  It  is  probable  that 
at  some  stage  in  this  process  we  should  come  to  a  form 
of  matter  of  such  minute  subdivision  that  its  particles 
possess  a  freedom  of  movement  in  four  dimensions.  This 
form  of  matter  I  speak  of  as  four-dimensional  ether,  and 
attribute  to  it  properties  approximating  to  those  of  a 
perfect  liquid. 

Deferring  the  detailed  discussion  of  this  form  of  matter 
to  Chapter  VI.,  we  will  now  examine  the  means  by  which 
a  plane  being  would  come  to  the  conclusion  that  three- 
dimensional  movements  existed  in  his  world,  and  point 
out  the  analogy  by  which  we  can  conclude  the  existence 
of  four-dimensional  movements  in  our  world.  Since  the 
dimensions  of  the  matter  in  his  world  are  small  in  the 
third  direction,  the  phenomena  in  which  he  would  detect 
the  motion  would  be  those  of  the  small  particles  of 
matter. 

Suppose  that  there  is  a  ring  in  his  plane.  We  can 
imagine  currents  flowing  round  the  ring  in  either  of  two 
opposite  directions.  These  would  produce  unlike  effects, 
and  give  rise  to  two  different  fields  of  influence.  If  the 


THE    SIGNIFICANCE   OF  A  FOUR-DIMENSIONAL   EXISTENCE  17 

ring  with  a  current  in  it  in  one  direction  be  taken  up 
and  turned  over,  and  put  down  again  on  the  plane,  it 
would  be  identical  with  the  ring  having  a  current  in  the 
opposite  direction.  An  operation  of  this  kind  would  be 
impossible  to  the  plane  being.  Hence  he  would  have 
in  his  space  two  irreconcilable  objects,  namely,  the  two 
fields  of  influence  due  to  the  two  rings  with  currents  in 
them  in  opposite  directions.  By  irreconcilable  objects 
in  the  plane  I  mean  objects  which  cannot  be  thought 
of  as  transformed  one  into  the  other  by  any  movement 
in  the  plane. 

Instead  of  currents  flowing  in  the  rings  we  can  imagine 
a  different  kind  of  current.  Imagine  a  number  of  small 
rings  strung  on  the  original  ring.  A  current  round  these 
secondary  rings  would  give  two  varieties  of  effect,  or  two 
different  fields  of  influence,  according  to  its  direction. 
These  two  varieties  of  current  could  be  turned  one  into 
the  other  by  taking  one  of  the  rings  up,  turning  it  over, 
and  putting  it  down  again  in  the  plane.  This  operation 
is  impossible  to  the  plane  being,  hence  in  this  case  also 
there  would  be  two  irreconcilable  fields  in  the  plane. 
Now,  if  the  plane  being  found  two  such  irreconcilable 
fields  and  could  prove  that  they  could  not  be  accounted 
for  by  currents  in  the  rings,  he  would  have  to  admit  the 
existence  of  currents  round  the  rings — that  is,  in  rings 
strung  on  the  primary  ring.  Thus  he  would  come  to 
admit  the  existence  of  a  three-dimensional  motion,  for 
such  a  disposition  of  currents  is  in  three  dimensions. 

Now  in  our  space  there  are  two  fields  of  different 
properties,  which  can  be  produced  by  an  electric  current 
flowing  in  a  closed  circuit  or  ring.  These  two  fields  can 
be  changed  one  into  the  other  by  reversing  the  currents,  but 
they  cannot  be  changed  one  into  the  other  by  any  turning 
about  of  the  rings  in  our  space  ;  for  the  disposition  of  the 
field  with  regard  to  the  ring  itself  is  different  when  we 

2 


18  THE    FOURTH   DIMENSION 

turn  the  ring,  over  and  when  we  reverse  the  direction  of 
the  current  in  the  ring. 

As  hypotheses  to  explain  the  differences  of  these  two 
fields  and  their  effects  we  can  suppose  the  following  kinds 
of  space  motions  : — First,  a  current  along  the  conductor  ; 
second,  a  current  round  the  conductor — that  is,  of  rings  of 
currents  strung  on  the  conductor  as  an  axis.  Neither  of 
these  suppositions  accounts  for  facts  of  observation. 

Hence  we  have  to  make  the  supposition  of  a  four- 
dimensional  motion.  We  find  that  a  four-dimensional 
rotation  of  the  nature  explained  in  a  subsequent  chapter, 
has  the  following  characteristics  : — First,  it  would  give  us 
two  fields  of  influence,  the  one  of  which  could  be  turned 
into  the  other  by  taking  the  circuit  up  into  the  fourth 
dimension,  turning  it  over,  and  putting  it  down  in  our 
space  again,  precisely  as  the  two  kinds  of  fields  in  the 
plane  could  be  turned  one  into  the  other  by  a  reversal  of 
the  current  in  our  space.  Second,  it  involves  a  phenome- 
non precisely  identical  with  that  most  remarkable  and 
mysterious  feature  of  an  electric  current,  namely  that  it 
is  a  field  of  action,  the  rim  of  which  necessarily  abuts  on  a 
continuous  boundary  formed  by  a  conductor.  Hence,  on 
the  assumption  of  a  four-dimensional  movement  in  the 
region  of  the  minute  particles  of  matter,  we  should  expect 
to  find  a  motion  analogous  to  electricity. 

Now,  a  phenomenon  of  such  universal  occurrence  as 
electricity  cannot  be  due  to  matter  and  motion  in  any 
very  complex  relation,  but  ought  to  be  seen  as  a  simple 
and  natural  consequence  of  their  properties.  I  infer  that 
the  difficulty  in  its  theory  is  due  to  the  attempt  to  explain 
a  four-dimensional  phenomenon  by  a  three-dimensional 
geometry. 

In  view  of  this  piece  of  evidence  we  cannot  disregard 
that  afforded  by  the  existence  of  symmetry.  In  this 
connection  I  will  allude  to  the  simple  way  of  producing 


THE    SIGNIFICANCE    OF  A   FOUR-DIMENSIONAL    EXISTENCE  19 

the  images  of  insects,  sometimes  practised  by  children. 
They  put  a  few  blots  of  ink  in  a  straight  line  on  a  piece  of 
paper,  fold  the  paper  along  the  blots,  and  on  opening  it  the 
lifelike  presentment  of  an  insect  is  obtained.  If  we  were 
to  find  a  multitude  of  these  figures,  we  should  conclude 
that  they  had  originated  from  a  process  of  folding  over ; 
the  chances  against  this  kind  of  reduplication  of  parts 
is  too  great  to  admit  of  the  assumption  that  they  had 
been  formed  in  any  other  way. 

The  production  of  the  symmetrical  forms  of  organised 
beings,  though  not  of  course  due  to  a  turning  over  of 
bodies  of  any  appreciable  size  in  four-dimensional  space, 
can  well  be  imagined  as  due  to  a  disposition  in  that 
manner  of  the  smallest  living  particles  from  which  they 
are  built  up.  Thus,  not  only  electricity,  but  life,  and  the 
processes  by  which  we  think  and  feel,1must  be  attributed 
to  that  region  of  magnitude  in  which  four-dimensional 
movements  take  place. 

I  do  not  mean,  however,  that  life  can  be  explained  as  a 
four-dimensional  movement.  It  seems  to  me  that  the 
whole  bias  of  thought,  which  tends  to  explain  the 
phenomena  of  life  and  volition,  as  due  to  matter  and 
motion  in  some  pecuHar  relation,  is  adopted  rather  in  the 
interests  of  the  explicability  of  things  than  with  any 
regard  to  probability. 

Of  course,  if  we  could  show  that  life  were  a  phenomenon 
of  motion,  we  should  be  able  to  explain  a  great  deal  that  is 
at  present  obscure.  But  there  are  two  great  difficulties  in 
the  way.  It  would  be  necessary  to  show  that  in  a  germ 
capable  of  developing  into  a  living  being,  there  were 
modifications  of  structure  capable  of  determining  in  the 
developed  germ  all  the  characteristics  of  its  form,  and  not 
only  this,  but  of  determining  those  of  all  the  descendants 
of  such  a  form  in  an  infinite  series.  Such  a  complexity  of 
mechanical  relations,  undeniable  though  it  be,  cannot 


20  THE    FOURTH   DIMENSION 

surely  be  the  best  way  of  grouping  the  phenomena  and 
giving  a  practical  account  of  them.  And  another  difficulty 
is  this,  that  no  amount  of  mechanical  adaptation  would 
give  that  element  of  consciousness  which  we  possess,  and 
which  is  shared  in  to  a  modified  degree  by  the  animal 
world. 

In  those  complex  structures  which  men  build  up  and 
direct,  such  as  a  ship  or  a  railway  train  (and  which,  if  seen 
by  an  observer  of  such  a  size  that  the  men  guiding  them 
were  invisible,  would  seem  to  present  some  of  the 
phenomena  of  life)  the  appearance  of  animation  is  not 
due  to  any  diffusion  of  life  in  the  material  parts  of  the 
structure,  but  to  the  presence  of  a  living  being. 

The  old  hypothesis  of  a  soul,  a  living  organism  within 
the  visible  one,  appears  to  me  much  more  rational  than  the 
attempt  to  explain  life  as  a  form  of  motion.  And  when  we 
consider  the  region  of  extreme  minuteness  characterised 
by  four-dimensional  motion  the  difficulty  of  conceiving 
such  an  organism  alongside  the  bodily  one  disappears. 
Lord  Kelvin  supposes  that  matter  is  formed  from  the 
ether.  We  may  very  well  suppose  that  the  living 
organisms  directing  the  material  ones  are  co-ordinate 
with  them,  not  composed  of  matter,  but  consisting  of 
etherial  bodies,  and  as  such  capable  of  motion  through 
the  ether,  and  able  to  originate  material  living  bodies 
throughout  the  mineral. 

Hypotheses  such  as  these  find  no  immediate  ground  for 
proof  or  disproof  in  the  physical  world.  Let  us,  therefore, 
turn  to  a  different  field,  and,  assuming  that  the  human 
soul  is  a  four-dimensional  being,  capable  in  itself  of  four 
dimensional  movements,  but  in  its  experiences  through 
the  senses  limited  to  three  dimensions,  ask  if  the  history 
of  thought,  of  these  productivities  which  characterise  man, 
correspond  to  our  assumption.  Let  us  pass  in  review 
those  steps  by  which  man,  presumably  a  four-dimensional 


THE    SIGNIFICANCE    OF  A  FOUR-DIMENSIONAL    EXISTENCE  21 

being,  despite  his  bodily  environment,  has  come  to  recog- 
nise the  fact  of  four-dimensional  existence. 

Deferring  this  enquiry  to  another  chapter,  I  will  here 
recapitulate  the  argument  in  order  to  show  that  our 
purpose  is  entirely  practical  and  independent  of  any 
philosophical  or  metaphysical  considerations. 

If  two  shots  are  fired  at  a  target,  and  the  second  bullet 
hits  it  at  a  different  place  to  the  first,  we  suppose  that 
there  was  some  difference  in  the  conditions  under  which 
the  second  shot  was  fired  from  those  affecting  the  first 
shot.  The  force  of  the  powder,  the  direction  of  aim,  the 
strength  of  the  wind,  or  some  condition  must  have  been 
different  in  the  second  case,  if  the  course  of  the  bullet  was 
not  exactly  the  same  as  in  the  first  case.  Corresponding 
to  every  difference  in  a  result  there  must  be  some  differ- 
ence in  the  antecedent  material  conditions.  By  tracing 
out  this  chain  of  relations  we  explain  nature. 

But  there  is  also  another  mode  of  explanation  which  we 
apply.  If  we  ask  what  was  the  cause  that  a  certain  ship 
was  built,  or  that  a  certain  structure  was  erected,  we  might 
proceed  to  investigate  the  changes  in  the  brain  cells  of 
the  men  who  designed  the  works.  Every  variation  in  one 
ship  or  building  from  another  ship  or  building  is  accom- 
panied by  a  variation  in  the  processes  that  go  on  in  the 
brain  matter  of  the  designers.  But  practically  this  would 
be  a  very  long  task. 

A  more  effective  mode  of  explaining  the  production  of 
the  ship  or  building  would  be  to  enquire  into  the  motives, 
plans,  and  aims  of  the  men  who  constructed  them.  We 
obtain  a  cumulative  and  consistent  body  of  knowledge 
much  more  easily  and  effectively  in  the  latter  way. 

Sometimes  we  apply  the  one,  sometimes  the  other 
mode  of  explanation. 

But  it  must  be  observed  that  the  method  of  explana- 
tion founded  on  aim,  purpose,  volition,  always  presupposes 


22  THE    FOURTH    DIMENSION 

a  mechanical  system  on  which  the  volition  and  aim 
works.  The  conception  of  man  as  willing  and  acting 
from  motives  involves  that  of  a  number  of  uniform  pro- 
cesses of  nature  which  he  can  modify,  and  of  which  he 
can  make  application.  In  the  mechanical  conditions  of 
the  three-dimensional  world,  the  only  volitional  agency 
which  we  can  demonstrate  is  the  human  agency.  But 
when  we  consider  the  four-dimensional  world  the 
conclusion  remains  perfectly  open. 

The  method  of  explanation  founded  on  purpose  and  aim 
does  not,  surely,  suddenly  begin  with  man  and  end  with 
him.  There  is  as  much  behind  the  exhibition  of  will  and 
motive  which  we  see  in  man  as  there  is  behind  the 
phenomena  of  movement ;  they  are  co-ordinate,  neither 
to  be  resolved  into  the  other.  And  the  commencement 
of  the  investigation  of  that  will  and  motive  which  lies 
behind  the  will  and  motive  manifested  in  the  three- 
dimensional  mechanical  field  is  in  the  conception  of  a 
soul — a  four-dimensional  organism,  which  expresses  its 
higher  physical  being  in  the  symmetry  of  the  body,  and 
gives  the  aims  and  motives  of  human  existence. 

Our  primary  task  is  to  form  a  systematic  knowledge  of 
the  phenomena  of  a  four-dimensional  world  and  find  those 
points  in  which  this  knowledge  must  be  called  in  to 
complete  our  mechanical  explanation  of  the  universe. 
But  a  subsidiary  contribution  towards  the  verification  of 
the  hypothesis  may  be  made  by  passing  in  review  the 
hiftory  of  human  thought,  and  enquiring  if  it  presents 
such  features  as  would  be  naturally  expected  on  this 
assumption. 


CHAPTER   IV 

THE   FIRST   CHAPTER  IN  THE   HISTORY 
OF  FOUR  SPACE 

PARMENIDES,  and  the  Asiatic  thinkers  with  whom  he  is 
in  close  affinity,  propound  a  theory  of  existence  which 
is  in  close  accord  with  a  conception  of  a  possible  relation 
between  a  higher  and  a  lower  dimensional  space.  This 
theory,  prior  and  in  marked  contrast  to  the  main  stream 
of  thought,  which  we  shall  afterwards  describe,  forms  a 
closed  circle  by  itself.  It  is  one  which  in  all  ages  has 
had  a  strong  attraction  for  pure  intellect,  and  is  the 
natural  mode  of  thought  for  those  who  refrain  from 
projecting  their  own  volition  into  nature  under  the  guise 
of  causality. 

According  to  Parmenides  of  the  school  of  Elea  the  all 
is  one,  unmoving  and  unchanging.  The  permanent  amid 
the  transient — that  foothold  for  thought,  that  solid  ground 
for  feeling  on  the  discovery  of  which  depends  all  our  life — 
is  no  phantom ;  it  is  the  image  amidst  deception  of  true 
being,  the  eternal,  the  unmoved,  the  one.  Thus  says 
Parmenides. 

But  how  explain  the  shifting  scene,  these  mutations 
of  things ! 

"Illusion,"  answers  Parmenides.  Distinguishing  be- 
tween truth  and  error,  he  tells  of  the  true  doctrine  of  the 
one — the  false  opinion  of  a  changing  world.  He  is  no 
less  memorable  for  the  manner  of  his  advocacy  than  for 

23 


24 


THE    FOURTH    DIMENSION 


the  cause  he  advocates.  It  is  as  if  from  his  firm  foothold 
of  being  he  could  play  with  the  thoughts  under  the 
burden  of  which  others  laboured,  for  from  him  springs 
that  fluency  of  supposition  and  hypothesis  which  forms 
the  texture  of  Plato's  dialectic. 

Can  the  mind  conceive  a  more  delightful  intellectual 
picture  than  that  of  Parmenides,  pointing  to  the  one,  the 
true,  the  unchanging,  and  yet  on  the  other  hand  ready  to 
discuss  all  manner  of  false  opinion,  forming  a  cosmogony 
too,  false  "  but  mine  own  "  after  the  fashion  of  the  time  ? 
In  support  of  the  true  opinion  he  proceeded  by  the 
negative  way  of  showing  the  self-contradictions  in  the 
ideas  of  change  and  motion.  It  is  doubtful  if  his  criticism, 
save  in  minor  points,  has  ever  been  successfully  refuted. 
To  express  his  doctrine  in  the  ponderous  modern  way  we 
must  make  the  statement  that  motion  is  phenomenal, 
not  real. 

Let  us  represent  his  doctrine. 

Imagine  a  sheet  of  still  water  into  which  a  slanting  stick 
is  being  lowered  with  a  motion  verti- 
cally downwards.  Let  1,2,3  (Fig.  13), 
be  three  consecutive  positions  of  the 
stick.  A,  B,  c,  will  be  three  consecutive 
positions  of  the  meeting  of  the  stick, 
with  the  surface  of  the  water.  As 
the  stick  passes  down,  the  meeting  will 
move  from  A  on  to  B  and  c. 

Suppose  now  all  the  water  to  be 
removed  except  a  film.  At  the  meet- 
ing of  the  film  and  the  stick  there 
will  be  an  interruption  of  the  film. 
If  we  suppose  the  film  to  have  a  pro- 


Fig.  13. 


perty,  like  that  of  a  soap  bubble,  of  closing  up  round  any 
penetrating  object,  then  as  the  stick  goes  vertically 
downwards  the  interruption  in  the  film  will  move  on. 


THE    FIRST    CHAPTER   IN    THE    HISTORY   OF    FOUR    SPACE   25 

If  we  pass  a  spiral  through   the  film  the  intersection 
will  give  a  point  moving  in  a  circle  shown  by  the  dotted 

lines    in    the   figure.      Suppose 
now  the  spiral  to  be  still  and 
.x^*^    ^J  the    film     to    move    vertically 

f  upwards,  the  whole   spiral   will 

Xf**"*"""*^  be   represented   in   the   film  of 

/^***  —  "^  the  consecutive  positions  of  the 

point  of  intersection.  In  the 
film  the  permanent  existence 
of  the  spiral  is  experienced  as 
a  time  series — the  record  of 
traversing  the  spiral  is  a  point 
Fi  J4  moving  in  a  circle.  If  now 

we  suppose  a  consciousness  con- 
nected with  the  film  in  such  a  way  that  the  intersection  of 
the  spiral  with  the  film  gives  rise  to  a  conscious  experience, 
we  see  that  we  shall  have  in  the  film  a  point  moving  in  a 
circle,  conscious  of  its  motion,  knowing  nothing  of  that 
real  spiral  the  record  of  the  successive  intersections  of 
which  by  the  film  is  the  motion  of  the  point. 

It  is  easy  to  imagine  complicated  structures  of  the 
nature  of  the  spiral,  structures  consisting  of  filaments, 
and  to  suppose  also  that  these  structures  are  distinguish- 
able from  each  other  at  every  section.  If  we  consider 
the  intersections  of  these  filaments  with  the  film  as  it 
passes  to  be  the  atoms  constituting  a  filmar  universe, 
we  shall  have  in  the  film  a  world  of  apparent  motion; 
we  shall  have  bodies  corresponding  to  the  filamentary 
structure,  and  the  positions  of  these  structures  with 
regard  to  one  another  will  give  rise  to  bodies  in  the 
film  moving  amongst  one  another.  This  mutual  motion 
is  apparent  merely.  The  reality  is  of  permanent  structures 
stationary,  and  all  the  relative  motions  accounted  for  by 
one  steady  movement  of  the  film  as  a  whole. 


26  THE   FOURTH   DIMENSION 

Thus  we  can  imagine  a  plane  world,  in  which  all  the 
variety  of  motion  is  the  phenomenon  of  structures  con- 
sisting of  filamentary  atoms  traversed  by  a  plane  of 
consciousness.  Passing  to  four  dimensions  and  our 
space,  we  can  conceive  that  all  things  and  movements 
in  our  world  are  the  reading  off  of  a  permanent  reality 
by  a  space  of  consciousness.  Each  atom  at  every  moment 
is  not  what  it  was,  but  a  new  part  of  that  endless  line 
which  is  itself.  And  all  this  system  successively  revealed 
in  the  time  which  is  but  the  succession  of  consciousness, 
separate  as  it  is  in  parts,  in  its  entirety  is  one  vast  unity. 
Kepresenting  Parmenides'  doctrine  thus,  we  gain  a  firmer 
hold  on  it  than  if  we  merely  let  his  words  rest,  grand  and 
massive,  in  our  minds.  And  we  have  gained  the  means  also 
of  representing  phases  of  that  Eastern  thought  to  which 
Parmenides  was  no  stranger.  Modifying  his  uncom- 
promising doctrine,  let  us  suppose,  to  go  back  to  the  plane 
of  consciousness  and  the  structure  of  filamentary  atoms, 
that  these  structures  are  themselves  moving — are  acting, 
living.  Then,  in  the  transverse  motion  of  the  film,  there 
would  be  two  phenomena  of  motion,  one  due  to  the  reading 
off  in  the  film  of  the  permanent  existences  as  they  are  in 
themselves,  and  another  phenomenon  of  motion  due  to 
the  modification  of  the  record  of  the  things  themselves,  by 
their  proper  motion  during  the  process  of  traversing  them. 

Thus  a  conscious  being  in  the  plane  would  have,  #s  it 
were,  a  two-fold  experience.  In  the  complete  traversing 
of  the  structure,  the  intersection  of  which  with  the  film 
gives  his  conscious  all,  the  main  and  principal  movements 
and  actions  which  he  went  through  would  be  the  record 
of  his  higher  self  as  it  existed  unmoved  and  uriacting. 
Slight  modifications  and  deviations  from  these  move- 
ments and  actions  would  represent  the  activity  and  self- 
determination  of  the  complete  being,  of  his  higher  self. 

It  is  admissible  to  suppose  that  the  consciousness  in 


THE    FIRST    CHAPTER    IN    THE    HISTORY    OF    FOUR    SPACE   27 

the  plane  has  a  share  in  that  volition  by  which  the 
complete  existence  determines  itself.  Thus  the  motive 
and  will,  the  initiative  and  life,  of  the  higher  being,  would 
be  represented  in  the  case  of  the  being  in  the  film  by  an 
initiative  and  a  will  capable,  not  of  determining  any  great 
things  or  important  movements  in  his  existence,  but  only 
of  small  and  relatively  insignificant  activities.  In  all  the 
main  features  of  his  life  his  experience  would  be  repre- 
sentative of  one  state  of  the  higher  being  whose  existence 
determines  his  as  the  film  passes  on.  But  in  his  minute 
and  apparently  unimportant  actions  he  would  share  in 
that  will  and  determination  by  which  the  whole  of  the 
being  he  really  is  acts  and  lives. 

An  alteration  of  the  higher  being  would  correspond  to 
a  different  life  history  for  him.  Let  us  now  make  the 
supposition  that  film  after  film  traverses  these  higher 
structures,  that  the  life  of  the  real  being  is  read  off  again 
and  again  in  successive  waves  of  consciousness.  There 
would  be  a  succession  of  lives  in  the  different  advancing 
planes  of  consciousness,  each  differing  from  the  preceding, 
andidiffering  in  virtue  of  that  will  and  activity  which  in 
the  preceding  had  not  been  devoted  to  the  greater  and 
apparently  most  significant  things  in  life,  but  the  minute 
and  apparently  unimportant.  In  all  great  things  the 
being  of  the  film  shares  in  the  existence  of  his  higher 
self,  as  it  is  at  any  one  time.  In  the  small  things  he 
shares  *in  that  volition  by  which  the  higher  being  alters 
and  changes,  acts  and  lives. 

Thus  we  gain  the  conception  of  a  life  changing  and 
developing  as  a  whole,  a  life  in  which  our  separation  and 
cessation  and  fugitiveness  are  merely  apparent,  but  which 
in  its  events  and  course  alters,  changes,  develops ;  and 
the  power  of  altering  and  changing  this  whole  lies  in  the 
will  and  power  the  limited  being  has  of  directing,  guiding, 
altering  himself  in  the  minute  things  of  his  existence. 


28  THE    FOUIITH    DIMENSION 

Transferring  our  conceptions  to  those  of  an  existence  in 
a  higher  dimensionality  traversed  by  a  space  of  con- 
sciousness, we  have  an  illustration  of  a  thought  which  has 
found  frequent  and  varied  expression.  When,  however, 
we  ask  ourselves  what  degree  of  truth  there  lies  in  it,  we 
must  admit  that,  as  far  as  we  can  see,  it  is  merely  sym- 
bolical. The  true  path  in  the  investigation  of  a  higher 
dimensionality  lies  in  another  direction. 

The  significance  of  the  Parmenidean  doctrine  lies  in 
this  that  here,  as  again  and  again,  we  find  that  those  con- 
ceptions which  man  introduces  of  himself,  which  he  does 
not  derive  from  the  mere  record  of  his  outward  experience, 
have  a  striking  and  significant  correspondence  to  the 
conception  of  a  physical  existence  in  a  world  of  a  higher 
space.  How  close  we  come  to  Parmenides'  thought  by 
this  manner  of  representation  it  is  impossible  to  say. 
What  I  want  to  point  out  is  the  adequateness  of  the 
illustration,  not  only  to  give  a  static  model  of  his  doctrine, 
but  one  capable  as  it  were,  of  a  plastic  modification  into  a 
correspondence  into  kindred  forms  of  thought.  Either  one 
of  two  things  must  be  true — that  four-dimensional  concep- 
tions give  a  wonderful  power  of  representing  the  thought 
of  the  East,  or  that  the  thinkers  of  the  East  must  have  been 
looking  at  and  regarding  four-dimensional  existence. 

Coming  now  to  the  main  stream  of  thought  we  must 
dwell  in  some  detail  on  Pythagoras,  not  because  of  his 
direct  relation  to  the  subject,  but  because  of  his  relation 
to  investigators  who  came  later. 

Pythagoras  invented  the  two-way  counting.  Let  us 
represent  the  single-way  counting  by  the  posits  aa, 
ab,  ac,  ad,  using  these  pairs  of  letters  instead  of  the 
numbers  1,  2,  3,  4.  I  put  an  a  in  each  case  first  for  a 
reason  which  will  immediately  appear. 

We  have  a  sequence  and  order.  There  is  no  con- 
ception of  distance  necessarily  involved.  The  difference 


THK    FIRST    CHAPTER   IN    THK    HISTORY    OF   FOUR    SPACE    29 

between  the  posits  is  one  of  order  not  of  distance — 
only  when  identified  with  a  number  of  equal  material 
things  in  juxtaposition  does  the  notion  of  distance  arise. 
Now,  besides  the  simple  series  I  can  have,  starting  from 
aa,  ba,  ca,  da,  from  ab,  66,  cb,  db,  and  so  on,  and  forming 

a  scheme : 

da  db  dc  dd 

ca  cb  cc  cd 

ba  bb  be  bd 

aa  ab  ao  ad 

This  complex  or  manifold  gives  a  two-way  order.     I  can 
represent   it  by  a   set  of  points,  if  I  am  on  my  guard 
«    .    .    •     against  assuming  any  relation  of  distance. 

•  •    •    •         Pythagoras   studied   this   two-fold  way  of 

•  •    .    •     counting  in  reference  to  material  bodies,  and 

•  •    •    •    discovered  that  most  remarkable  property  of 

the  combination  of  number  and  matter  that 
Fig.  15.  ,  . 

bears  his  name. 

The  Pythagorean  property  of  an  extended  material 
system  can  be  exhibited  in  a  manner  which  will  be  of 
use  to  us  afterwards,  and  which  therefore  I  will  employ 
now  instead  of  using  the  kind  of  figure  which  he  himself 
employed. 

Consider  a  two-fold  field  of  points  arranged  in  regular 
rows.     Such  a  field  will  be  presupposed  in  the  following 
argument. 
It  is  evident  that  in  fig.  1 6  four 

•  of  the  points  determine  a  square, 

•  which  square  we  may  take  as  the 

•  unit  of   measurement  for   areas. 

But  we  can   also   measure  areas 
Fig.  16.  .  ,, 

in  another  way. 

Fig.  16  (1)  shows  four  points  determining  a  square. 
But  four  squares  also  meet  in  a  point,  fig.  16  (2). 
Hence  a  point  at  the  corner  of  a  square  belongs  equally 
to  four  squares. 


30  THE   FOU11TH   DIMENSION 

Thus  we  may  say  that  the  point  value  of  the  square 
shown  is  one  point,  for  if  we  take  the  square  in  fig.  1G  (1) 
it  has  four  points,  but  each  of  these  belong  equally  to 
four  other  squares.  Hence  one  fourth  of  each  of  them 
belongs  to  the  square  (1)  in  fig.  16.  Thus  the  point 
value  of  the  square  is  one  point. 

The  result  of  counting  the  points  is  the  same  as  that 
arrived  at  by  reckoning  the  square  units  enclosed. 

Hence,  if  we  wish  to  measure  the  area  of  any  square 
we  can  take  the  number  of  points  it  encloses,  count  these 
as  one  each,  and  take  one-fourth  of  the  number  of  points 
at  its  corners. 

Now  draw  a  diagonal  square  as  shown  in  fig.  17.  It 
contains  one  point  and  the  four  corners  count  for  one 

*  point   more ;    hence   its  point  value  is   2. 
.    The  value  is  the  measure  of  its  area — the 

*  size  of  this  square  is  two  of  the  unit  squares. 
.     U-.1     .     .        Looking  now  at  the  sides  of  this  figure 

...    we  see  that  there  is  a  unit  square  on  each 
i'ig.  i7.         Of  them — the  two  squares  contain  no  points, 
but  have  four  corner  points  each,  which  gives  the  point 
value  of  each  as  one  point. 

Hence  we  see  that  the  square  on  the  diagonal  is  equal 
to  the  squares  on  the  two  sides;  or  as  it  is  generally 
expressed,  the  square  on  the  hypothenuse  is  equal  to 
the  sum  of  the  squares  on  the  sides. 

Noticing  this  fact  we  can  proceed  to  ask  if  it  is  always 
true.     Drawing  the  square  shown  in  fig.  18,  we  can  count 
the  number  of  its  points.     There  are  five 
altogether.      There  are  four  points   inside 

*  the  square  on  the  diagonal,  and  hence,  with 

*  the   four   points   at   its  corners  the  point 

*  value  is  5 — that  is,  the  area  is  5.     Now 
the  squares  on  the  sides  are  respectively 

of  the  area  4  and   1.     Hence  in  this  case  also  the  square 


THE   FIRST   CHAPTER   IN   THE   HISTORY   OF   FOUR    SPACE  31 

on  the  diagonal  is  equal  to  the  sum  of  the  square  on 
the  sides.  This  property  of  matter  is  one  of  the  first 
great  discoveries  of  applied  mathematics.  We  shall  prove 
afterwards  that  it  is  not  a  property  of  space.  For  the 
present  it  is  enough  to  remark  that  the  positions  in 
which  the  points  are  arranged  is  entirely  experimental. 
It  is  by  means  of  equal  pieces  of  some  material,  or  the 
same  piece  of  material  moved  from  one  place  to  another, 
that  the  points  are  arranged. 

Pythagoras  next  enquired  what  the  relation  must  be 
so  that  a  square  drawn  slanting-wise  should  be  equal  to 
one  straight-wise.  He  found  that  a  square  whose  side  is 
five  can  be  placed  either  rectangularly  along  the  lines 
of  points,  or  in  a  slanting  position.  And  this  square  is 
equivalent  to  two  squares  of  sides  4  and  3. 

Here  he  came  upon  a  numerical  relation  embodied  in 
a  property  of  matter.  Numbers  immanent  in  the  objects 
produced  the  equality  so  satisfactory  for  intellectual  appre- 
hension. And  he  found  that  numbers  when  immanent 
in  sound — when  the  strings  of  a  musical  instrument 
were  given  certain  definite  proportions  of  length — were 
no  less  captivating  to  the  ear  than  the  equality  of  squares 
was  to  the  reason.  What  wonder  then  that  he  ascribed 
an  active  power  to  number  ! 

We  must  remember  that,  sharing  like  ourselves  the 
search  for  the  permanent  in  changing  phenomena,  the 
Greeks  had  not  that  conception  of  the  permanent  in 
matter  that  we  have.  To  them  material  things  were  not 
permanent.  In  fire  solid  things  would  vanish  ;  absolutely 
disappear.  Kock  and  earth  had  a  more  stable  existence, 
but  they  too  grew  and  decayed.  The  permanence  of 
matter,  the  conservation  of  energy,  were  unknown  to 
them.  And  that  distinction  which  we  draw  so  readily 
between  the  fleeting  and  permanent  causes  of  sensation, 
between  a  sound  and  a  material  object,  for  instance,  had 


- 


32  THE    FOUiiTH    DIMENSION 

not  the  same  meaning  to  them  which  it  has  for  us. 
Let  us  but  imagine  for  a  moment  that  material  things 
are  fleeting,  disappearing,  and  we  shall  enter  with  a  far 
better  appreciation  into  that  search  for  the  permanent 
which,  with  the  Greeks,  as  with  us,  is  the  primary 
intellectual  demand. 

What  is  that  which  amid  a  thousand  forms  is  ever  the 
same,  which  we  can  recognise  under  all  its  vicissitudes, 
of  which  the  diverse  phenomena  are  the  appearances  ? 

To  think  that  this  is  number  is  not  so  very  wide  of 
the  mark.  With  an  intellectual  apprehension  which  far 
outran  the  evidences  for  its  application,  the  atomists 
asserted  that  there  were  everlasting  material  particles, 
which,  by  their  union,  produced  all  the  varying  forms  and 
states  of  bodies.  But  in  view  of  the  observed  facts  of 
nature  as  then  known,  Aristotle,  with  perfect  reason, 
refused  to  accept  this  hypothesis. 

He  expressly  states  that  there  is  a  change  of  quality, 
and  that  the  change  due  to  motion  is  only  one  of  the 
possible  modes  of  change. 

With  no  permanent  material  world  about  us,  with 
the  fleeting,  the  unpermanent,  all  around  we  should,  I 
think,  be  ready  to  follow  Pythagoras  in  his  identification 
of  number  with  that  principle  which  subsists  amidst 
all  changes,  which  in  multitudinous  forms  we  apprehend 
immanent  in  the  changing  and  disappearing  substance 
of  things. 

And  from  the  numerical  idealism  of  Pythagoras  there 
is  but  a  step  to  the  more  rich  and  full  idealism  of  Plato. 
That  which  is  apprehended  by  the  sense  of  touch  we 
put  as  primary  and  real,  and  the  other  senses  we  say 
are  merely  concerned  with  appearances.  But  Plato  took 
them  all  as  valid,  as  giving  qualities  of  existence.  That 
the  qualities  were  not  permanent  in  the  world  as  given 
to  the  senses  forced  him  to  attribute  to  them  a  different 


TfiE   FIRST   CHAPTER   Itf   THE    HISTORY  OF   tOtJR    SPACE  S3 

kind  of  permanence.  He  formed  the  conception  of  a 
world  of  ideas,  in  which  all  that  really  is,  all  that  affects 
us  and  gives  the  rich  and  wonderful  wealth  of  our 
experience,  is  not  fleeting  and  transitory,  but  eternal; 
And  of  this  real  and  eternal  we  see  in  the  things  about 
us  the  fleeting  and  transient  images. 

And  this  world  of  ideas  was  no  exclusive  one,  wherein 
was  no  place  for  the  innermost  convictions  of  the  soul  and 
its  most  authoritative  assertions.  Therein  existed  justice, 
beauty — the  one,  the  good,  all  that  the  soul  demanded 
to  be.  The  world  of  ideas,  Plato's  wonderful  creation 
preserved  for  man,  for  his  deliberate  investigation  and 
their  sure  development,  all  that  the  rude  incom- 
prehensible changes  of  a  harsh  experience  scatters  and 
destroys. 

Plato  believed  in  the  reality  of  ideas.  He  meets  us 
fairly  and  squarely.  Divide  a  line  into  two  parts,  he 
says ;  one  to  represent  the  real  objects  in  the  world,  the 
other  to  represent  the  transitory  appearances,  such  as  the 
image  in  still  water,  the  glitter  of  the  sun  on  a  bright 
surface,  the  shadows  on  the  clouds. 

A  B 

1 . 

Real  things:  Appearances: 

e.g.,  the  sun.  e.y.,  the  reflection  of  the  sun. 

Take  another  line  and  divide  it  into  two  parts,  one 
representing  our  ideas,  the  ordinary  occupants  of  our 
minds,  such  as  whiteness,  equality,  and  the  other  repre- 
senting our  true  knowledge,  which  is  of  eternal  principles, 
such  "as  beauty,  goodness. 

A1  B1 

Eternal  principles,  Appearances  in  the  mind, 

as  beauty  as  whiteness,  equality 

Then  as  A  is  to  B,  so  is  A1  to  B1. 

That  is,  the  soul  can  proceed,  going  away  from  real 

3 


34  TSE  FOURTH  DIMENSION 

things  to  a  region  of  perfect  certainty,  where  it  beholds 
what  is,  not  the  scattered  reflections ;  beholds  the  sun,  not 
the  glitter  on  the  sands  ;  true  being,  not  chance  opinion. 

Now,  this  is  to  us,  as  it  was  to  Aristotle,  absolutely 
inconceivable  from  a  scientific  point  of  view.  We  can 
understand  that  a  being  is  known  in  the  fulness  of  his 
relations ;  it  is  in  his  relations  to  his  circumstances  that 
a  man's  character  is  known  ;  it  is  in  his  acts  under  his 
conditions  that  his  character  exists.  We  cannot  grasp  or 
conceive  any  principle  of  individuation  apart  from  the 
fulness  of  the  relations  to  the  surroundings. 

But  suppose  now  that  Plato  is  talking  about  the  higher 
man — the  four-dimensional  being  that  is  limited  in  our 
external  experience  to  a  three-dimensional  world.  Do  not 
his  words  begin  to  have  a  meaning  ?  Such  a  being 
would  have  a  consciousness  of  motion  which  is  not  as 
the  motion  he  can  see  with  the  eyes  of  the  body.  He, 
in  his  own  being,  knows  a  reality  to  which  the  outward 
matter  of  this  too  solid  earth  is  flimsy  superficiality.  He 
too  knows  a  mode  of  being,  the  fulness  of  relations,  in 
which  can  only  be  represented  in  the  limited  world  of 
sense,  as  the  painter  unsubstantially  portrays  the  depths 
of  woodland,  plains,  and  air.  Thinking  of  such  a  being 
in  man,  was  not  Plato's  line  well  divided  ? 

It  is  noteworthy  that,  if  Plato  omitted  his  doctrine  of 
the  independent  origin  of  ideas,  he  would  present  exactly 
the  four-dimensional  argument;  a  real  thing  as  we  think 
it  is  an  idea.  A  plane  being's  idea  of  a  square  object  is 
the  idea  of  an  abstraction,  namely,  a  geometrical  square. 
Similarly  our  idea  of  a  solid  thing  is  an  abstraction,  for  in 
our  idea  there  is  not  the  four-dimensional  thickness  which 
is  necessary,  however  slight,  to  give  reality.  The  argu- 
ment would  then  run,  as  a  shadow  is  to  a  solid  object,  so 
is  the  solid  object  to  the  reality.  Thus  A  and  B'  would 
be  identified. 


THE    FIRST   CHAPTER  IN   THE   HISTORY   OF   FOUR    SPACE   35 

In  the  allegory  which  I  have  already  alluded  to,  Plato 
in  almost  as  many  words  shows  forth  the  relation  between 
existence  in  a  superficies  and  in  solid  space.  And  he 
uses  this  relation  to  point  to  the  conditions  of  a  higher 
being. 

He  imagines  a  number  of  men  prisoners,  chained  so 
that  they  look  at  the  wall  of  a  cavern  in  which  they  are 
confined,  with  their  backs  to  the  road  and  the  light. 
Over  the  road  pass  men  and  women,  figures  and  pro- 
cessions, but  of  all  this  pageant  all  that  the  prisoners 
behold  is  the  shadow  of  it  on  the  wall  whereon  they  gaze. 
Their  own  shadows  and  the  shadows  of  the  things  in  the 
world  are  all  that  they  see,  and  identifying  themselves 
with  their  shadows  related  -as  shadows  to  a  world  of 
shadows,  they  live  in  a  kind  of  dream. 

Plato  imagines  one  of  their  number  to  pass  out  from 
amongst  them  into  the  real  space  world,  and  then  return- 
ing to  tell  them  of  their  condition. 

Here  he  presents  most  plainly  the  relation  between 
existence  in  a  plane  world  and  existence  in  a  three- 
dimensional  world.  And  he  uses  this  illustration  as  a 
type  of  the  manner  in  which  we  are  to  proceed  to  a 
higher  state  from  the  three-dimensional  life  we  know. 

It  must  have  hung  upon  the  weight  of  a  shadow  which 
path  he  took ! — whether  the  one  we  shall  follow  toward 
the  higher  solid  and  the  four-dimensional  existence,  or 
the  one  .which  makes  ideas  the  higher  realities,  and  the 
direct  perception  of  them  the  contact  with  the  truer 
world. 

Passing  on  to  Aristotle,  we  will  touch  on  the  points 
which  most  immediately  concern  our  enquiry. 

Just,  as  a  scientific  man  of  the  present  day  in 
reviewing  the  speculations  of  the  ancient  world  would 
treat  them  with  a  curiosity  half  amused  but  wholly 
respectful,  asking  of  each  and  all  wherein  lay  their 


36  THE   FOURTH   DIMENSION 

relation  to  fact,  so  Aristotle,  in  discussing  the  philosophy 
of  Greece  as  he  found  it,  asks,  above  all  other  things : 
"  Does  this  represent  the  world  ?  In  this  system  is  there 
an  adequate  presentation  of  what  is  ?  " 

He  finds  them  all  defective,  some  for  the  very  reasons 
which  we  esteem  them  most  highly,  as  when  he  criticises 
the  Atomic  theory  for  its  reduction  of  all  change  to  motion. 
But  in  the  lofty  march  of  his  reason  he  never  loses  sight 
of  the  whole ;  and  that  wherein  our  views  differ  from  his 
lies  not  so  much  in  a  superiority  of  our  point  of  view,  as 
in  the  fact  which  he  himself  enunciates — that  it  is  im- 
possible for  one  principle  to  be  valid  in  all  branches  of 
enquiry.  The  conceptions  of  one  method  of  investigation 
are  not  those  of  another ;  and  our  divergence  lies  in  our 
exclusive  attention  to  the  conceptions  useful  in  one  way 
of  apprehending  nature  rather  than  in  any  possibility  we 
find  in  our  theories  of  giving  a  view  of  the  whole  tran- 
scending that  of  Aristotle. 

He  takes  account  of  everything ;  he  does  not  separate 
matter  and  the  manifestation  of  matter ;  he  fires  all 
together  in  a  conception  of  a  vast  world  process  in 
which  everything  takes  part — the  motion  of  a  grain  of 
dust,  the  unfolding  of  a  leaf,  the  ordered  motion  of  the 
spheres  in  heaven — all  are  parts  of  one  whole  which 
he  will  not  separate  into  dead  matter  and  adventitious 
modifications. 

And  just  as  our  theories,  as  representative  of  actuality, 
fall  before  his  unequalled  grasp  of  fact,  so  the  doctrine 
of  ideas  fell.  It  is  not  an  adequate  account  of  exist- 
ence, as  Plato  himself  shows  in  his  "  Parmenides " ; 
it  only  explains  things  by  putting  their  doubles  beside 
them. 

For  his  own  part  Aristotle  invented  a  great  marching 
definition  which,  with  a  kind  of  power  of  its  own,  cleaves 
its  way  through  phenomena  to  limiting  conceptions  on 


THE    FIRST    CHAPTER   IN   THE    HISTORY   OF    FOUR   SPACE  37 

either  hand,  towards  whose  existence  all  experience 
points. 

In  Aristotle's  definition  of  matter  and  form  as  the 
constituent  of  reality,  as  in  Plato's  mystical  vision  of  the 
kingdom  of  ideas,  the  existence  of  the  higher  Jimension- 
ality  is  implicitly  involved. 

Substance  according  to  Aristotle  is  relative,  not  absolute. 
In  everything  that  is  there  is  the  matter  of  which  it 
is  composed,  the  form  which  it  exhibits ;  but  these  are 
indissolubly  connected,  and  neither  can  be  thought 
without  the  other. 

The  blocks  of  stone  out  of  which  a  house  is  built  are  the 
material  for  the  builder  ;  but,  as  regards  the  quarry  men, 
they  are  the  matter  of  the  rocks  with  the  form  he  has 
imposed  on  them.  Words  are  the  final  product  of  the 
grammarian,  but  the  mere  matter  of  the  orator  or  poet. 
The  atom  is,  with  us,  that  out  of  which  chemical  substances 
are  built  up,  but  looked  at  from  another  point  of  view  is 
the  result  of  complex  processes. 

Nowhere  do  we  find  finality.  The  matter  in  one  sphere 
is  the  matter,  plus  form,  of  another  sphere  of  thought. 
Making  an  obvious  application  to  geometry,  plane  figures 
exist  as  the  limitation  of  different  portions  of  the  plane 
by  one  another.  In  the  bounding  lines  the  separated 
matter  of  the  plane  shows  its  determination  into  form. 
And  as  the  plane  is  the  matter  relatively  to  determinations 
in  the  plane,  so  the  plane  itself  exists  in  virtue  of  the 
determination  of  space.  A  plane  is  that  wherein  formless 
space  has  form  superimposed  on  it,  and  gives  an  actuality 
of  real  relations.  We  cannot  refuse  to  carry  this  process 
of  reasoning  a  step  farther  back,  and  say  that  space  itself 
is  that  which  gives  form  to  higher,  space.  As  a  line  is 
the  determination  of  a  plane,  and  a  plane  of  a  solid,  so 
solid  space  itself  is  the  determination  of  a  higher  space. 

As  a.  line  by  itself  is  inconceivable  without  that  plane 


38  THE   FOUKTH   DIMENSION 

which  it  separates,  so  the  plane  is  inconceivable  without 
the  solids  which  it  limits  on  either  hand.  And  so  space 
itself  cannot  be  positively  defined.  It  is  the  negation 
of  the  possibility  of  movement  in  more  than  three 
dimensions.  The  conception  of  space  demands  that  of 
a  higher  space.  As  a  surface  is  thin  and  unsubstantial 
without  the  substance  of  which  it  is  the  surface,  so  matter 
itself  is  thin  without  the  higher  matter. 

Just  as  Aristotle  invented  that  algebraical  method  of 
representing  unknown  quantities  by  mere  symbols,  not  by 
lines  necessarily  determinate  in  length  as  was  the  habit 
of  the  Greek  geometers,  and  so  struck  out  the  path 
towards  those  objectifications  of  thought  which,  like 
independent  machines  for  reasoning,  supply  the  mathe- 
matician with  his  analytical  weapons,  so  in  the  formulation 
o£vthe  doctrine  of  matter  and  form,  of  potentiality  and 
actuality,  of  the  relativity  of  substance,  he  produced 
another  kind  of  objectification  of  mind — a  definition 
which  had  a  vital  force  and  an  activity  of  its  own. 

In  none  of  his  writings,  as  far  as  we  know,  did  he  carry  it 
to  its  legitimate  conclusion  on  the  side  of  matter,  but  in 
the  direction  of  the  formal  qualities  he  was  led  to  his 
limiting  conception  of  that  existence  of  pure  form  which 
lies  beyond  all  known  determination  of  matter.  The 
unmoved  mover  of  all  things  is  Aristotle's  highest 
principle.  Towards  it,  to  partake  of  its  perfection  all 
things  move.  The  universe,  according  to  Aristotle,  is  an 
active  process — he  does  not  adopt  the  illogical  conception 
that  it  was  once  set  in  motion  and  has  kept  on  ever  since. 
There  is  room  for  activity,  will,  self-determination,  in 
Aristotle's  system,  and  for  the  contingent  and  accidental 
as  well.  We  do  not  follow  him,  because  we  are  accus- 
tomed to  find  in  nature  infinite  series,  and  do  not  feel 
obliged  to  pass  on  to  a  belief  in  the  ultimate  limits  to 
which  they  seem  to  point, 


THE   FIRST    CHAPTER   IN   THE    HISTORY  OF   FOUR   SPACE  39 

But  apart  from  the  pushing  to  the  limit,  as  a  relative 
principle  this  doctrine  of  Aristotle's  as  to  the  relativity  of 
substance  is  irrefragible  in  its  logic.  He  was  the  first  to 
show  the  necessity  of  that  path  of  thought  which  when 
followed  leads  to  a  belief  in  a  four-dimensional  space. 

Antagonistic  as  he  was  to  Plato  in  his  conception 
of  the  practical  relation  of  reason  to  the  world  of 
phenomena,  yet  in  one  point  he  coincided  with  him. 
And  in  this  he  showed  the  candour  of  his  intellect.  He 
was  more  anxious  to  lose  nothing  than  to  explain  every- 
thing. And  that  wherein  so  many  have  detected  an 
inconsistency,  an  inability  to  free  himself  from  the  school 
of  Plato,  appears  to  us  in  connection  with  our  enquiry 
as  an  instance  of  the  acuteness  of  his  observation.  For 
beyond  all  knowledge  given  by  the:  senses  Aristotle  held 
that  there  is  an  active  intelligence,  a  mind  not  the  passive 
recipient  of  impressions  from  without,  but  an  active  and 
originative  being,  capable  of  grasping  knowledge  at  first 
hand.  In  the  active  soul  Aristotle  recognised  something 
in  man  not  produced  by  his  physical  surroundings,  some- 
thing which  creates,  whose  activity  is  a  knowledge 
underived  from  sense.  This,  he  says,  is  the  immortal  and 
undying  being  in  man. 

Thus  we  see  that  Aristotle  was  not  far  from  the 
recognition  of  the  four-dimensional  existence,  both 
without  and  within  man,  and  the  process  of  adequately 
realising  the  higher  dimensional  figures  to  which  we 
shall  come  subsequently  is  a  simple  reduction  to  practice 
of  his  hypothesis  of  a  soul. 

The  next  step  in  the  unfolding  of  the  drama  of  the 
recognition  of  the  soul  as  connected  with  our  scientific 
conception  of  the  world,  and,  at  the  same  time,  the 
recognition  of  that  higher  of  which  a  three-dimensional 
world  presents  the  superficial  appearance,  took  place  many 
centuries  later,  {f  we  pass  over  the  intervening  time 


40  ,         THE   FOURTH   DIMENSION 

without  a  word  it  is  because  the  soul  was  occupied  with 
the  assertion  of  itself  in  other  ways  than  that  of  knowledge. 
When  it  took  up  the  task  in  earnest  of  knowing  this 
material  world  in  which  it  found  itself,  and  of  directing 
the  course  of  inanimate  nature,  from  that  most  objective 
aim  came,  reflected  back  as  from  a  mirror,  its  knowledge 
Of  itself. 


CHAPTER   V 

THE   SECOND   CHAPTER  IN   THE   HISTORY 
OF   FOUR  SPACE 

LOBATCHEWSKY,   BOLYAI,   AND  GAUSS 

BEFORE  entering  on  a  description  of  the  work  of 
Lobatchewsky  and  Bolyai  it  will  not  be  out  of  place 
to  give  a  brief  account  of  them,  the  materials  for  which 
are  to  be  found  in  an  article  by  Franz  Schmidt  in  the 
forty-second  volume  of  the  Mathematische  Annalen, 
and  in  Engel's  edition  of  Lobatchewsky. 

Lobatchewsky  was  a  man  of  the  most  complete  and 
wonderful  talents.  As  a  youth  he  was  full  of  vivacity, 
carrying  his  exuberance  so  far  as  to  fall  into  serious 
trouble  for  hazing  a  professor,  and  other  freaks.  Saved 
by  the  good  offices  of  the  mathematician  Bartels,  who 
appreciated  his  ability,  he  managed  to  restrain  himself 
within  the  bounds  of  prudence.  Appointed  professor  at 
his  own  University,  Kasan,  he  entered  on  his  duties  under 
the  regime  of  a  pietistic  reactionary,  who  surrounded 
himself  with  sycophants  and  hypocrites.  Esteeming 
probably  the  interests  of  his  pupils  as  higher  than  any 
attempt  at  a  vain  resistance,  he  made  himself  the  tyrant's 
right-hand  man,  doing  an  incredible  amount  of  teaching 
and  performing  the  most  varied  official  duties.  Amidst 
all  his  activities  he  found  time  to  make  important  con- 
tributions to  science.  His  theory  of  parallels  is 


42  THE    FOURTH    DIMENSION. 

closely  connected  with  his  name,  but  a  study  of  his 
writings  shows  that  he  was  a  man  capable  of  carrying 
on  mathematics  in  its  main  lines  of  advance,  and  of  a 
judgment  equal  to  discerning  what  these  lines  were. 
Appointed  rector  of  his  University,  he  died  at  an 
advanced  age,  surrounded  by  friends,  honoured,  with  the 
results  of  his  beneficent  activity  all  around  him.  To  him 
no  subject  came  amiss,  from  the  foundations  of  geometry 
to  the  improvement  of  the  stoves  by  which  the  peasants 
warmed  their  houses. 

He  was  born  in  1793.  His  scientific  work  was 
unnoticed  till,  in  1867,  Houel,  the  French  mathematician,' 
drew  attention  to  its  importance. 

Johann  Bolyai  de  Bolyai  was  born  in  Klausenburg, 
a  town  in  Transylvania,  December  loth,  1802. 

His  father,  Wolfgang  Bolyai,  a  professor  in  the 
Reformed  College  of  Maros  Vasarhely,  retained  the  ardour 
in  mathematical  studies  which  had  made  him  a  chosen 
companion  of  Gauss  in  their  early  student  days  at 
Gottingen. 

He  found  an  eager  pupil  in  Johann.  He  relates  that 
the  boy  sprang  before  him  like  a  devil.  As  soon  as  he 
had  enunciated  a  problem  the  child  would  give  the 
solution  and  command  him  to  go  on  further.  As  a 
thirteen-year-old  boy  his  father  sometimes  sent  him  to  fill 
his  place  when  incapacitated  from  taking  his  classes. 
The  pupils  listened  to  him  with  more  attention  than  to 
his  father  for  they  found  him  clearer  to  understand. 

In  a  letter  to  Gauss  Wolfgang  Bolyai  writes  : — 

"  My  boy  is  strongly  built.  He  has  learned  to  recognise 
many  constellations,  and  the  ordinary  figures  of  geometry. 
He  makes  apt  applications  of  his  notions,  drawing  for 
instance  the  positions  of  the  stars  with  their  constellations. 
Last  winter  in  the  country,  seeing  Jupiter  he  asked : 
is  it  that  we  can  gee  him  from  here  as  well  as  from 


THE   SECOND   CHAPTER  IN  THE   HISTORY  OF  FOUR   SPACE  43 

the  town  ?  He  must  be  far  off.'  And  as  to  three 
different  places  to  which  he  had  been  he  asked  me  to  tell 
him  about  them  in  one  word.  I  did  not  know  what  he 
meant,  and  then  he  asked  me  if  one  was  in  a  line  with 
the  other  and  all  in  a  row,  or  if  they  were  in  a  triangle. 

"  He  enjoys  cutting  paper  figures  with  a  pair  of  scissors, 
and  without  my  ever  having  told  him  about  triangles 
remarked  that  a  right-angled  triangle  which  he  had  cut 
out  was  half  of  an  oblong.  I  exercise  his  body  with  care, 
he  can  dig  well  in  the  earth  with  his  little  hands.  The 
blossom  can  fall  and  no  fruit  left.  When  he  is  fifteen 
•I  want  to  send  him  to  you  to  be  your  pupil." 

In  Johann's  autobiography  he  says : — 

"  My  father  called  my  attention  to  the  imperfections 
and  gaps  in  the  theory  of  parallels.  He  told  me  he  had 
gained  more  satisfactory  results  than  his  predecessors, 
but  had  obtained  no  perfect  and  satisfying  conclusion. 
None  of  his  assumptions  had  the  necessary  degree  of 
geometrical  certainty,  although  they  sufficed  to  prove  the 
eleventh  axiom  and  appeared  acceptable  on  first  sight. 

"  He  begged  of  me,  anxious  not  without  a  reason,  to 
hold  myself  aloof  and  to  shun  all  investigation  on  this 
subject,  if  I  did  not  wish  to  live  all  my  life  in  vain." 

Johann,  in  the  failure  of  his  father  to  obtain  any 
response  from  Gauss,  in  answer  to  a  letter  in  which  he 
asked  the  great  mathematician  to  make  of  his  son  "  an 
apostle  of  truth  in  a  far  land,"  entered  the  Engineering 
School  at  Vienna.  He  writes  from  Temesvar,  where  he 
was  appointed  sub-lieutenant  September,  1823  : — 

"  Temesvar,  November  3rd,  1823. 
"DEAR  GOOD  FATHER, 

"I  have  so  overwhelmingly  much  to  write 
about  my  discovery  that  I  know  no  other  way  of  checking 
myself  than  taking  a  quarter  of  a  sheet  only  to  write  on, 
I  want  an  answer  to  my  four-sheet  letter, 


44  THE   FOURTH   DIMENSION 

"  I  am  unbroken  in  my  determination  to  publish  a 
work  on  Parallels,  as  soon  as  I  have  put  rny  material  in 
order  and  have  the  means. 

"  At  present  I  have  not  made  any  discovery,  but 
the  way  I  have  followed  almost  certainly  promises  me 
the  attainment  of  my  object  if  any  possibility  of  it 
exists. 

"  I  have  not  got  my  object  yet,  but  I  have  produced 
such  stupendous  things  that  I  was  overwhelmed  myself, 
and  it  would  be  an  eternal  shame  if  they  were  lost. 
When  you  see  them  you  will  find  that  it  is  so.  Now 
I  can  only  say  that  I  have  made  a  new  world  out  of 
nothing.  Everything  that  I  have  sent  you  before  is  a 
house  of  cards  in  comparison  with  a  tower.  I  am  con- 
vinced that  it  will  be  no  less  to  my  honour  than  if  I  had 
already  discovered  it." 

The  discovery  of  which  Johann  here  speaks  was 
published  as  an  appendix  to  Wolfgang  Bolyai's  Tentamen. 

Sending  the  book  to  Gauss,  Wolfgang  writes,  after  an 
interruption  of  eighteen  years  in  his  correspondence : — 

"  My  son  is  first  lieutenant  of  Engineers  and  will  soon 
be  captain.  He  is  a  fine  youth,  a  good  violin  player, 
a  skilful  fencer,  and  brave,  but  has  had  many  duels,  and 
is  wild  even  for  a  soldier.  Yet  he  is  distinguished — light 
in  darkness  and  darkness  in  light.  He  is  an  impassioned 
mathematician  with  extraordinary  capacities.  ...  He 
will  think  more  of  your  judgment  on  his  work  than  that 
of  all  Europe." 

Wolfgang  received  no  answer  from  Gauss  to  this  letter, 
but  sending  a  second  copy  of  the  book  received  the 
following  reply : — 

"You  have  rejoiced  me,  my  unforgotten  friend,  by  your 
letters.  I  delayed  answering  the  first  because  I  wanted 
to  wait  for  the  arrival  of  the  promised  little  book, 

"Now  something  about  your  son's  wor^, 


THE  SECOND  CHAPTER  IN  THE  HISTORY  Of  FOUR   SPACE  45 

"  If  I  begin  with  saying  that  '  I  ought  not  to  praise  it,' 
you  will  be  staggered  for  a  moment.  But  I  cannot  say 
anything  else.  To  praise  it  is  to  praise  myself,  for  the 
path  your  son  has  broken  in  upon  and  the  results  to  which 
he  has  been  led  are  almost  exactly  the  same  as  my  own 
reflections,  some  of  which  date  from  thirty  to  thirty-five 
years  ago. 

"  In  fact  I  am  astonished  to  the  uttermost.  My  inten- 
tion was  to  let  nothing  be  known  in  my  lifetime  about 
my  own  work,  of  which,  for  the  rest,  but  little  is  com- 
mitted to  writing.  Most  people  have  but  little  perception 
of  the  problem,  and  I  have  found  very  few  who  took  any 
interest  in  the  views  I  expressed  to  them.  To  be  able  to 
do  that  one  must  first  of  all  have  had  a  real  live  feeling 
of  what  is  wanting,  and  as  to  that  most  men  are  com- 
pletely in  the  dark. 

"  Still  it  was  my  intention  to  commit  everything  to 
writing  in  the  course  of  time,  so  that  at  least  it  should 
not  perish  with  me. 

"I  am  deeply  surprised  that  this  task  can  be  spared 
me,  and  I  am  most  of  all  pleased  in  this  that  it  is  the  son 
of  my  old  friend  who  has  in  so  remarkable  a  manner 
preceded  me." 

The  impression  which  we  receive  from  Gauss's  in- 
explicable silence  towards  his  old  friend  is  swept  away 
by  this  letter.  Hence  we  breathe  the  clear  air  of  the 
mountain  tops.  Gauss  would  not  have  failed  to  perceive 
the  vast  significance  of  his  thoughts,  sure  to  be  all  the 
greater  in  their  effect  on  future  ages  from  the  want  of 
comprehension  of  the  present.  Yet  there  is  not  a  word 
or  a  sign  in  his  writing  to  claim  the  thought  for  himself. 
He  published  no  single  line  on  the  subject.  By  the 
measure  of  what  he  thus  silently  relinquishes,  by  such  a 
measure  of  a  world-transforming  thought,  we  can  appre- 
ciate his  greatness. 


46  THE  FOURTH  DIMENSION 

It  is  a  long  step  from  Gauss's  serenity  to  the  disturbed 
and  passionate  life  of  Johann  Bolyai — he  and  Galois, 
the  two  most  interesting  figures  in  the  history  of  mathe- 
matics. For  Bolyai,  the  wild  soldier,  the  duellist,  fell 
at  odds  with  the  world.  It  is  related  of  him  that  he  was 
challenged  by  thirteen  officers  of  his  garrison,  a  thing  not 
unlikely  to  happen  considering  how  differently  he  thought 
from  every  one  else.  He  fought  them  all  in  succession — 
making  it  his  only  condition  that  he  should  be  allowed 
to  play  on  his  violin  for  an  interval  between  meeting  each 
opponent.  He  disarmed  or  wounded  all  his  antagonists. 
It  can  be  easily  imagined  that  a  temperament  such  as 
his  was  one  not  congenial  to  his  military  superiors.  He 
was  retired  in  1833. 

His  epoch-making  discovery  awoke  no  attention.  He 
seems  to  have  conceived  the  idea  that  his  father  had 
betrayed  him  in  some  inexplicable  way  by  his  communi- 
cations with  Gauss,  and  he  challenged  the  excellent 
Wolfgang  to  a  duel.  He  passed  his  life  in  poverty, 
many  a  time,  says  his  biographer,  seeking  to  snatch 
himself  from  dissipation  and  apply  himself  again  to 
mathematics.  But  his  efforts  had  no  result.  He  died 
January  27th,  1860,  fallen  out  with  the  world  and  with 
himself. 


METAGEOMETRY 

The  theories  which  are  generally  connected  with  the 
names  of  Lobatchewsky  and  Bolyai  bear  a  singular  and 
curious  relation  to  the  subject  of  higher  space. 

In  order  to  show  what  this  relation  is,  I  must  ask  the 
reader  to  be  at  the  pains  to  count  carefully  the  sets  of 
points  by  which  I  shall  estimate  the  volumes  of  certain 
figures. 


THE   SECOND   CHAPTER  IN  THE  HISTORY  OF  FOUR   SPACE  47 

No  mathematical  processes  beyond  this  simple  one  of 

•  '    •    •    •   counting  will  be  necessary. 
Let   us    suppose  we   have    before    us    in 

•  •     .    .     .    fig.  19  a  plane  covered  with  points  at  regular 
.    .    •    .    .    intervals,  so  placed  that  every  four  deter- 
.....   mine  a  square. 

.big.  iy.  Now   it   is   evident  that   as   four   points 

determine  a  square,  so  four  squares  meet  in  a  point. 

Thus,  considering  a  point  inside  a  square  as 
belonging  to  it,  we  may  say  that  a  point  on 
the  corner  of  a   square  belongs  to  it  and  to 
.     .    .    four  others  equally :  belongs  a  quarter  of   it 
Fig.  20.      to  each  square. 
Thus  the  square  ACDE  (fig.  21)  contains  one  point,  and 


D 

•'E^C'- 

D 

E*      •      1 

c  • 

SX. 

^.    •   J 

A    'B 

'  A     ' 

B 

Fig.  21.  Fig.  22. 

has  four  points  at  the  four  corners.  Since  one-fourth  of 
each  of  these  four  belongs  to  the  square,  the  four  together 
count  as  one  point,  and  the  point  value  of  the  square  is 
two  points — the  one  inside  and  the  four  at  the  corner 
make  two  points  belonging  to  it  exclusively. 

Now  the  area  of  this  square  is  two  unit  squares,  as  can 
be  seen  by  drawing  two  diagonals  in  fig.  22. 

We  also  notice  that  the  square  in  question  is  equal  to 
the  sum  of  the  squares  on  the  sides  AB,  BC,  of  the  right- 
angled  triangle  ABC.  Thus  we  recognise  the  proposition 
that  the  square  on  the  hypothenuse  is  equal  to  the  sum 
of  the  squares  on  the  two  sides  of  a  right-angled  triangle. 

Now  suppose  we  set  ourselves  the  question  of  deter- 
mining the  whereabouts  in  the  ordered  system  of  points, 


48 


FOURTH  DIMENSION 


the  end  of  a  line  would  come  when  it  turned  about  a 
point  keeping  one  extremity  fixed  at  the  point. 

We  can  solve  this  problem  in  a  particular  case.  If  we 
can  find  a  square  lying  slantwise  amongst  the  dots  which  is 
equal  to  one  which  goes  regularly,  we  shall  know  that  the 
two  sides  are  equal,  and  that  the  slanting  side  is  equal  to  the 
straight-way  side.  Thus  the  volume  and  shape  of  a  figure 
remaining  unchanged  will  be  the  test  of  its  having  rotated 
about  the  point,  so  that  we  can  say  that  its  side  in  its  first 
position  would  turn  into  its  side  in  the  second  position. 

Now,  such  a  square  can  be  found  in  the  one  whose  side 
is  five  units  in  length. 


Ing.  23. 

In  fig.  23,  in  the  square  on  AB,  there  are — 
9  points  interior         .        .        . 

4  at  the  corners 

4  sides  with  3  on  each  side,  considered  as 
1£  on  each  side,  because  belonging 
equally  to  two  squares  .  .  , 


The  total 
on  BC. 


is   16.     There  are    9   points   in  the  square 


TBE   feECOtfb   CHAPTER  IN  THE  HISTORY  OF  FOUR  SPACE  49 

In  the  square  on  AC  there  are — 

24  points  inside 24 

4  at  the  corners        .....      .1 

or  25  altogether. 

Hence  we  see  again  that  the  square  on  the  hypothenuse 
is  equal  to  the  squares  on  the  sides. 

Now  take  the  square  AFHG,  which  is  larger  than  the 
square  on  AB.  It  contains  25  points. 

16  inside 16 

16  on  the  sides,  counting  as    .        .        .8 
4  on  the  corners 1 

making  25  altogether. 

If  two  squares  are  equal  we  conclude  the  sides  are 
equal.  Hence,  the  line  AF  turning  round  A  would 
move  so  that  it  would  after  a  certain  turning  coincide 
with  AC. 

This  is  preliminary,  but  it  involves  all  the  mathematical 
difficulties  that  will  present  themselves. 

There  are  two  alterations  of  a  body  by  which  its  volume 
is  not  changed. 

One  is  the  one  we  have  just  considered,  rotation,  the 
other  is  what  is  called  shear. 

Consider  a  book,  or  heap  of  loose  pages.  They  can  be 

slid  so  that  each  one  slips 


y       over   the   preceding   one, 


a  b  and    the   whole    assumes 

the  shape  b  in  fig.  24. 

This  deformation  is  not  shear  alone,  but  shear  accom- 
panied by  rotation. 

Shear  can  be  considered  as  produced  in  another  way. 
Take  the  square   ABCD   (fig.  25),   and  suppose  that  it 
is  pulled  out  from  along  one  of  its  diagonals  both  ways, 
and  proportionately  compressed  along  the  other  diagonal. 
It  will  assume  the  shape  in  fig.  26. 

4 


50 


THE  FOURTH  DIMENSION 


This  compression  and  expansion  along  two  lines  at  right 
angles  is  what  is  called  shear;  it  is  equivalent  to  the 
sliding  illustrated  above,  combined  with  a  turning  round. 


In  pure  shear  a  body  is  compressed  and  extended  in 
two  directions  at  right  angles  to  each  other,  so  that  its 
volume  remains  unchanged. 

Now  we  know  that  our  material  bodies  resist  shear — 
shear  does  violence  to  the  internal  arrangement  of  their 
particles,  but  they  turn  as  wholes  without  such  internal 
resistance. 

But  there  is  an  exception.  In  a  liquid  shear  and 
rotation  take  place  equally  easily,  there  is  no  more 
resistance  against  a  shear  than  there  is  against  a 
rotation. 

Now,  suppose  all  bodies  were  to  be  reduced  to  the  liquid 
state,  in  which  they  yield  to  shear  and  to  rotation  equally 
easily,  and  then  were  to  be  reconstructed  as  solids,  but  in 
such  a  way  that  shear  and  rotation  had  interchanged 
places. 

That  is  to  say,  let  us  suppose  that  when  they  had 
become  solids  again  they  would  shear  without  offering 
any  internal  resistance,  but  a  rotation  would  do  violence 
to  their  internal  arrangement. 

That  is,  we  should  have  a  world  in  which  shear  would 
have  taken  the  place  of  rotation. 


A  shear  does  not  alter  the  volume  of  a  body :  thus  an 
inhabitant  living  in  such  a  world  would  look  on  a  body 
sheared  as  we  look  on  a  body  rotated.  He  would  say 
that  it  was  of  the  same  shape,  but  had  turned  a  bit 
round. 

Let  us  imagine  a  Pythagoras  in  this  world  going  to 
work  to  investigate,  as  is  his  wont. 

Fig.    27    represents    a    square    unsheared.      Fig.    28 


Fig.  27. 


Fig.  28. 


represents  a  square  sheared.  It  is  not  the  figure  into 
which  the  square  in  fig.  27  would  turn,  but  the  result  of 
shear  on  some  square  not  drawn.  It  is  a  simple  slanting 
placed  figure,  taken  now  as  we  took  a  simple  slanting 
placed  square  before.  Now,  since  bodies  in  this  world  of 
shear  offer  no  internal  resistance  to  shearing,  and  keep 
their  volume  when  sheared,  an  inhabitant  accustomed  to 
them  would  not  consider  that  they  altered  their  shape 
under  shear.  He  would  call  ACDE  as  much  a  square  as 
the  square  in  fig.  27.  We  will  call  such  figures  shear 
squares.  Counting  the  dots  in  ACDE,  we  find — 

2  inside          =     2 
4  at  corners  =     1 

or  a  total  of  3. 

Now,  the  square  on  the  side  AB  has  4  points,  that  on  BC 
has  1  point.  Here  the  shear  square  on  the  hypothenuse 
has  not  5  points  but  3 ;  it  is  not  the  sum  of  the  squares  on 
the  sides,  but  the  difference. 


52 


THE  FOURTH  DIMENSION 

This  relation  always  holds.      Look  at 
fig.  29. 

Shear  square  on  hypothenuse — 


7  internal 
4  at  corners 


Fig.  29. 


Square  on   one  side — which  the  reader  can  draw  for 

himself — 


4  internal 
8  on  sides    . 
4  at  corners 


Fig.  29  bis. 


and  the  square  on  the  other 
side  is  1.  Hence  in  this 
case  again  the  difference  is 
equal  to  the  shear  square  on 
the  hypothenuse,  9—1  =  8. 
Thus  in  a  world  of  shear 
the  square  on  the  hypothen- 
use would  be  equal  to  the 
difference  of  the  squares  on 
the  sides  of  a  right-angled 
triangle. 

In  fig.  29  bis  another  shear  square  is  drawn  on  which 
the  above  relation  can  be  tested. 

What  now  would  be  the  position  a  line  on  turning  by 
shear  would  take  up  ? 

We  must  settle  this  in  the  same  way  as  previously  with 
our  turning. 

Since  a  body  shear,  d  remains  the  same,  we  must  find  two 
equal  bodies,  one  in  the  straight  way,  one  in  the  slanting 
way,  which  have  the  same  volume.  Then  the  side  of  one 
will  by  turning  become  the  side  of  the  other,  for  the  two 
figures  are  each  what  the  other  becomes  by  a  shear  turning. 


THE  SECOND  CHAPTER  IN  THE  HISTORY  OF  FOUR  SPACE  53 


We  can  solve  the  problem  in  a  particular  case — 

In  the  figure  ACDE 
(fig.  30)  there  are — 

15  inside    .        .     15 
4  at  corners     r      1 

a  total  of  16. 

Now  in  the  square  ABGF, 
there  are  16 — 

9  inside     .        .    9 
12  on  sides  .        .    6 
4  at  corners       .     1 

16 

Hence  the  square  on  AB 
would,  by  the  shear  turn- 
ing, become  the  shear  square 

ACDE. 


Fig.  30. 


And  hence  the  inhabitant  of  this  world  would  say  that 
the  line  AB  turned  into  the  line  AC.  These  two  lines 
would  be  to  him  two  lines  of  equal  length,  one  turned 
a  little  way  round  from  the  other. 

That  is,  putting  shear  in  place  of  rotation,  we  -get  a 
different  kind  of  figure,  as  the  result  of  the  shear  rotation, 
from  what  we  got  with  our  ordinary  rotation.  And  as  a 
consequence  we  get  a  position  for  the  end  of  a  line  of 
invariable  length  when  it  turns  by  the  shear  rotation, 
different  from  the  position  which  it  would  assume  on 
turning  by  our  rotation. 

A  real  material  rod  in  the  shear  world  would,  on  turning 
about  A,  pass  from  the  position  AB  to  th*e  position  AC. 
We  say  that  its  length  alters  when  it  becomes  AC,  but  this 
transformation  of  AB  would  seem  to  an  inhabitant  of  the 
shear  world  like  a  turning  of  AB  without  altering  in 
length. 

If  now  we  suppose  a  communication  of  ideas  that  takes 
place  between  one  of  ourselves  and  an  inhabitant  of  the 


54  THE   FOURTH   DIMENSION 

shear  world,  there  would  evidently  be  a  difference  between 
his  views  of  distance  and  ours. 

We  should  say  that  his  line  AB  increased  in  length  in 
turning  to  AC.  He  would  say  that  our  line  AF  (fig.  23) 
decreased  in  length  in  turning  to  AC.  He  would  think 
that  what  we  called  an  equal  line  was  in  reality  a  shorter 
one. 

We  should  say  that  a  rod  turning  round  would  have  its 
extremities  in  the  positions  we  call  at  equal  distances. 
So  would  he — but  the  positions  would  be  different.  He 
could,  like  us,  appeal  to  the  properties  of  matter.  His 
rod  to  him  alters  as  little  as  ours  does  to  us. 

Now,  is  there  any  standard  to  which  we  could  appeal,  to 
say  which  of  the  two  is  right  in  this  argument  ?  There 
is  no  standard. 

We  should  say  that,  with  a  change  of  position,  the 
configuration  and  shape  of  his  objects  altered.  He  would 
say  that  the  configuration  and  shape  of  our  objects  altered 
in  what  we  called  merely  a  change  of  position.  Hence 
distance  independent  of  position  is  inconceivable,  or 
practically  distance  is  solely  a  property  of  matter. 

There  is  no  principle  to  which  either  party  in  this 
controversy  could  appeal.  There  is  nothing  to  connect 
the  definition  of  distance  with  our  ideas  rather  than  with 
his,  except  the  behaviour  of  an  actual  piece  of  matter. 

For  the  study  of  the  processes  which  go  on  in  our  world 
the  definition  of  distance  given  by  taking  the  sum  of  the 
squares  is  of  paramount  importance  to  us.  But  as  a  ques- 
tion of  pure  space  without  making  any  unnecessary 
assumptions  the  shear  world  is  just  as  possible  and  just  as 
interesting  as  our  world. 

It  was  the  geometry  of  such  conceivable  worlds  that 
Lobatchewsky  and  Bolyai  studied. 

This  kind  of  geometry  has  evidently  nothing  to  do 
directly  with  four-dimensional  space. 


THE   SECOND   CHAPTER  IN  THE   HISTORY  OF  FOUR  SPACE  55 

But  a  connection  arises  in  this  way.  It  is  evident  that, 
instead  of  taking  a  simple  shear  as  I  have  done,  and 
defining  it  as  that  change  of  the  arrangement  of  the 
particles  of  a  solid  which  they  will  undergo  without 
offering  any  resistance  due  to  their  mutual  action,  I 
might  take  a  complex  motion,  composed  of  a  shear  and 
a  rotation  together,  or  some  other  kind  of  deformation. 

Let  us  suppose  such  an  alteration  picked  out  and 
defined  as  the  one  which  means  simple  rotation,  then  the 
type,  according  to  which  all  bodies  will  alter  by  this 
rotation,  is  fixed. 

Looking  at  the  movements  of  this  kind,  we  should  say 
that  the  objects  were  altering  their  shape  as  well  as 
rotating.  But  to  the  inhabitants  of  that  world  they 
would  seem  to  be  unaltered,  and  our  figures  in  their 
motions  would  seem  to  them  to  alter. 

In  such  a  world  the  features  of  geometry  are  different. 
We  have  seen  one  such  difference  in  the  case  of  our  illus- 
tration of  the  world  of  shear,  where  the  square  on  the 
hypothenuse  was  equal  to  the  difference,  not  the  sum,  of 
the  squares  on  the  sides. 

In  our  illustration  we  have  the  same  laws  of  parallel 
lines  as  in  our  ordinary  rotation  world,  but  in  general  the 
laws  of  parallel  lines  are  different. 

In  one  of  these  worlds  of  a  different  constitution  of 
matter  through  one  point  there  can  be  two  parallels  to 
a  given  line,  in  another  of  them  there  can  be  none,  that 
is,  although  a  line  be  drawn  parallel  to  another  it  will 
meet  it  after  a  time. 

Now  it  was  precisely  in  this  respect  of  parallels  that 
Lobatchewsky  and  Bolyai  discovered  these  different 
worlds.  They  did  not  think  of  them  as  worlds  of  matter, 
but  they  discovered  that  space  did  not  necessarily  mean 
that  our  law  of  parallels  is  true.  They  made  the 
distinction  between  laws  of  space  and  laws  of  matter, 


56  THE   FOURTH   DIMENSION 

although  that  is  not  the  form  in  which  they  stated  their 
results. 

The  way  in  which  they  were  led  to  these  results  was  the 
following.  Euclid  had  stated  the  existence  of  parallel  lines 
as  a  postulate — putting  frankly  this  unproved  proposition 
—  that  one  line  and  only  one  parallel  to  a  given  straight 
line  can  be  drawn,  as  a  demand,  as  something  that  must 
be  assumed.  The  words  of  his  ninth  postulate  are  these : 
"  If  a  straight  line  meeting  two  other  straight  lines 
makes  the  interior  angles  on  the  same  side  of  it  equal 
to  two  right  angles,  the  two  straight  lines  will  never 
meet." 

The  mathematicians  of  later  ages  did  not  like  this  bald 
assumption,  and  not  being  able  to  prove  the  proposition 
they  called  it  an  axiom — the  eleventh  axiom. 

Many  attempts  were  made  to  prove  the  axiom  ;  no  one 
doubted  of  its  truth,  but  no  means  could  be  found  to 
demonstrate  it.  At  last  an  Italian,  Sacchieri,  unable  to 
find  a  proof,  said :  "Let  us  suppose  it  not  true."  He  deduced 
the  results  of  there  being  possibly  two  parallels  to  one 
given  line  through  a  given  point,  but  feeling  the  waters 
too  deep  for  the  human  reason,  he  devoted  the  latter  half 
of  his  book  to  disproving  what  he  had  assumed  in  the  first 
part. 

Then  Bolyai  and  Lobatchewsky  with  firm  step  entered 
on  the  forbidden  path.  There  can  be  no  greater  evidence 
of  the  indomitable  nature  of  the  human  spirit,  or  of  its 
manifest  destiny  to  conquer  all  those  limitations  which 
bind  it  down  within  the  sphere  of  sense  than  this  grand 
a<sf>rtir.r.  t^f  "Rnlvai  and  Lobatchewsky. 

^ ft  Take  a  line  AB  and  a  point  c.  We 

say  and  see  and  know  that  through  c 

^ Q  can  only  be  drawn  one  line  parallel 

i'ig.  3i.  to  AB. 

But  Bolyai  said  :  "  I  will  draw  two."     Let  CD  be  parallel 


THE  SECOND   CHAPTER  IN   THE  HISTORY  OF  FOUR   SPACE  57 

to  AB,  that  is,  not  meet  AB  however  far  produced,  and  let 

lines  beyond  CD  also  not  meet 
AB;  let  there  be  a  certain 
region  between  CD  and  CE, 
in  which  no  line  drawn  meets 
AB.  CE  and  CD  produced 

lackwards  through  c  will  give  a  similar  region  on  the 
other  side  of  c. 

Nothing  so  triumphantly,  one  may  almost  say  so 
insolently,  ignoring  of  sense  had  ever  been  written  before. 
Men  had  struggled  against  the  limitations  of  the  body, 
fought  them,  despised  them,  conquered  them.  But  no 
one  had  ever  thought  simply  as  if  the  body,  the  bodily 
eyes,  the  organs  of  vision,  all  this  vast  experience  of  space, 
had  never  existed.  The  age-long  contest  of  the  ^oul  with 
the  body,  the  struggle  for  mastery,  had  come  to  a  cul- 
mination. Bolyai  and  Lobatchewsky  simply  thought  as 
if  the  body  was  not.  The  struggle  for  dominion,  the  strife 
and  combat  of  the  soul  were  over;  they  had  mastered, 
and  the  Hungarian  drew  his  line. 

Can  we  point  out  any  connection,  as  in  the  case  of 
Parmenides,  between  these  speculations  and  higher 
space  ?  Can  we  suppose  it  was  any  inner  perception  by 
the  soul  of  a  motion  not  known  to  the  senses,  which  re- 
sulted in  this  theory  so  free  from  the  bonds  of  sense  ?  No 
such  supposition  appears  to  be  possible. 

Practically,  however,  metageometry  had  a  great  in- 
fluence in  bringing  the  higher  space  to  the  front  as  a 
working  hypothesis.  This  can  be  traced  to  the  tendency 
the  mind  has  to  move  in  the  direction  of  least  resistance. 
The  results  of  the  new  geometry  could  not  be  neglected, 
the  problem  of  parallels  had  occupied  a  place  too  prominent 
in  the  development  of  mathematical  thought  for  its  final 
solution  to  be  neglected.  But  this  utter  independence  of 
all  mechanical  considerations,  this  perfect  cutting  loose 


58  THE   FOURTH   DIMENSION 

from  the  familiar  intuitions,  was  so  difficult  that  almost 
any  other  hypothesis  was  more  easy  of  acceptance,  and 
when  Beltrami  showed  that  the  geometry  of  Lobatchewsky 
and  Bolyai  was  the  geometry  of  shortest  lines  drawn  on 
certain  curved  surfaces,  the  ordinary  definitions  of  measure- 
ment being  retained,  attention  was  drawn  to  the  theory  of 
a  higher  space.  An  illustration  of  Beltrami's  theory  is 
furnished  by  the  simple  consideration  of  hypothetical 
beings  living  on  a  spherical  surface. 

Let  A  BCD  be  the  equator  of  a  globe,  and  AP,  BF, 
meridian  lines  drawn  to  the  pole,  p. 
The  lines  AB,  AP,  BP  would  seem  to  be 
perfectly  straight  to  a  person  moving 
on  the  surface  of  the  sphere,  and 
unconscious  of  its  curvature.  Now 
AP  and  BP  both  make  right  angles 
with  AB.  Hence  they  satisfy  the 
Fig.  33.  definition  of  parallels.  Yet  they 

meet  in  P.  Hence  a  being  living  on  a  spherical  surface, 
and  unconscious  of  its  curvature,  would  find  that  parallel 
lines  would  meet.  He  would  also  find  that  the  angles 
in  a  triangle  were  greater  than  two  right  angles.  In 
the  triangle  PAB,  for  instance,  the  angles  at  A  and  B 
are  right  angles,  so  the  three  angles  of  the  triangle 
PAB  are  greater  than  two  right  angles. 

Now  in  one  of  the  systems  of  metageometry  (for  after 
Lobatchewsky  had  shown  the  way  it  was  found  that  other 
systems  were  possible  besides  his)  the  angles  of  a  triangle 
are  greater  than  two  right  angles. 

Thus  a  being  on  a  sphere  would  form  conclusions  about 
his  space  which  are  the  same  as  he  would  form  if  he  lived 
on  a  plane,  the  matter  in  which  had  such  properties  as 
are  presupposed  by  one  of  these  systems  of  geometry. 
Beltrami  also  discovered  a  certain  surface  on  which  there 
could  be  drawn  more  than  one  "straight"  line  through  a 


THE   SECOND  CHAPTER  IN  THE  HISTORY  OF  FOUR  SPACE  59 

point  which  would  not  meet  another  given  line.  I  use 
the  word  straight  as  equivalent  to  the  line  having  the 
property  of  giving  the  shortest  path  between  any  two 
points  on  it.  Hence,  without  giving  up  the  ordinary 
methods  of  measurement,  it  was  possible  to  find  conditions 
in  which  a  plane  being  would  necessarily  have  an  ex- 
perience corresponding  to  Lobatchewsky's  geometry. 
And  by  the  consideration  of  a  higher  space,  and  a  solid 
curved  in  such  a  higher  space,  it  was  possible  to  account 
for  a  similar  experience  in  a  space  of  three  dimensions. 

Now,  it  is  far  more  easy  to  conceive  of  a  higher  dimen- 
sionality to  space  than  to  imagine  that  a  rod  in  rotating 
does  not  move  so  that  its  end  describes  a  circle.  Hence, 
a  logical  conception  having  been  found  harder  than  that 
of  a  four  dimensional  space,  thought  turned  to  the  latter 
as  a  simple  explanation  of  the  possibilities  to  which 
Lobatchewsky  had  awakened  it.  Thinkers  became  accus- 
tomed to  deal  with  the  geometry  of  higher  space — it  was 
Kant,  says  Veronese,  who  first  used  the  expression  of 
"  different  spaces  " — and  with  familiarity  the  inevitable- 
ness  of  the  conception  made  itself  felt. 

From  this  point  it  is  but  a  small  step  to  adapt  the 
ordinary  mechanical  conceptions  to  a  higher  spatial 
existence,  and  then  the  recognition  of  its  objective 
existence  could  be  delayed  no  longer.  Here,  too,  as  in  so 
many  cases,  it  turns  out  that  the  order  and  connection  of 
our  ideas  is  the  order  and  connection  of  things. 

What  is  the  significance  of  Lobatchewsky's  and  Bolyai's 
work  ? 

It  must  be  recognised  as  something  totally  different 
from  the  conception  of  a  higher  space ;  it  is  applicable  to 
spaces  of  any  number  of  dimensions.  By  immersing  the 
conception  of  distance  in  matter  to  which  it  properly 
belongs,  it  promises  to  be  of  the  greatest  aid  in  analysis 
for  the  effective  distance  of  any  two  particles  is  the 


60  THE    FOURTH    DIMENSION 

product  of  complex  material  conditions  and  cannot  be 
measured  by  hard  and  fast  rules.  Its  ultimate  signi- 
ficance is  altogether  unknown.  It  is  a  cutting  loose 
from  the  bonds  of  sense,  not  coincident  with  the  recognition 
of  a  higher  dimensionality,  but  indirectly  contributory 
thereto. 

Thus,  finally,  we  have  come  to  accept  what  Plato  held 
in  the  hollow  of  his  hand ;  what  Aristotle's  doctrine  of 
the  relativity  of  substance  implies.  The  vast  universe,  too, 
has  its  higher,  and  in  recognising  it  we  find  that  the 
directing  being  within  us  no  longer  stands  inevitably 
outside  our  systematic  knowledge. 


THE  HIGHER  WORLD 

IT  is  indeed  strange,  the  manner  in  which  we  must  begin 
to  think  about  the  higher  world. 

Those  simplest  objects  analogous  to  those  which  are 
about  us  on  every  side  in  our  daily  experience  such  as  a 
door,  a  table,  a  wheel  are  remote  and  uncognisable  in  the 
world  of  four  dimensions,  while  the  abstract  ideas  of 
rotation,  stress  and  strain,  elasticity  into  which  analysis 
resolves  the  familiar  elements  of  our  daily  experience  are 
transferable  and  applicable  with  no  difficulty  whatever. 
Thus  we  are  in  the  unwonted  position  qf  being  obliged 
to  construct  the  daily  and  habitual  experience  of  a  four- 
dimensional  being,  from  a  knowledge  of  the  abstract 
theories  of  the  space,  the  matter,  the  motion  of  it ; 
instead  of,  as  in  our  case,  passing  to  the  abstract  theories 
from  the  richness  of  sensible  things. 

What  would  a  wheel  be  in  four  dimensions?  What 
the  shafting  for  the  transmission  of  power  which  a 
four-dimensional  being  would  use. 

The  four-dimensional  wheel,  and  the  four-dimensional 
shafting  are  what  will  occupy  us  for  these  few  pages.  And 
it  is  no  futile  or  insignificant  enquiry.  For  in  the  attempt 
to  penetrate  into  the  nature  of  the  higher,  to  grasp  within 
our  ken  that  which  transcends  all  analogies,  because  what 
we  know  are  merely  partial  views  of  it,  the  purely 
material  and  physical  path  affords  a  means  of  approach 

61 


FOURTH   DIMENSION 

pursuing  which  we  are  in  less  likelihood  of  error  than  if 
we  use  the  more  frequently  trodden  path  of  framing 
conceptions  which  in  their  elevation  and  beauty  seem  to 
us  ideally  perfect. 

For  where  we  are  concerned  with  our  own  thoughts,  the 
development  of  our  own  ideals,  we  are  as  it  were  on  a 
curve,  moving  at  any  moment  in  a  direction  of  tangency. 
Whither  we  go,  what  we  set  up  and  exalt  as  perfect, 
represents  not  the  true  trend  of  the  curve,  but  our  own 
direction  at  the  present — a  tendency  conditioned  by  the 
past,  and  by  a  vital  energy  of  motion  essential  but 
only  true  when  perpetually  modified.  That  eternal  cor- 
rector of  our  aspirations  and  ideals,  the  material  universe 
draws  sublimely  away  from  the  simplest  things  we  can 
touch  or  handle  to  the  infinite  depths  of  starry  space, 
in  one  and  all  uninfluenced  by  what  we  think  or  feel, 
presenting  unmoved  fact  to  which,  think  it  good  or 
think  it  evil,  we  can  but  conform,  yet  out  of  all  that 
impassivity  with  a  reference  to  something  beyond  our 
individual  hopes  and  fears  supporting  us  and  giving  us 
our  being. 

And  to  this  great  being  we  come  with  the  question  : 
"  You,  too,  what  is  your  higher  ?  " 

Or  to  put  it  in  a  form  which  will  leave  our  conclusions  in 
the  shape  of  no  barren  formula,  and  attacking  the  problem 
on  its  most  assailable  side  :  "  What  is  the  wheel  and  the 
shafting  of  the  four-dimensional  mechanic  ?  " 

In  entering  on  this  enquiry  we  must  make  a  plan  of 
procedure.  The  method  which  I  shall  adopt  is  to  trace 
out  the  steps  of  reasoning  by  which  a  being  confined 
to  movement  in  a  two-dimensional  world  could  arrive  at  a 
conception  of  our  turning  and  rotation,  and  then  to  apply 
an  analogous  process  to  the  consideration  of  the  higher 
movements.  The  plane  being  must  be  imagined  as  no 
abstract  figure,  but  as  a  real  body  possessing  all  three 


63 

dimensions.  His  limitation  to  a  plane  must  be  the  result 
of  physical  conditions. 

We  will  therefore  think  of  him  as  of  a  figure  cut  out  of 
paper  placed  on  a  smooth  plane.  Sliding  over  this  plane, 
and  coming  into  contact  with  other  figures  equally  thin 
as  he  in  the  third  dimension,  he  will  apprehend  them  only 
by  their  edges.  To  him  they  will  be  completely  bounded 
by  lines.  A  "  solid  "  body  will  be  to  him  a  two-dimensional 
extent,  the  interior  of  which  can  only  be  reached  by 
penetrating  through  the  bounding  lines. 

Now  such  a  plane  being  can  think  of  our  three- 
dimensional  existence  in  two  ways. 

First,  he  can  think  of  it  as  a  series  of  sections,  each  like 
the  solid  he  knows  of  extending  in  a  direction  unknown 
to  him,  which  stretches  transverse  to  his  tangible 
universe,  which  lies  in  a  direction  at  right  angles  to  every 
motion  which  he  made. 

Secondly,  relinquishing  the  attempt  to  think  of  the 
three-dimensional  solid  body  in  its  entirety  he  can  regard 
it  as  consisting  of  a  number  of  plane  sections,  each  of  them 
in  itself  exactly  like  the  two-dimensional  bodies  he  knows, 
but  extending  away  from  his  two-dimensional  space. 

A  square  lying  in  his  space  he  regards  as  a  solid 
bounded  by  four  lines,  each  of  which  lies  in  his  space. 

A  square  standing  at  right  angles  to  his  plane  appears 
to  him  as  simply  a  line  in  his  plane,  for  all  of  it  except 
the  line  stretches  in  the  third  dimension. 

He  can  think  of  a  three-dimensional  body  as  consisting 
of  a  number  of  such  sections,  each  of  which  starts  from  a 
line  in  his  space. 

Now,  since  in  his  world  he  can  make  any  drawing  or 
model  which  involves  only  two  dimensions,  he  can  represent 
each  such  upright  section  as  it  actually  is,  and  can  repre- 
sent a  turning  from  a  known  into  the  unknown  dimension 
as  a  turning  from  one  to  another  of  his  known  dimensions. 


64  THE  FOURTH  DIMENSION 

To  see  the  whole  he  must  relinquish  part  of  that  which 
he  has,  and  take  the  whole  portion  by  portion. 

Consider  now  a  plane  being  in  front  of  a  square,  fig.  34. 
The  square  can  turn  about  any  point 
in  the  plane — say  the  point  A.  But  it 
cannot  turn  about  a  line,  as  AB.  For, 
in  order  to  turn  about  the  line  AB, 
the  square  must  leave  the  plane  and 
move  in  the  third  dimension.  This 


"  &  motion  is  out  of  his  range  of  observa- 

tion,  and   is   therefore,  except  for  a 
process  of  reasoning,  inconceivable  to  him. 

Rotation  will  therefore  be  to  him  rotation  about  a  point. 
Rotation  about  a  line  will  be  inconceivable  to  him. 

The  result  of  rotation  about  a  line  he  can  appprehend. 
He  can  see  the  first  and  last  positions  occupied  in  a  half 
revolution  about  the  line  AC.  The  result  of  such  a  half  revo- 
lution is  to  place  the  square  ABCD  on  the  left  hand  instead 
of  on  the  right  hand  of  the  line  AC.  It  would  correspond 
to  a  pulling  of  the  whole  body  ABCD  through  the  line  AC, 
or  to  the  production  of  a  solid  body  which  was  the  exact 
reflection  of  it  in  the  line  AC.  It  would  be  as  if  the  square 
ABCD  turned  into  its  image,  the  line  AB  acting  as  a  mirror. 
Such  a  reversal  of  the  positions  of  the  parts  of  the  square 
would  be  impossible  in  his  space.  The  occurrence  of  it 
would  be  a  proof  of  the  existence  of  a  higher  dimensionality. 
Let  him  now,  adopting  the  conception  of  a  three- 
dimensional  body  as  a  series  of 
sections  lying,  each  removed  a  little 
farther  than  the  preceding  one,  in 
„  direction  at  right  angles  to  his 

plane,  regard  a  cube,  fig.  36,  as  a 
series    of    sections,   each    like   the 


0         *    square   which   forms    its    base,   all 
Fig.  35.  rigidly  connected  together. 


filGttBR   WOELt) 


65 


tf  DOW  he  turns  the  square  about  the  point  A  in  the 
plane  of  xy,  each  -parallel  section  turns  with  the  square 
he  moves.  In  each  of  the  sections  there  is  a  point  at 
rest,  that  vertically  over  A.  Hence  he  would  conclude 
that  in  the  turning  of  a  three-dimensional  body  there  is 
one  line  which  is  at  rest.  That  is  a  three-dimensional 
turning  in  a  turning  about  a  line. 

In  a  similar  way  let  us  regard  ourselves  as  limited  to  a 
three-dimensional  world  by  a  physical  condition.  Let  us 
imagine  that  there  is  a  direction  at  right  angles  to  every 
direction  in  which  we  can  move,  and  that  we  are  pre- 
vented from  passing  in  this  direction  by  a  vast  solid,  that 
against  which  in  every  movement  we  make  we  slip  as 
the  plane  being  slips  against  his  plane  sheet. 

We  can  then  consider  a  four-dimensional  body  as  con- 
sisting of  a  series  of  sections,  each  parallel  to  our  space, 
and  each  a  little  farther  off  than  the  preceding  on  the 
unknown  dimension. 

Take  the  simplest  four-dimensional  body — one  which 
begins  as  a  cube,  fig.  36,  in  our 
space,  and  consists  of  sections,  each 
a  cube  like  fig.  36,  lying  away  from 
our  space.  If  we  turn  the  cube 
which  is  its  base  in  our  space 
about  a  line,  if,  e.g.,  in  fig.  36  we 
turn  the  cube  about  the  line  AB, 
not  only  it  but  each  of  the  parallel 
cubes  moves  about  a  line.  The 


t 

f 

H 

\ 

\ 

\ 

8 
G 

D 

\ 

A  C 

Fig.  36. 

cube  we  see  moves  about  the  line  AB,  the  cube  beyond  it 
about  a  line  parallel  to  AB  and  so  on.  Hence  the  whole 
four-dimensional  body  moves  about  a  plane,  for  the 
assemblage  of  these  lines  is  our  way  of  thinking  about  the 
plane  which,  starting  from  the  line  AB  in  our  space,  runs 
off  in  the  unknown  direction. 


(>6  1HE   FOURta   DIMENSION 


In  this  case  all  that  we  see  of  the  pjane  about  which 
the  turning  takes  place  is  the  line  AB. 

But  it  is  obvious  that  the  axis  plane  may  lie  in  our 
space.  A  point  near  the  plane  determines  with  it  a  three- 
dimensional  space.  When  it  begins  to  rotate  round  the 
plane  it  does  not  move  anywhere  in  this  three-dimensional 
space,  but  moves  out  of  it.  A  point  can  no  more  rotate 
round  a  plane  in  three-dimensional  space  than  a  point 
can  move  round  a  line  in  two-dimensional  space. 

We  will  now  apply  the  second  of  the  modes  of  repre- 
sentation to  this  case  of  turning  about  a  plane,  building 
up  our  analogy  step  by  step  from  the  turning  in  a  plane 
about  a  point  and  that  in  space  about  a  line,  and  so  on. 

In  order  to  reduce  our  considerations  to  those  of  the 
greatest  simplicity  possible,  let  us  realise  how  the  plane 
being  would  think  of  the  motion  by  which  a  square  is 
turned  round  a  line. 

Let,  fig.  34,  ABCD  be  a  square  on  his  plane,  and  repre- 
sent the  two  dimensions  of  his  space  by  the  axes  AX  Ay. 

Now  the  motion  by  which  the  square  is  turned  over 
about  the  line  AC  involves  the  third  dimension. 

He  cannot  represent  the  motion  of  the  whole  square  in 
its  turning,  but  he  can  represent  the  motions  of  parts  of 
it.  Let  the  third  axis  perpendicular  to  the  plane  of  the 
paper  be  called  the  axis  of  z.  Of  the  three  axes  x,  y,  z, 
the  plane  being  can  represent  any  two  in  his  space.  Let 
him  then  draw,  in  fig.  35,  two  axes,  x  and  z.  Here  he  has 
in  his  plane  a  representation  of  what  exists  in  the  plane 
which  goes  off  perpendicularly  to  his  space. 

In  this  representation  the  square  would  not  be  shown, 
for  in  the  plane  of  xz  simply  the  line  AB  of  the  square  is 
contained. 

The  plane  being  then  would  have  before  him,  in  fig.  35, 
the  representation  of  one  line  AB  of  his  square  and  two 
axes,  x  and  z,  at  right  angles.  Now  it  would  be  obvious 


THE   HIGHER  WORLD  67 

to  him  that,  by  a  turning  such  as  he  knows,  by  a  rotation 
about  a  point,  the  line  AB  can  turn  round  A,  and  occu- 
pying all  the  intermediate  positions,  such  as  ABi,  come 
after  half  a  revolution  to  lie  as  AX  produced  through  A. 

Again,  just  as  he  can  represent  the  vertical  plane 
through  AB,  so  he  can  represent  the  vertical  plane 
through  A'B',  fig.  34,  and  in  a  like  manner  can  see  that 
the  line  A'B'  can  turn  about  the  point  A'  till  it  lies  in  the 
opposite  direction  from  that  which  it  ran  in  at  first. 

Now  these  two  turnings  are  not  inconsistent.  In  his 
plane,  if  AB  turned  about  A,  and  A'B'  about  A',  the  con- 
sistency of  the  square  would  be  destroyed,  it  would  be  an 
impossible  motion  for  a  rigid  body  to  perform.  But  in 
the  turning  which  he  studies  portion  by  portion  there  is 
nothing  inconsistent.  Each  line  in  the  square  can  turn 
in  this  way,  hence  he  would  realise  the  turning  of  the 
whole  square  as  the  sum  of  a  number  of  turnings  of 
isolated  parts.  Such  turnings,  if  they  took  place  in  his 
plane,  would  be  inconsistent,  but  by  virtue  of  a  third 
dimension  they  are  consistent,  and  the  result  of  them  all 
is  that  the  square  turns  about  the  line  AC  and  lies  in  a 
position  in  which  it  is  the  mirror  image  of  what  it  was  in 
its  first  position.  Thus  he  can  realise  a  turning  about  a 
line  by  relinquishing  one  of  his  axes,  and  representing  his 
body  part  by  part. 

Let  us  apply  this  method  to  the  turning  of  a  cube  so  as 
to  become  the  mirror  image  of  itself.  In  our  space  we  can 
construct  three  independent  axes,  x,  y,  z,  shown  in  fig.  36. 
Suppose  that  there  is  a  fourth  axis,  w,  at  right  angles  to 
each  and  every  one  of  them.  We  cannot,  keeping  all 
three  axes,  a;,  y,  z,  represent  iv  in  our  space ;  but  if  we 
relinquish  one  of  our  three  axes  we  can  let  the  fourth  axis 
take  its  place,  and  we  can  represent  what  lies  in  the 
space,  determined  by  the  two  axes  we  retain  and  the 
fourth  axis. 


B 


B 


Let  us  suppose  that  we  let  the  y  axis  drop,  and  that 
we  represent  the  w  axis  as  occupy- 
ing its  direction.  We  have  in  fig. 
37  a  drawing  of  what  we  should 
then  see  of  the  cube.  The  square 
ABCD,  remains  unchanged,  for  that 

- : £ is   in    the   plane    of    xz,   and   we 

still  have  that  plane.     But  from 
this  plane  the  cube  stretches  out 

in  the  direction  of  the  y  axis.     Now  the  y  axis  is  gone, 
and  fo  we  have  no  more  of  the  cube  than  the  face  ABCD. 
Considering  now  this  face  ABCD,  we 
see  that  it  is  free  to  turn  about  the 
line  AB.     It  can  rotate  in  the  a?  to  w 
direction  about  this  line.     In  fig.  38 
it  is  shown  on  its  way,  and   it   can 
evidently  continue  this  rotation  till 
A  *     it   lies   on    the   other   side   of  the  z 

axis  in  the  plane  of  xz. 

We  can  also  take  a  section  parallel  to  the  face  ABCD, 
and  then  letting  drop  all  of  our  space  except  the  plane  of 
that  section,  introduce  the  w  axis,  running  in  the  old  y 
direction.  This  section  can  be  represented  by  the  same 
drawing,  fig.  38,  and  we  see  that  it  can  rotate  about  the 
line  on  its  left  until  it  swings  half  way  round  and  runs  in 
the  opposite  direction  to  that  which  it  ran  in  before. 
These  turnings  of  the  different  sections  are  not  incon- 
sistent, and  taken  all  together  they  will  bring  the  cubt- 
from  the  position  shown  in  fig.  36  to  that  shown  in 
fig.  41. 

Since  we  have  three  axes  at  our  disposal  in  our  space, 
we  are  not  obliged  to  represent  the  w  axis  by  any  particular 
one.  We  may  let  any  axis  we  like  disappear,  and  let  th« 
fourth  axis  take  its  place. 

In  fig.  36  suppose  the  z  axis  to  go.     We  have  then 


THE   HIGHER   WORLD 


w 


simply  the  plane  of  xy  and  the  square  base  of  the 
cube  ACEG,  fig.  39,  is  all  that  could 
be  seen  of  it.  Let  now  the  w  axis 
take  the  place  of  the  z  axis  and 
we  have,  in  fig.  39  again,  a  repre- 
sentation of  the  space  of  xyiv,  in 
A  C  which  all  that  exists  of  the  cube  is 

its  square  base.    Now,  by  a  turning 

of  x  to  w,  this  base  can  rotate  around  the  line  AE,  it  is 
shown  on  its  way  in  fig.  40,  and 
finally  it  will,  after  half  a  revolution, 
lie  on  the  other  side  of  the  y  axis. 
In  a  similar  way  we  may  rotate 
sections  parallel  to  the  base  of  the 
xw  rotation,  and  each  of  them  comes 
to  run  in  the  opposite  direction  from 
that  which  they  occupied  at  first. 

Thus  again  the  cube  comes  from  the  position  of  fig.  36. 

to  that  of  fig.  41.  In  this  x 
to  w  turning,  we  see  that  it 
takes  place  by  the  rotations  of 
sections  parallel  to  the  front 
face  about  lines  parallel  to  AB, 
or  else  we  may  consider  it  as 
consisting  of  the  rotation  of 
sections  parallel  to  the  base 
about  lines  parallel  to  AE.  It 


H 


C  A        * 

2-posificn      I -position 

Fig.  41. 

is  a  rotation  of  the  whole  cube  about  the  plane  ABEF. 
Two  separate  sections  could  not  rotate  about  two  separate 
lines  in  our  space  without  conflicting,  but  their  motion  is 
consistent  when  we  consider  another  dimension.  Just, 
then,  as  a  plane  being  can  think  of  rotation  about  a  line  as 
a  rotation  about  a  number  of  points,  these  rotations  not 
interfering  as  they  would  if  they  took  place  in  his  two- 
(Ijrnensjonal  space,  so  we  can  think  of  a  rotation  about  a 


70 


THE   FOURTH   DIMENSION 


plane  as  the  rotation  of  a  number  of  sections  of  a  body 
about  a  number  of  lines  in  a  plane,  these  rotations  not 
being  inconsistent  in  a  four-dimensional  space  as  they  are 
in  three-dimensional  space. 

We  are  not  limited  to  any  particular  direction  for  the 
lines  in  the  plane  about  which  we  suppose  the  rotation 
of  the  particular  sections  to  take  place.  Let  us  draw 
the  section  of  the  cube,  fig.  36,  through  A,  F,  C,  H,  forming  a 
sloping  plane.  Now  since  the  fourth  dimension  is  at 
right  angles  to  every  line  in  our  space  it  is  at  right 
angles  to  this  section  also.  We  can  represent  our  space 
by  drawing  an  axis  at  right  angles  to  the  plane  ACEG,  our 
space  is  then  determined  by  the  plane  ACEG,  and  the  per- 
pendicular axis.  If  we  let  this  axis  drop  and  suppose  the 
fourth  axis,  w,  to  take  its  place,  we  have  a  representation  of 
the  space  which  runs  off  in  the  fourth  dimension  from  the 
plane  ACEG.  In  this  space  we  shall  see  simply  the  section 
ACEG  of  the  cube,  and  nothing  else,  for  one  cube  does  not 
extend  to  any  distance  in  the  fourth  dimension. 

If,  keeping  this  plane,  we  bring  in  the  fourth  dimension, 
we  shall  have  a  space  in  which  simply  this  section  of 
the  cube  exists  and  nothing  else.  The  section  can  turn 
about,  the  line  AF,  and  parallel  sections  can  turn  about 

parallel  lines.  Thus  in  con- 
sidering the  rotation  about 
a  plane  we  can  draw  any 
lines  we  like  and  consider 
the  rotation  as  taking  place 
in  sections  about  them. 

To  bring  out  this  point 
more  clearly  let  us  take  two 
parallel  lines,  A  and  B,  in 
the  space  of  xyz,  and  let  CD 
and  EF  be  two  rods  running 
If  we 


8 


* 

Fig.  42. 

above  a.n4  below  the  plane  pf  xy,  from  these  lines. 


THE   HIGHER  WORLD  7] 

turn  these  rods  in  our  space  about  the  lines  A  and  B,  as 
the  upper  end  of  one,  F,  is  going  down,  the  lower  end  of 
the  other,  c,  will  be  coming  up.  They  will  meet  and 
conflict.  But  it  is  quite  possible  for  these  two  rods 
each  of  them  to  turn  about  the  two  lines  without  altering 
their  relative  distances. 

To  see  this  suppose  the  y  axis  to  go,  and  let  the  w  axis 
take  its  place.     We  shall  see  the  lines  A  and  B  no  longer, 
for  they  run  in  the  y  direction  from  the  points  G  and  H. 
Fig.  43  is  a  picture  of  the  two  rods  seen  in  the  space 
of  xzw.     If  they  rotate  in  the 
direction  shown  by  the  arrows — 
in  the  z   to  w  direction — they 
move   parallel  to   one  another, 
keeping  their  relative  distances. 
Each  will  rotate  about  its  own 
line,  but  their  rotation  will  not 
be  inconsistent  with  their  form- 
ing part  of  a  rigid  body. 

Now  we  have  but  to  suppose 
a  central  plane  with  rods  crossing 
it  at  every  point,  like  CD  and  EF  cross  the  plane  of  xy, 
to  have  an  image  of  a  mass  of  matter  extending  equal 
distances  on  each  side  of  a  diametral  plane.  As  two  of 
these  rods  can  rotate  round,  so  can  all,  and  the  whole 
mass  of  matter  can  rotate  round  its  diametral  plane. 

This  rotation  round  a  plane  corresponds,  in  four 
dimensions,  to  the  rotation  round  an  axis  in  three 
dimensions.  Rotation  of  a  body  round  a  plane  is  the 
analogue  of  rotation  of  a  rod  round  an  axis. 

In  a  plane  we  have  rotation  round  a  point,  in  three- 
space  rotation  round  an  axis  line,  in  four-space  rotation 
round  an  axis  plane. 

The  four-dimensional  being's  shaft  by  which  he  trans- 
mits power  is  a  disk  rotating  round  its  central  plane— 


72  THE   FOURTH    DIMENSION 

the  whole  contour  corresponds  to  the  ends  of  an  axis 
of  rotation  in  our  space.  He  can  impart  the  rotation  at 
any  point  and  take  it  off  at  any  other  point  on  the  contour, 
just  as  rotation  round  a  line  can  in  three-space  be  imparted 
at  one  end  of  a  rod  and  taken  off  at  the  other  end. 

A  four-dimensional  wheel  can  easily  be  described  from 
the  analogy  of  the  representation  which  a  plane  being 
would  form  for  himself  of  one  of  our  wheels. 

Suppose  a  wheel  to  move  transverse  to  a  plane,  so  that 
the  whole  disk,  which  I  will  consider  to  be  solid  and 
without  spokes,  came  at  the  same  time  into  contact  with 
the  plane.  It  would  appear  as  a  circular  portion  of  plane 
matter  completely  enclosing  another  and  smaller  portion — 
the  axle. 

This  appearance  would  last,  supposing  the  motion  of 
the  wheel  to  continue  until  it  had  traversed  the  plane  by 
the  extent  of  its  thickness,  when  there  would  remain  in 
the  plane  only  the  small  disk  which  is  the  section  of  the 
axle.  There  would  be  no  means  obvious  in  the  plane 
at  first  by  which  the  axle  could  be  reached,  except  by 
going  through  the  substance  of  the  wheel.  But  the 
possibility  of  reaching  it  without  destroying  the  substance 
of  the  wheel  would  be  shown  by  the  continued  existence 
of  the  axle  section  after  that  of  the  wheel  had  disappeared. 

In  a  similar  way  a  four-dimensional  wheel  moving 
transverse  to  our  space  would  appear  first  as  a  solid  sphere, 
completely  surrounding  a  smaller  solid  sphere.  The 
outer  sphere  would  represent  the  wheel,  and  would  last 
until  the  wheel  has  traversed  our  space  by  a  distance 
equal  to  its  thickness.  Then  the  small  sphere  alone 
would  remain,  representing  the  section  of  the  axle.  The 
large  sphere  could  move  round  the  small  one  quite  freely. 
Any  line  in  space  could  be  taken  as  an  axis,  and  round 
this  line  the  outer  sphere  could  rotate,  while  the  inner 
sphere  remained  still.  But  in  all  these  directions  of 


THE   HIGHER   WORLD  73 

revolution  there  would  be  in  reality  one  line  which 
remained  unaltered,  that  is  the  line  which  stretches  away 
in  the  fourth  direction,  forming  the  axis  of  the  axle.  The 
four-dimensional  wheel  can  rotate  in  any  number  of  planes, 
but  all  these  planes  are  such  that  there  is  a  line  at  right 
angles  to  them  all  unaffected  by  rotation  in  them. 

An  objection  is  sometimes  experienced  as  to  this  mode 
of  reasoning  from  a  plane  world  to  a  higher  dimensionality. 
How  artificial,  it  is  argued,  this  conception  of  a  plane 
world  is.  If  any  real  existence  confined  to  a  superficies 
could  be  shown  to  exist,  there  would  be  an  argument  for 
one  relative  to  which  our  three-dimensional  existence  is 
superficial.  But,  both  on  the  one  side  and  the  other  of 
the  space  we  are  familiar  with,  spaces  either  with  less 
or  more  than  three  dimensions  are  merely  arbitrary 
conceptions. 

In  reply  to  this  I  would  remark  that  a  plane  being 
having  one  less  dimension  than  our  three  would  have  one- 
third  of  our  possibilities  of  motion,  while  we  have  only 
one-fourth  less  than  those  of  the  higher  space.  It  may 
very  well  be  that  there  may  be  a  certain  amount  of 
freedom  of  motion  which  is  demanded  as  a  condition  of  an 
organised  existence,  and  that  no  material  existence  is 
possible  with  a  more  limited  dimensionality  than  ours. 
This  is  well  seen  if  we  try  to  construct  the  mechanics  of  a 
two-dimensional  world.  No  tube  could  exist,  for  unless 
joined  together  completely  at  one  end  two  parallel  lines 
would  be  completely  separate.  The  possibility  of  an 
organic  structure,  subject  to  conditions  such  as  this,  is 
highly  problematical ;  yet,  possibly  in  the  convolutions 
of  the  brain  there  may  be  a  mode  of  existence  to  be 
described  as  two-dimensional. 

We  have  but  to  suppose  the  increase  in  surface  and 
the  diminution  in  mass  carried  on  to  a  certain  extent 
to  fi,nd  a  region  which,  though  without  mobility  of  the 


74  THE   FOURTH  DIMENSION 

constituents,  would  have  to  be  described  as  two-dimensional. 

But,  however  artificial  the  conception  of  a  plane  being 
may  be,  it  is  none  the  less  to  be  used  in  passing  to  the 
conception  of  a  greater  dimensionality  than  ours,  and 
hence  the  validity  of  the  first  part  of  this  objection 
altogether  disappears  directly  we  find  evidence  for  such  a 
state  of  being. 

The  second  part  of  the  objection  has  more  weight. 
How  is  it  possible  to  conceive  that  in  a  four-dimensional 
space  any  creatures  should  be  confined  to  a  three- 
dimensional  existence? 

In  reply  I  would  say  that  we  know  as  a  matter  of  fact 
that  life  is  essentially  a  phenomenon  of  surface.  The 
amplitude  of  the  movements  which  we  can  make  is  much 
greater  along  the  surface  of  the  earth  than  it  is  up 
or  down. 

Now  we  have  but  to  conceive  the  extent  of  a  solid 
surface  increased,  while  the  motions  possible  tranverse  to 
it  are  diminished  in  the  same  proportion,  to  obtain  the 
image  of  a  three-dimensional  world  in  four-dimensional 
space. 

And  as  our  habitat  is  the  meeting  of  air  and  earth  on 
the  world,  so  we  must  think  of  the  meeting  place  of  two 
as  affording  the  condition  for  our  universe.  The  meeting 
of  what  two  ?  What  can  that  vastness  be  in  the  higher 
space  which  stretches  in  such  a  perfect  level  that  our 
astronomical  observations  fail  to  detect  the  slightest 
curvature  ? 

The  perfection  of  the  level  suggests  a  liquid — a  lake 
amidst  what  vast  scenery  ! — whereon  the  matter  of  the 
universe  floats  speck-like. 

But  this  aspect  of  the  problem  is  like  what  are  called 
in  mathematics  boundary  conditions. 

We  can  trace  out  all  the  consequences  of  four-dimen- 
sional movements  down  to  their  last  detail.  Then,  knowing 


THE   HIGHER   WORLD.  75 

the  mode  of  action  which  would  be  characteristic  of  the 
mioutest  particles,  if  they  were  free,  we  can  draw  con- 
clusions from  what  they  actually  do  of  what  the  constraint 
on  them  is.  Of  the  two  things,  the  material  conditions  and 
the  motion,  one  is  known,  and  the  other  can  be  inferred. 
If  the  place  of  this  universe  is  a  meeting  of  two,  there 
would  be  a  one-sideness  to  space.  If  it  lies  so  that  what 
stretches  away  in  one  direction  in  the  unknown  is  unlike 
what  stretches  away  in  the  other,  then,  as  far  as  the 
movements  which  participate  in  that  dimension  are  con- 
cerned, there  would  be  a  difference  as  to  which  way  the 
motion  took  place.  This  would  be  shown  in  the  dissimi- 
larity of  phenomena,  which,  so  far  as  all  three-space 
movements  are  concerned,  were  perfectly  symmetrical. 
To  take  an  instance,  merely,  for  the  sake  of  precising 
our  ideas,  not  for  any  inherent  probability  in  it ;  if  it  could 
be  shown  that  the  electric  current  in  the  positive  direction 
were  exactly  like  the  electric  current  in  the  negative 
direction,  except  for  a  reversal  of  the  components  of  the 
motion  in  three-dimensional  space,  then  the  dissimilarity 
of  the  discharge  from  the  positive  and  negative  poles 
would  be  an  indication  of  a  one-sideness  to  our  space. 
The  only  cause  of  difference  in  the  two  discharges  would 
be  due  to  a  component  in  the  fourth  dimension,  which 
directed  in  one  direction  transverse  to  our  space,  met  with 
a  different  resistance  to  that  which  it  met  when  directed 
in  the  opposite  direction. 


CHAPTER   VII 
THE  EVIDENCES  FOR  A  FOURTH  DIMENSION 

THE  method  necessarily  to  be  employed  in  the  search  for 
the  evidences  of  a  fourth  dimension,  consists  primarily  in 
the  formation  of  the  conceptions  of  four-dimensional 
shapes  and  motions.  When  we  are  in  possession  of  these 
it  is  possible  to  call  in  the  aid  of  observation,  without 
them  we  may  have  been  all  our  lives  in  the  familiar 
presence  of  a  four-dimensional  phenomenon  without  ever 
recognising  its  nature. 

To  take  one  of  the  conceptions  we  have  already  formed, 
the  turning  of  a  real  thing  into  its  mirror  image  would  be 
an  occurrence  which  it  would  be  hard  to  explain,  except  on 
the  assumption  of  a  fourth  dimension. 

We  know  of  no  such  turning.  But  there  exist  a  multi- 
tude of  forms  which  show  a  certain  relation  to  a  plane, 
a  relation  of  symmetry,  which  indicates  more  than  an  acci- 
dental juxtaposition  of  parts.  In  organic  life  the  universal 
type  is  of  right-  and  left-handed  symmetry,  there  is  a  plane 
on  each  side  of  which  the  parts  correspond.  Now  we  have 
seen  that  in  four  dimensions  a  plane  takes  the  place  of  a 
line  in  three  dimensions.  In  our  space,  rotation  about  an 
axis  is  the  type  of  rotation,  and  the  origin  of  bodies  sym- 
metrical about  a  line  as  the  earth  is  symmetrical  about  an 
axis  can  easily  be  explained.  But  where  there  is  symmetry 
a.bout  a  plane  no  simple  physical  motion,  such  as  we 


EVIDENCES   FOR   A   FotJRtS   DIMENSION 

are  accustomed  to,  suffices  to  explain  it.  In  our  space  a 
symmetrical  object  must  be  built  up  by  equal  additions 
on  each  side  of  a  central  plane.  Such  additions  about 
such  a  plane  are  as  little  likely  as  any  other  increments. 
The  probability  against  the  existence  of  symmetrical 
form  in  inorganic  nature  is  overwhelming  in  our  space, 
and  in  organic  forms  they  would  be  as  difficult  of  produc- 
tion as  any  other  variety  of  configuration.  To  illustrate 
this  point  we  may  take  the  child's  amusement  of  making 
from  dots  of  ink  on  a  piece  of  paper  a  life-like  repre- 
sentation of  an  insect  by  simply  folding  the  paper 
over.  The  dots  spread  out  on  a  symmetrical  line,  and 
give  the  impression  of  a  segmented  form  with  antenna 
and  legs. 

Now  seeing  a  number  of  such  figures  we  should 
naturally  infer  a  folding  over.  Can,  then,  a  folding  over 
in  four-dimensional  space  account  for  the  symmetry  of 
organic  forms  ?  The  folding  cannot  of  course  be  of  the 
bodies  we  see,  but  it  may  be  of  those  minute  constituents, 
the  ultimate  elements  of  living  matter  which,  turned  in  one 
way  or  the  other,  become  right-  or  left-handed,  and  so 
produce  a  corresponding  structure. 

There  is  something  in  life  not  included  in  our  concep- 
tions of  mechanical  movement.  Is  this  something  a  four- 
dimensional  movement? 

If  we  look  at  it  from  the  broadest  point  of  view,  there  is 
something  striking  in  the  fact  that  where  life  comes  in 
there  arises  an  entirely  different  set  of  phenomena  to 
those  of  the  inorganic  world. 

The  interest  and  values  of  life  as  we  know  it  in  our- 
selves, as  we  know  it  existing  around  us  in  subordinate 
forms,  is  entirely  and  completely  different  to  anything 
which  inorganic  nature  shows.  And  in  living  beings  we 
have  a  kind  of  form,  a  disposition  of  matter  which  is 
entirely  different  from  that  shown  in  inorganic  matter. 


78  THE  FOURTH   DIMENSION 

Right-  and  left-handed  symmetry  does  not  occur  in  the 
configurations  of  dead  matter.  We  have  instances  of 
symmetry  about  an  axis,  but  not  about  a  plane.  It  can 
be  argued  that  the  occurrence  of  symmetry  in  two  dimen- 
sions involves  the  existence  of  a  three-dimensional  process, 
as  when  a  stone  falls  into  water  and  makes  rings  of  ripples, 
or  as  when  a  mass  of  soft  material  rotates  about  an  axis. 
It  can  be  argued  that  symmetry  in  any  number  of  dimen- 
sions is  the  evidence  of  an  action  in  a  higher  dimensionality. 
Thus  considering  living  beings,  there  is  an  evidence  both 
in  their  structure,  and  their  different  mode  of  activity,  of  a 
something  coming  in  from  without  into  the  inorganic 
world. 

And  the  objections  which  will  readily  occur,  such  as 
those  derived  from  the  forms  of  twin  crystals  and  the 
theoretical  structure  of  chemical  molecules,  do  not  in- 
validate the  argument ;  for  in  these  forms  too  the 
presumable  seat  of  the  activity  producing  them  lies  in  that 
very  minute  region  in  which  we  necessarily  place  the  seat 
of  a  four-dimensional  mobility. 

In  another  respect  also  the  existence  of  symmetrical  forms 
is  noteworthy.  It  is  puzzling  to  conceive  how  two  shapes 
exactly  equal  can  exist  which  are  not  superposible.  Such 
a  pair  of  symmetrical  figures  as  the  two  hands,  right  and 
left,  show  either  a  limitation  in  our  power  of  movement, 
by  which  we  cannot  superpose  the  one  on  the  other,  or  a 
definite  influence  and  compulsion  of  space  on  matter, 
inflicting  limitations  which  are  additional  to  those  of  the 
proportions  of  the  parts. 

We  will,  however,  put  aside  the  arguments  to  be  drawn 
from  the  consideration  of  symmetry  as  inconclusive, 
retaining  one  valuable  indication  which  they  afford.  If 
it  is  in  virtue  of  a  four-dimensional  motion  that  sym- 
metry exists,  it  is  only  in  the  very  minute  particles 
of  bodies  that  that  motion  is  to  be  found,  for  there  is 


THE   EVIDENCES   FOR  A  FOURTH   DIMENSION  79 

no  such  thing  as  a  bending  over  in  four  dimensions  of 
any  object  of  a  size  which  we  can  observe.  The  region 
of  the  extremely  minute  is  the  one,  then,  which  we 
shall  have  to  investigate.  We  must  look  for  some 
phenomenon  which,  occasioning  movements  of  the  kind 
we  know,  still  is  itself  inexplicable  as  any  form  of  motion 
which  we  know. 

Now  in  the  theories  of  the  actions  of  the  minute 
particles  of  bodies  on  one  another,  and  in  the  motions  of 
the  ether,  mathematicians  have  tacitly  assumed  that  the 
mechanical  principles  are  the  same  as  those  which  prevail 
in  the  case  of  bodies  which  can  be  observed,  it  has  been 
assumed  without  proof  that  the  conception  of  motion  being 
three-dimensional,  holds  beyond  the  region  from  observa- 
tions in  which  it  was  formed. 

Hence  it  is  not  from  any  phenomenon  explained  by 
mathematics  that  we  can  derive  a  proof  of  four  dimensions. 
Every  phenomenon  that  has  been  explained  is  explained 
as  three-dimensional.  And,  moreover,  since  in  the  region 
of  the  very  minute  we  do  not  find  rigid  bodies  acting 
on  each  other  at  a  distance,  but  elastic  substances  and 
continuous  fluids  such  as  ether,  we  shall  have  a  double 
task. 

We  must  form  the  conceptions  of  the  possible  move- 
ments of  elastic  and  liquid  four-dimensional  matter,  before 
we  can  begin  to  observe.  Let  us,  therefore,  take  the  four- 
dimensional  rotation  about  a  plane,  and  enquire  what  it 
becomes  in  the  case  of  extensible  fluid  substances.  If 
four-dimensional  movements  exist,  this  kind  of  rotation 
must  exist,  and  the  finer  portions  of  matter  must  exhibit 
it. 

Consider  for  a  moment  a  rod  of  flexible  and  extensible 
material.  It  can  turn  about  an  axis,  even  if  not  straight ; 
a  ring  of  india  rubber  can  turn  inside  out. 

What  would  this  be  in  the  case  of  four  dimensions  ? 


80 


tHE    FOtJRtti 


Let   us   consider   a   sphere    of    our   three-dimensional 

matter  having  a  definite 
thickness.  To  represent 
this  thickness  let  us  sup- 
pose that  from  every  point 
of  the  sphere  in  fig.  44  rods 
project  both  ways,  in  and 
out,  like  D  and  F.  We  can 
only  see  the  external  por- 
tion, because  the  internal 
parts  are  hidden  by  the 
sphere. 

In  this  sphere  the  axis 
of  x  is  supposed  to  come 
towards  the  observer,  the 


Fig.  44. 

Axis  ofx  running 
the  observer. 


axis  of  z  to  run  up,  the  axis  of  y  to  go  to  the  right. 

Now   take  the   section    determined   by  the   zy   plane. 

This  will  be  a  circle  as 
shown  in  fig.  45.  If  we 
let  drop  the  x  axis,  this 
circle  is  all  we  have  of 
the  sphere.  Letting  the 
w  axis  now  run  in  the 
place  of  the  old  x  axis 
we  have  the  space  yzw, 
and  in  this  space  all  that 
we  have  of  the  sphere  is 
the  circle.  Fig.  45  then 
represents  all  that  there 
is  of  the  sphere  in  the 
space  of  yzw.  In  this  space  it  is  evident  that  the  rods 
CD  and  EF  can  turn  round  the  circumference  as  an  axis. 
If  the  matter  of  the  spherical  shell  is  sufficiently  exten- 
sible to  allow  the  particles  c  and  E  to  become  as  widely 
separated  as  they  would  be  in  the  positions  D  and  F,  then 


45< 


THE   EVIDENCES   FOR   A   FOUfcTfi   DIMENSION  81 

the  strip  of  matter  represented  by  CD  and  EF  and  a 
multitude  of  rods  like  them  can  turn  round  the  circular 
circumference. 

Thus  this  particular  section  of  the  sphere  can  turn 
inside  out,  and  what  holds  for  any  one  section  holds  for 
all.  Hence  in  four  dimensions  the  whole  sphere  can,  if 
extensible  turn  inside  out.  Moreover,  any  part  of  it — 
a  bowl-shaped  portion,  for  instance — can  turn  inside  out, 
and  so  on  round  and  round. 

This  is  really  no  more  than  we  had  before  in  the 
rotation  about  a  plane,  except  that  we  see  that  the  plane 
can,  in  the  case  of  extensible  matter,  be  curved,  and  still 
play  the  part  of  an  axis. 

If  we  suppose  the  spherical  shell  to  be  of  four-dimen- 
sional matter,  our  representation  will  be  a  little  different. 
Let  us  suppose  there  to  be  a  small  thickness  to  the  matter 
in  the  fourth  dimension.  This  would  make  no  difference 
in  fig.  44,  for  that  merely  shows  the  view  in  the  xyz 
space.  But  when  the  x  axis  is  let  drop,  and  the  w  axis 
comes  in,  then  the  rods  CD  and  EF  which  represent  the 
matter  of  the  shell,  will  have  a  certain  thickness  perpen- 
dicular to  the  plane  of  the  paper  on  which  they  are  drawn. 
If  they  have  a  thickness  in  the  fourth  dimension  they  will 
show  this  thickness  when  looked  at  from  the  direction  of 
the  iv  axis. 

Supposing  these  rods,  then,  to  be  small  slabs  strung  on 
the  circumference  of  the  circle  in  fig.  45,  we  see  that 
there  will  not  be  in  this  case  either  any  obstacle  to  their 
turning  round  the  circumference.  We  can  have  a  shell 
of  extensible  material  or  of  fluid  material  turning  inside 
out  in  four  dimensions. 

And  we  must  remember  that  in  four  dimensions  there 
is  no  such  thing  as  rotation  round  an  axis.  If  we  want  to 
investigate  the  motion  of  fluids  in  four  dimensions  we 
must  take  a  movement  about  an  axis  in  our  space,  and 

6 


82  tHE   FOUKTS   DIMENSION 

find  the  corresponding  movement  about  a  plane  in 
four  space. 

Now,  of  all  the  movements  which  take  place  in  fluids, 
the  most  important  from  a  physical  point  of  view  is 
vortex  motion. 

A  vortex  is  a  whirl  or  eddy — it  is  shown  in  the  gyrating 
wreaths  of  dust  seen  on  a  summer  day ;  it  is  exhibited  on 
a  larger  scale  in  the  destructive  march  of  a  cyclone. 

A  wheel  whirling  round  will  throw  off  the  water  on  it. 
But  when  this  circling  motion  takes  place  in  a  liquid 
itself  it  is  strangely  persistent.  There  is,  of  course,  a 
certain  cohesion  between  the  particles  of  water  by  which 
they  mutually  impede  their  motions.  But  in  a  liquid 
devoid  of  friction,  such  that  every  particle  is  free  from 
lateral  cohesion  on  its  path  of  motion,  it  can  be  shown 
that  a  vortex  or  eddy  separates  from  the  mass  of  the 
fluid  a  certain  portion,  which  always  remain  in  that 
vortex. 

The  shape  of  the  vortex  may  alter,  but  it  always  con- 
sists of  the  same  particles  of  the  fluid. 

Now,  a  very  remarkable  fact  about  such  a  vortex  is  that 
the  ends  of  the  vortex  cannot  remain  suspended  and 
isolated  in  the  fluid.  They  must  always  run  to  the 
boundary  of  the  fluid.  An  eddy  in  water  that  remains 
half  way  down  without  coming  to  the  top  is  impossible. 

The  ends  of  a  vortex  must  reach  the  boundary  of  a 
fluid — the  boundary  may  be  external  or  internal — a  vortex 
may  exist  between  two  objects  in  the  fluid,  terminating 
one  end  on  each  object,  the  objects  being  internal 
boundaries  of  the  fluid.  Again,  a  vortex  may  have  its 
ends  linked  together,  so  that  it  forms  a  ring.  Circular 
vortex  rings  of  this  description  are  often  seen  in  puffs  of 
smoke,  and  that  the  smoke  travels  on  in  the  ring  is  a 
proof  that  the  vortex  always  consists  of  the  same  particles 
of  ai: 


THE  EVIDENCES  FOR  A  FOURTH  DIMENSION  83 

Let  us  now  enquire  what  a  vortex  would  be  in  a  four- 
dimensional  fluid. 

We  must  replace  the  line  axis  by  a  plane  axis.  We 
should  have  therefore  a  portion  of  fluid  rotating  round 
a  plane. 

We  have  seen  that  the  contour  of  this  plane  corresponds 
with  the  ends  of  the  axis  line.  Hence  such  a  four- 
dimensional  vortex  must  have  its  rim  on  a  boundary  of 
the  fluid.  There  would  be  a  region  of  vorticity  with  a 
contour.  If  such  a  rotation  were  started  at  one  part  of  a 
circular  boundary,  its  edges  would  run  round  the  boundary 
in  both  directions  till  the  whole  interior  region  was  filled 
with  the  vortex  sheet. 

A  vortex  in  a  three-dimensional  liquid  may  consist  of  a 
number  of  vortex  filaments  lying  together  producing  a 
tube,  or  rod  of  vorticity. 

In  the  same  way  we  can  have  in  four  dimensions  a 
number  of  vortex  sheets  alongside  each  other,  each  of  which 
can  be  thought  of  as  a  bowl-shaped  portion  of  a  spherical 
shell  turning  inside  out.  The  rotation  takes  place  at  any 
point  not  in  the  space  occupied  by  the  shell,  but  from 
that  space  to  the  fourth  dimension  and  round  back  again. 

Is  there  anything  analogous  to  this  within  the  range 
of  our  observation  ? 

An  electric  current  answers  this  description  in  every 
respect.  Electricity  does  not  flow  through  a  wire.  Its  effect 
travels  both  ways  from  the  starting  point  along  the  wire. 
The  spark  which  shows  its  passing  midway  in  its  circuit 
is  later  than  that  which  occurs  at  points  near  its  starting 
point  on  either  side  of  it. 

Moreover,  it  is  known  that  the  action  of  the  current 
is  not  in  the  wire.  It  is  in  the  region  enclosed  by  the 
wire,  this  is  the  field  of  force,  the  locus  of  the  exhibition 
of  the  effects  of  the  current. 

And  the  necessity  of  a  conducting  circuit  for  a  current  is 


84  THE  FOURTH  DIMENSION 

exactly  that  which  we  should  expect  if  it  were  a  four-dimen- 
sional vortex.  According  to  Maxwell  every  current  forms 
a  closed  circuit,  and  this,  from  the  four-dimensional  point 
of  view,  is  the  same  as  saying  a  vortex  must  have  its  ends 
on  a  boundary  of  the  fluid. 

Thus,  on  the  hypothesis  of  a  fourth  dimension,  the  rota- 
tion of  the  fluid  ether  would  give  the  phenomenon  of  an 
electric  current.  We  must  suppose  the  ether  to  be  full  of 
movement,  for  the  more  we  examine  into  the  conditions 
which  prevail  in  the  obscurity  of  the  minute,  the  more  we 
find  that  an  unceasing  and  perpetual  motion  reigns.  Thus 
we  may  say  that  the  conception  of  the  fourth  dimension 
means  that  there  must  be  a  phenomenon  which  presents 
the  characteristics  of  electricity. 

We  know  now  that  light  is  an  electro-magnetic  action, 
and  that  so  far  from  being  a  special  and  isolated  pheno- 
menon this  electric  action  is  universal  in  the  realm  of  the 
minute.  Hence,  may  we  not  conclude  that,  so  far  from 
the  fourth  dimension  being  remote  and  far  away,  being  a 
thing  of  symbolic  import,  a  term  for  the  explanation  of 
dubious  facts  by  a  more  obscure  theory,  it  is  really  the 
most  important  fact  within  our  knowledge.  Our  three- 
dimensional  world  is  superficial.  These  processes,  which 
really  lie  at  the  basis  of  all  phenomena  of  matter, 
escape  our  observation  by  their,  minuteness,  but  reveal 
to  our  intellect  an  amplitude  of  motion  surpassing  any 
that  we  can  see.  In  such  shapes  and  motions  there  is  a 
realm  of  the  utmost  intellectual  beauty,  and  one  to 
which  our  symbolic  methods  apply  with  a  better  grace 
than  they  do  to  those  of  three  dimensions. 


CHAPTER  VIII 

THE     USE     OF    FOUR     DIMENSIONS    IN 
THOUGHT 

HAVING  held  before  ourselves  this  outline  of  a  conjecture 
of  the  world  as  four-dimensional,  having  roughly  thrown 
together  those  facts  of  movement  which  we  can  see  apply 
to  our  actual  experience,  let  us  pass  to  another  branch 
of  our  subject. 

The  engineer  uses  drawings,  graphical  constructions, 
in  a  variety  of  manners.  He  has,  for  instance,  diagrams 
which  represent  the  expansion  of  steam,  the  efficiency 
of  his  valves.  These  exist  alongside  the  actual  plans  of 
his  machines.  They  are  not  the  pictures  of  anything 
really  existing,  but  enable  him  to  think  about  the  relations 
which  exist  in  his  mechanisms. 

And  so,  besides  showing  us  the  actual  existence  of  that 
world  which  lies  beneath  the  one  of  visible  movements, 
four-dimensional  space  enables  us  to  make  ideal  con- 
structions which  serve  to  represent  the  relations  of  things, 
and  throw  what  would  otherwise  be  obscure  into  a  definite 
and  suggestive  form. 

From  amidst  the  great  variety  of  instances  which  lies 
before  me  I  will  select  two,  one  dealing  with  a  subject 
of  slight  intrinsic  interest,  which  however  gives  within 
a  limited  field  a  striking  example  of  the  method 

85 


86  THE   FOURTH   DIMENSION 

of  drawing  conclusions  and  the  use  of  higher  space 
figures.* 

The  other  instance  is  chosen  on  account  of  the  bearing 
it  has  on  our  fundamental  conceptions.  In  it  I  try  to 
discover  the  real  meaning  of  Kant's  theory  of  experience. 

The  investigation  of  the  properties  of  numbers  is  much 
facilitated  by  the  fact  that  relations  between  numbers  are 
themselves  able  to  be  represented  as  numbers — e.g.,  12, 
and  3  are  both  numbers,  and  the  relation  between  them 
is  4,  another  number.  The  way  is  thus  opened  for  a 
process  of  constructive  theory,  without  there  being  any 
necessity  for  a  recourse  to  another  class  of  concepts 
besides  that  which  is  given  in  the  phenomena  to  be 
studied. 

The  discipline  of  number  thus  created  is  of  great  and 
varied  applicability,  but  it  is  not  solely  as  quantitative 
that  we  learn  to  understand  the  phenomena  of  nature. 
It  is  not  possible  to  explain  the  properties  of  matter 
by  number  simply,  but  all  the  activities  of  matter  are 
energies  in  space.  They  are  numerically  definite  and  also, 
we  may  say,  directedly  definite,  i.e.  definite  in  direction. 

Is  there,  then,  a  body  of  doctrine  about  space  which,  like 
that  of  number,  is  available  in  science  ?  It  is  needless 
to  answer :  Yes ;  geometry.  But  there  is  a  method 
lying  alongside  the  ordinary  methods  of  geometry,  which 
tacitly  used  and  presenting  an  analogy  to  the  method 
of  numerical  thought  deserves  to  be  brought  into  greater 
prominence  than  it  usually  occupies. 

The  relation  of  numbers  is  a  number. 

Can  we  say  in  the  same  way  that  the  relation  of 
shapes  is  a  shape  ? 

We  can. 

*  It  is  suggestive  also  in  another  respect,  because  it  shows  very 
clearly  that  in  our  processes  of  thought  there  are  in  play  faculties  other 
than  logical;  in  it  the  origin  of  the  idea  which  proves  to  be  justified  is 
drawn  from  the  consideration  of  symmetry,  a  branch  of  the  beautiful. 


THE   USE   OF   FOUR   DIMENSIONS   IN  THOUGHT  87 

To  take  an  instance  chosen   on   account  of  its   ready 

availability.  Let  us  take 
two  right-angled  triangles  of 
a  given  hypothenuse,  but 
having  sides  of  different 
lengths  (fig.  46).  These 
triangles  are  shapes  which  have  a  certain  relation  to  each 
other.  Let  us  exhibit  their  relation  as  a  figure. 

Draw  two  straight  lines  at  right  angles  to  each  other, 
the  one  HL  a  horizontal  level,  the 
other  VL  a  vertical  level  (fig.  47). 
By  means  of  these  two  co-ordin- 
ating lines  we  can  represent  a 
double  set  of  magnitudes  ;  one  set 
j  as  distances  to  the  right  of  the  ver- 

Fig.  47.  tical  level,  the  other  as  distances 

above  the  horizontal  level,  a  suitable  unit  being  chosen. 

Thus  the  line  marked  7  will  pick  out  the  assemblage 
of  points  whose  distance  from  the  vertical  level  is  7, 
and  the  line  marked  1  will  pick  out  the  points  whose 
distance  above  the  horizontal  level  is  1.  The  meeting 
point  of  these  two  lines,  7  and  1,  will  define  a  point 
which  with  regard  to  the  one  set  of  magnitudes  is  7, 
with  regard  to  the  other  is  1.  Let  us  take  the  sides  of 
our  triangles  as  the  two  sets  of  magnitudes  in  question. 
Then  the  point  7,  1,  will  represent  the  triangle  whose 
sides  are  7  and  1.  Similarly  the  point  5,  5  —  5,  that 
is,  to  the  ricrht  of  the  vertical  level  and  5  above  the 
.5,5  horizontal  level  —  will  represent  the 

triangle  whose  sides  are  5  and  5 


Thus  we  have  obtained  a  figure 
consisting  of  the  two  points  7,  1, 
Fig.  48.  and  5?  5^  representative  of  our  two 

triangles.     But  we  can  go  further,  and,  drawing  an  arc 


88  THE   FOURTH   DIMENSION 

of  a  circle  about  o,  the  meeting  point  of  the  horizontal 
and  vertical  levels,  which  passes  through  7,  1,  and  5,  5, 
assert  that  all  the  triangles  which  are  right-angled  and 
have  a  hypothenuse  whose  square  is  50  are  represented 
by  the  points  on  this  arc. 

Thus,  each  individual  of  a  class  being  represented  by  a 
point,  the  whole  class  is  represented  by  an  assemblage  of 
points  forming  a  figure.  Accepting  this  representation 
we  can  attach  a  definite  and  calculable  significance  to  the 
expression,  resemblance,  or  similarity  between  two  indi- 
viduals of  the  class  represented,  the  difference  being 
measured  by  the  length  of  the  line  between  two  repre- 
sentative points.  It  is  needless  to  multiply  examples,  or 
to  show  how,  corresponding  to  different  classes  of  triangles, 
we  obtain  different  curves. 

A  representation  of  this  kind  in  which  an  object,  a 
thing  in  space,  is  represented  as  a  point,  and  all  its  pro- 
perties are  left  out,  their  effect  remaining  only  in  the 
relative  position  which  the  representative  point  bears 
to  the  representative  points  of  the  other  objects,  may  be 
called,  after  the  analogy  of  Sir  William  K.  Hamilton's 
hodograph,  a  "Poiograph." 

Representations  thus  made  have  the  character  of 
natural  objects;  they  have  a  determinate  and  definite 
character  of  their  own.  Any  lack  of  completeness  in  them 
is  probably  due  to  a  failure  in  point  of  completeness 
of  those  observations  which  form  the  ground  of  their 
construction. 

Every  system  of  classification  is  a  poiograph.  In 
Mendeleeff's  scheme  of  the  elements,  for  instance,  each 
element  is  represented  by  a  point,  and  the  relations 
between  the  elements  are  represented  by  the  relations 
between  the  points. 

So  far  I  have  simply  brought  into  prominence  processes 
and  considerations  with  which  we  are  all  familiar.  But 


THE   USE   OF  FOUR   DIMENSIONS   IN   THOUGHT  89 

it  is  worth  while  to  bring  into  the  full  light  of  our  atten- 
tion our  habitual  assumptions  and  processes.  It  often 
happens  that  we  find  there  are  two  of  them  which  have 
a  bearing  on  each  other,  which,  without  this  dragging  into 
the  light,  we  should  have  allowed  to  remain  without 
mutual  influence. 

There  is  a  fact  which  it  concerns  us  to  take  into  account 
in  discussing  the  theory  of  the  poiograph. 

With  respect  to  our  knowledge  of  the  world  we  are 
far  from  that  condition  which  Laplace  imagined  when  he 
asserted  that  an  all-knowing  mind  could  determine  the 
future  condition  of  every  object,  if  he  knew  the  co-ordinates 
of  its  particles  in  space,  and  their  velocity  at  any 
particular  moment. 

On  the  contrary,  in  the  presence  of  any  natural  object, 
we  have  a  great  complexity  of  conditions  before  us, 
which  we  cannot  reduce  to  position  in  space  and  date 
in  time. 

There  is  mass,  attraction  apparently  spontaneous,  elec- 
trical and  magnetic  properties  which  must  be  superadded 
to  spatial  configuration.  To  cut  the  list  short  we  must 
say  that  practically  the  phenomena  of  the  world  present 
us  problems  involving  many  variables,  which  we  must 
take  as  independent. 

From  this  it  follows  that  in  making  poiographs  we 
must  be  prepared  to  use  space  of  more  than  three  dimen- 
sions. If  the  symmetry  and  completeness  of  our  repre- 
sentatation  is  to  be  of  use  to  us  we  must  be  prepared  to 
appreciate  and  criticise  figures  of  a  complexity  greater 
than  of  those  in  three  dimensions.  It  is  impossible  to  give 
an  example  of  such  a  poiograph  which  will  not  be  merely 
trivial,  without  going  into  details  of  some  kind  irrelevant 
to  our  subject.  I  prefer  to  introduce  the  irrelevant  details 
rather  than  treat  this  part  of  the  subject  perfunctorily. 

To  take  an  instance  of  a  poiograph  which  does  not  lead 


90 


THE   FOURTH   DIMENSION 


us  into  the  complexities  incident  on  its  application  in 
classificatory  science,  let  us  follow  Mrs.  Alicia  Boole  Stott 
in  her  representation  of  the  syllogism  by  its  means.  She 
will  be  interested  to  find  that  the  curious  gap  she  detected 
has  a  significance. 

A  syllogism  consists  of  two  statements,  the  major  and 
the  minor  premiss,  with  the  conclusion  that  can  be  drawn 
from  them.  Thus,  to  take  an  instance,  fig.  49.  It  is 
evident,  from  looking  at  the  successive  figures  that,  if  we 
know  that  the  region  M  lies  altogether  within  the  region 
p,  and  also  know  that  the  region  s  lies  altogether  within 
the  region  M,  we  can  conclude  that  the  region  s  lies 
altogether  within  the  region  P.  M  is  P, 
major  premiss;  s  is  M,  minor  premiss;  s 
is  p,  conclusion.  Given  the  first  two  data 
we  must  conclude  that  s  lies  in  p.  The 
conclusion  s  is  P  involves  two  terms,  s  and 
p,  which  are  respectively  called  the  subject 
and  the  predicate,  the  letters  s  and  P 
being  chosen  with  reference  to  the  parts 
the  notions  they  designate  play  in  the 
conclusion,  s  is  the  subject  of  the  con- 
clusion, P  is  the  predicate  of  the  conclusion. 
The  major  premiss  we  take  to  be,  that 
which  does  not  involve  s,  and  here  we 
always  write  it  first. 

There  are  several  varieties  of  statement 
possessing  different  degrees  of  universality  and  manners  of 
assertiveness.  These  different  forms  of  statement  are 
called  the  moods. 

We  will  take  the  major  premiss  as  one  variable,  as  a 
thing  capable  of  different  modifications  of  the  same  kind, 
the  minor  premiss  as  another,  and  the  different  moods  we 
will  consider  as  defining  the  variations  which  these 
variables  undergo. 


Fig.  49. 


THE   USE   OF   FOUR   DIMENSIONS   IN   THOUGHT 


91 


There  are  four  moods : — 

1.  The  universal  affirmative ;  all  M  is  p,  called  mood  A. 

2.  The  universal  negative  ;  no  M  is  P,  mood  E. 

3.  The  particular  affirmative ;  some  M  is  p,  mood  I. 

4.  The  particular  negative ;  some  M  is  not  p,  mood  o. 
The  dotted  lines  in  3  and  4,  fig.  50,  denote  that  it  is 

not  known  whether  or  no  any  objects  exist,  corresponding 


4. 
Mood  o. 


Fig.  50. 


to  the  space  of  which  the  dotted  line  forms  one  delimiting 
boundary  ;  thus,  in  mood  I  we  do  not  know  if  there  are 
any  M'S  which  are  not  P,  we  only  know  some  M'S  are  P. 
Representing  the  first  premiss  in  its  various  moods  by 
regions   marked   by  vertical  lines  to 
the  right  of  PQ,  we  have  in  fig.  51, 
running  up  from  the  four  letters  AEIO, 
four  column?,  each  of  which  indicates 
that  the  major  premiss  is  in  the  mood 
denoted  by  the  respective  letter.     In 
the  first  column  to  the  right  of  PQ  is 


Q 
o 


o  s 


R     PA  E    I 

Fig.  51. 

the  mood  A.  Now  above  the  line  RS  let  there  be  marked 
off  four  regions  corresponding  to  the  four  moods  of  the 
minor  premiss.  Thus,  in  the  first  row  above  RS  all  the 
region  between  RS  and  the  first  horizontal  line  above  it 
denotes  that  the  minor  premiss  is  in  the  mood  A.  The 


92 


THE   FOURTH   DIMENSION 


M 

L.U.. 

i 

n 

letters  E,  i,  o,  in  the  Same  way  show  the  mood  character- 
ising the  minor  premiss  in  the  rows  opposite  these  letters. 
We  have  still  to  exhibit  the  conclusion.     To  do  this  we 
must  consider  the  conclusion  as  a  third  variable,  character- 
ised in  its  different  varieties  by  four  moods — this  being 
the  syllogistic  classification.     The  introduction  of  a  third 
variable  involves  a  change  in  our  system  of  representation. 
Before  we  started  with  the  regions  to  the  right  of  a 
certain  line  as  representing  successively  the  major  premiss 
in  its  moods ;  now  we  must  start  with  the  regions  to  the 
right   of  a    certain  plane.      Let  LMNR 
be  the  plane  face  of  a  cube,  fig.  52,  and 
let  the  cube  be  divided  into  four  parts 
by   vertical   sections    parallel   to   LMNR. 
The  variable,  the  major  premiss,  is  re- 
A  c  '  ~     'presented    by    the    successive     regions 
Fig.  52.  which  occur  to  the  right  of  the  plane 

LMNR — that  region  to  which  A  stands  opposite,  that 
slice  of  the  cube,  is  significative  of  the  mood  A.  This 
whole  quarter-part  of  the  cube  represents  that  for  every 
part  of  it  the  major  premiss  is  in  the  mood  A. 

In  a  similar  manner  the  next  section,  the  second  with 
the  letter  E  opposite  it,  represents  that  for  every  one  of 
the  sixteen  small  cubic  spaces  in  it,  the  major  premiss  is 
in  the  mood  E.  The  third  and  fourth  compartments  made 
by  the  vertical  sections  denote  the  major  premiss  in  the 
moods  i  and  o.  But  the  cube  can  be  divided  in  other 
ways  by  other  planes.  Let  the  divisions,  of  which  four 
stretch  from  the  front  face,  correspond  to  the  minor 
premiss.  The  first  wall  of  sixteen  cubes,  facing  the 
observer,  has  as  its  characteristic  that  in  each  of  the  small 
cubes,  whatever  else  may  be  the  case,  the  minor  premiss  is 
in  the  mood  A.  The  variable — the  minor  premiss — varies 
through  the  phases  A,  E,  I,  O,  away  from  the  front  face  of  the 
cube,  or  the  front  plane  of  which  the  front  face  is  a  part. 


•THE   USE  Of  FOtJR   DIMENSIONS   IN  fHOUGHT  S3 

And  now  we  can  represent  the  third  variable  in  a  precisely 
similar  way.  We  can  take  the  conclusion  as  the  third 
variable,  going  through  its  four  phases  from  the  ground 
plane  upwards.  Each  of  the  small  cubes  at  the  base  of 
the  whole  cube  has  this  true  about  it,  whatever  else  may 
be  the  case,  that  the  conclusion  is,  in  it,  in  the  mood  A. 
Thus,  to  recapitulate,  the  first  wall  of  sixteen  small  cubes, 
the  first  of  the  four  walls  which,  proceeding  from  left  to 
right,  build  up  the  whole  cube,  is  characterised  in  each 
part  of  it  by  this,  that  the  major  premiss  is  in  the  mood  A. 

The  next  wall  denotes  that  the  major  premiss  is  in  the 
mood  E,  and  so  on.  Proceeding  from  the  front  to  the 
back  the  first  wall  presents  a  region  in  every  part  of 
which  the  minor  premiss  is  in  the  mood  A.  The  second 
wall  is  a  region  throughout  which  the  minor  premiss  is  in 
the  mood  E,  and  so  on.  In  the  layers,  from  the  bottom 
upwards,  the  conclusion  goes  through  its  various  moods 
beginning  with  A  in  the  lowest,  E  in  the  second,  I  in  the 
third,  0  in  the  fourth. 

In  the  general  case,  in  which  the  variables  represented 
in  the  poiograph  pass  through  a  wide  range  of  values,  the 
planes  from  which  we  measure  their  degrees  of  variation 
in  our  representation  are  taken  to  be  indefinitely  extended. 
In  this  case,  however,  all  we  are  concerned  with  is  the 
finite  region. 

We  have  now  to  represent,  by  some  limitation  of  the 
complex  we  have  obtained,  the  fact  that  not  every  com- 
bination of  premisses  justifies  any  kind  of  conclusion. 
This  can  be  simply  effected  by  marking  the  regions  in 
which  the  premisses,  being  such  as  are  defined  by  the 
positions,  a  conclusion  which  is  valid  is  found. 

Taking  the  conjunction  of  the  major  premiss,  all  M  is 
p,  and  the  minor,  all  s  is  M,  we  conclude  that  all  s  is  P. 
Hence,  that  region  must  be  marked  in  which  we  have  the 
conjunction  of  major  premiss  in  mood  A ;  minor  premiss, 


94 


THE  fOtlKTH  DIMENSION 


mood  A  ;  conclusion,  mood  A.     This  is  the  cube  occupying 
the  lowest  left-hand  corner  of  the  large  cube. 

Proceeding  in  this  way,  we  find  that  the  regions  which 
must  be  marked  are  those  shown  in  fig.  53. 
To  discuss  the  case  shown  in  the  marked 
cube  which  appears  at  the  top  of  fig.  53. 
Here  the  major  premiss  is  in  the  second 
wall  to  the  right — it  is  in  the  mood  E  and 
is  of  the  type  no  M  is  P.  The  minor 
premiss  is  in  the  mood  characterised  by 
the  third  wall  from  the  front.  It  is  of 
the  type  some  s  is  M.  From  these  premisses  we  draw 
the  conclusion  that  some  s  is  not  P,  a  conclusion  in  the 
mood  o.  Now  the  mood  0  of  the  conclusion  is  represented 
in  the  top  layer.  Hence  we  see  that  the  marking  is 
correct  in  this  respect. 

It  would,  of  course,  be  possible  to  represent  the  cube  on 

a  plane  by  means  of  four 
squares,  as  in  fig.  54,  if  we 
consider  each  square  to  re- 
present merely  the  beginning 
of  the  region  it  stands  for. 
Thus  the  whole  cube  can  be 
represented  by  four  vertical 
squares,  each  standing  for  a 
kind  of  vertical  tray,  and  the 


Fig.  54. 


markings  would  be  as  shown.  In  No.  1  the  major  premiss 
is  in  mood  A  for  the  whole  of  the  region  indicated  by  the 
vertical  square  of  sixteen  divisions ;  in  No.  2  it  is  in  the 
mood  E,  and  so  on. 

A  creature  confined  to  a  plane  would  have  to  adopt  some 
such  disjunctive  way  of  representing  the  whole  cube.  He 
would  be  obliged  to  represent  that  which  we  see  as  a 
whole  in  separate  parts,  and  each  part  would  merely 
represent,  would  not  be,  that  solid  content  which  we  see. 


THEXUSE  OF  FOUR  DIMENSIONS  IN  THOUGH*       95 

The  view  of  these  four  squares  which  the  plane  creature 
would  have  would  not  be  such  as  ours.  He  would  not 
see  the  interior  of  the  four  squares  represented  above,  but 
each  would  be  entirely  contained  within  its  outline,  the 
internal  boundaries  of  the  separate  small  squares  he  could 
not  see  except  by  removing  the  outer  squares. 

We  are  now  ready  to  introduce  the  fourth  variable 
involved  in  the  syllogism. 

In  assigning  letters  to  denote  the  terms  of  the  syllogism 
we  have  taken  s  and  p  to  represent  the  subject  and 
predicate  in  the  conclusion,  and  thus  in  the  conclusion 
their  order  is  invariable.  But  in  the  premisses  we  have 
taken  arbitrarily  the  order  all  M  is  P,  and  all  s  is  M. 
There  is  no  reason  why  M  instead  of  P  should  not  be  the 
predicate  of  the  major  premiss,  and  so  on. 

Accordingly  we  take  the  order  of  the  terms  in  the  pre- 
misses as  the  fourth  variable.  Of  this  order  there  are  four 
varieties,  and  these  varieties  are  called  figures. 

Using  the  order  in  which  the  letters  are  written  to 
denote  that  the  letter  first  written  is  subject,  the  one 
written  second  is  predicate,  we  have  the  following  pos- 
sibilities : — 

1st  Figure.        2nd  Figure.        3rd  Figure,        4th  Figure. 
Major      MP  PM  MP  PM 

Minor      SM  SM  MS  MS 

There  are  therefore  four  possibilities  with  regard  to 
this  fourth  variable  as  with  regard  to  the  premisses. 

We  have  used  up  our  dimensions  of  space  in  represent- 
ing the  phases  of  the  premisses  and  the  conclusion  in 
respect  of  mood,  and  to  represent  in  an  analogous  manner 
the  variations  in  figure  we  require  a  fourth  dimension. 

Now  in  bringing  in  this  fourth  dimension  we  must 
make  a  change  in  our  origins  of  measurement  analogous 
to  that  which  we  made  in  passing  from  the  plane  to  the 
solid. 


96  ftfE   FOUlVrti   DIMENSION 


This  fourth  dimension  is  supposed  to  run  at  right 
angles  to  any  of  the  three  space  dimensions,  as  the  third 
space  dimension  runs  at  right  angles  to  the  two  dimen- 
sions of  a  plane,  and  thus  it  gives  us  the  opportunity  of 
generating  a  new  kind  of  volume.  If  the  whole  cube 
moves  in  this  dimension,  the  solid  itself  traces  out  a  path, 
each  section  of  which,  made  at  right  angles  to  the 
direction  in  which  it  moves,  is  a  solid,  an  exact  repetition 
of  the  cube  itself. 

The  cube  as  we  see  it  is  the  beginning  of  a  solid  of  such 
a  kind.  It  represents  a  kind  of  tray,  as  the  square  face  of 
the  cube  is  a  kind  of  tray  against  which  the  cube  rests. 

Suppose  the  cube  to  move  in  this  fourth  dimension  in 
four  stages,  and  let  the  hyper-solid  region  traced  out  in 
the  first  stage  of  its  progress  be  characterised  by  this,  that 
the  terms  of  the  syllogism  are  in  the  first  figure,  then  we 
can  represent  in  each  of  the  three  subsequent  stages  the 
remaining  three  figures.  Thus  the  whole  cube  forms 
the  basis  from  which  we  measure  the  variation  in  figure. 
The  first  figure  holds  good  for  the  cube  as  we  see  it,  and 
for  that  hyper-solid  which  lies  within  the  first  stage  ; 
the  second  figure  holds  good  in  the  second  stage,  and 
so  on. 

Thus  we  measure  from  the  whole  cube  as  far  as  figures 
are  concerned. 

But  we  saw  that  when  we  measured  in  the  cube  itself 
having  three  variables,  namely,  the  two  premisses  and 
the  conclusion,  we  measured  from  three  planes.  The  base 
from  which  we  measured  was  in  every  case  the  same. 

Hence,  in  measuring  in  this  higher  space  we  should 
have  bases  of  the  same  kind  to  measure  from,  we  should 
have  solid  bases. 

The  first  solid  base  is  easily  seen,  it  is  the  cube  itself. 
The  other  can  be  found  from  this  consideration. 

That  soli(J  from  which  we  measure  figure  is  that  in 


"tHE   USE   Of   tfOtift   blMENSlONS   IN   THOUGflt  9* 

which  the  remaining  variables  run  through  their  full 
range  of  varieties. 

Now,  if  we  want  to  measure  in  respect  of  the  moods  of 
the  major  premiss,  we  must  let  the  minor  premiss,  the 
conclusion,  run  through  their  range,  and  also  the  order 
of  the  terms.  That  is  we  must  take  as  basis  of  measure- 
ment in  respect  to  the  moods  of  the  major  that  which 
represents  the  variation  of  the  moods  of  the  minor,  the 
conclusion  and  the  variation  of  the  figures. 

Now  the  variation  of  the  moods  of  the  minor  and  of  the 
conclusion  are  represented  in  the  square  face  on  the  left 
of  the  cube.  Here  are  all  varieties  of  the  minor  premiss 
and  the  conclusion.  The  varieties  of  the  figures  are 
represented  by  stages  in  a  motion  proceeding  at  right 
angles  to  all  space  directions,  at  right  angles  consequently 
to  the  face  in  question,  the  left-hand  face  of  the  cube. 

Consequently  letting  the  left-hand  face  move  in  this 
direction  we  get  a  cube,  and  in  this  cube  all  the  varieties 
of  the  minor  premiss,  the  conclusion,  and  the  figure  are 
represented. 

Thus  another  cubic  base  of  measurement  is  given  to 
the  cube,  generated  by  movement  of  the  left-hand  square 
in  the  fourth  dimension. 

We  find  the  other  bases  in  a  similar  manner,  one  is  the 
cube  generated  by  the  front  square  moved  in  the  fourth 
dimension  so  as  to  generate  a  cube.  From  this  cube 
variations  in  the  mood  of  the  minor  are  measured.  The 
fourth  base  is  that  found  by  moving  the  bottom  square  of 
the  cube  in  the  fourth  dimension.  In  this  cube  the 
variations  of  the  major,  the  minor,  and  the  figure  are  given. 
Considering  this  as  a  basis  in  the  four  stages  proceeding 
from  it,  the  variation  in  the  moods  of  the  conclusion  are 
given. 

Any  one  of  these  cubic  bases  can  be  represented  in  space, 
and  then  the  higher  solid  generated  from  them  lies  out  of 


98  THE   FOURTH   DIMENSION 

our  space.  It  can  only  be  represented  by  a  device  analogous 
to  that  by  which  the  plane  being  represents  a  cube. 

He  represents  the  cube  shown  above,  by  taking  four 
square  sections  and  placing  them  arbitrarily  at  convenient 
distances  the  one  from  the  other. 

So  we  must  represent  this  higher  solid  by  four  cubes : 
each  cube  represents  only  the  beginning  of  the  correspond- 
ing higher  volume. 

It  is  sufficient  for  us,  then,  if  we  draw  four  cubes,  the 
first  representing  that  region  in  which  the  figure  is  of  the 
first  kind,  the  second  that  region  in  which  the  figure  is 
of  the  second  kind,  and  so  on.  These  cubes  are  the 
beginnings  merely  of  the  respective  regions — they  are 
the  trays,  as  it  were,  against  which  the  real  solids  must 
be  conceived  as  resting,  from  which  they  start.  The  first 
one,  as  it  is  the  beginning  of  the  region  of  the  first  figure, 
is  characterised  by  the  order  of  the  terms  in  the  premisses 
being  that  of  the  first  figure.  The  second  similarly  has 
the  terms  of  the  premisses  in  the  order  of  the  second 
figure,  and  so  on. 

These  cubes  are  shown  below. 

For  the  sake  of  showing  the  properties  of  the  method 
of  representation,  not  for  the  logical  problem,  I  will  make 
a  digression.  I  will  represent  in  space  the  moods  of  the 
minor  and  of  the  conclusion  and  the  different  figures, 
keeping  the  major  always  in  mood  A.  Here  we  have 
three  variables  in  different  stages,  the  minor,  the  con- 
clusion, and  the  figure.  Let  the  square  of  the  left-hand 
side  of  the  original  cube  be  imagined  to  be  standing  by 
itself,  without  the  solid  part  of  the  cube,  represented  by 
(2)  fig.  55.  The  A,  E,  I,  o,  which  run  away  represent  the 
moods  of  the  minor,  the  A,  E,  I,  o,  which  run  up  represent 
the  moods  of  the  conclusion.  The  whole  square,  since  it 
is  the  beginning  of  the  region  in  the  major  premiss,  mood 
A,  is  to  be  considered  as  in  major  premiss,  mood  A. 


THE  USE  OF  FOUR  DIMENSIONS  IN   THOUGHT 


From  this  square,  let  it  be  supposed  that  that  direc- 
tion in  which  the  figures  are  represented  runs  to  the 
left  hand.  Thus  we  have  a  cube  (1)  running  from  the 
square  above,  in  which  the  square  itself  is  hidden,  but 
the  letters  A,  E,  I,  o,  of  the  conclusion  are  seen.  In  this 
cube  we  have  the  minor  premiss  and  the  conclusion  in  all 
their  moods,  and  all  the  figures  represented.  With  regard 
to  the  major  premiss,  since  the  face  (2)  belongs  to  the  first 
wall  from  the  left  in  the  original  arrangement,  and  in  this 


(D 


Fig.  55. 


4321 


arrangement  was  characterised  by  the  major  premiss  in  the 
mood  A,  we  may  say  that  the  whole  of  the  cube  we  now 
have  put  up  represents  the  mood  A  of  the  major  premiss. 

Hence  the  small  cube  at  the  bottom  to  the  right  in  1, 
nearest  to  the  spectator,  is  major  premiss,  mood  A;  minor 
premiss,  mood  A;  conclusion,  mood  A;  and  figure  the  first. 
The  cube  next  to  it,  runn'ng  to  the  left,  is  major  premiss, 
mood  A ;  minor  premiss,  mood  A ;  conclusion,  mood  A ; 
figure  2. 

So  in  this  cube  we  have  the  representations  of  all  the 
combinations  which  can  occur  when  the  major  premiss, 
remaining  in  the  mood  A,  the  minor  premiss,  the  conclu- 
sion, and  the  figures  pass  through  their  varieties. 

In  this  case  there  is  no  room  in  space  for  a  natural 
representation  of  the  moods  of  the  major  premiss.  To 
represent  them  we  must  suppose  as  before  that  there  is  a 
fourth  dimension,  and  starting  from  this  cube  as  base  in 
the  fourth  direction  in  four  equal  stages,  all  the  first  volume 
corresponds  to  major  premiss  A,  the  s^pond  to  major 


100 


'J'HE   FOURTH   DIMENSION 


premiss,  mood  E,  the  next  to  the  mood  I,  and  the  last 
to  mood  o. 

The  cube  we  see  is  as  it  were  merely  a  tray  against 
which  the  four-dimensional  figure  rests.  Its  section  at 
any  stage  is  a  cube.  But  a  transition  in  this  direction 
being  transverse  to  the  whole  of  our  space  is  represented 
by  no  space  motion.  We  can  exhibit  successive  stages  of 
the  result  of  transference  of  the  cube  in  that  direction, 
but  cannot  exhibit  the  product  of  a  transference,  however- 
small,  in  that  direction. 

To  return  to  the  original  method  of  representing  our 
variables,  consider  fig.  56.  These  four  cubes  represent 
four  sections  of  the  figure  derived  from  the  first  of  them 


Fig.  5(:. 

by  moving  it  in  the  fourth  dimension.  The  first  por- 
tion of  the  motion,  which  begins  with  1,  traces  out  a 
more  than  solid  body,  which  is  all  in  the  first  figure. 
The  beginning  of  this  body  is  shown  in  1.  The  next 
portion  of  the  motion  traces  out  a  more  than  solid  body, 
all  of  which  is  in  the  second  figure;  the  beginning  of 
this  body  is  shown  in  2;  3  and  4  follow  on  in  like 
manner.  Here,  then,  in  one  four-dimensional  figure  we 
have  all  the  combinations  of  the  four  variables,  major 
premiss,  minor  premiss,  figure,  conclusion,  represented, 
each  variable  going  through  its  four  varieties.  The  dis- 
connected cubes  drawn  are  our  representation  in  space  by 
means  of  disconnected  sections  of  this  higher  body. 


THE   USE   OF  FOUR    DIMENSIONS   IN   THOUGHT         101 

Now  it  is  only  a  limited  number  of  conclusions  which 
are  true — their  truth  depends  on  the  particular  combina- 
tions of  the  premisses  and  figures  which  they  accompany. 
The  total  figure  thus  represented  may  be  called  the 
universe  of  thought  in  respect  to  these  four  constituents, 
and  out  of  the  universe  of  possibly  existing  combinations 
it  is  the  province  of  logic  to  select  those  which  corre- 
spond to  the  results  of  our  reasoning  faculties. 

We  can  go  over  each  of  the  premisses  in  each  of  the 
moods,  and  find  out  what  conclusion  logically  follows. 
But  this  is  done  in  the  works  on  logic ;  most  simply  and 
clearly  I  believe  in  "  Jevon's  Logic."  As  we  are  only  con- 
cerned with  a  formal  presentation  of  the  results  we  will 
make  use  of  the  mnemonic  lines  printed  below,  in  which 
the  words  enclosed  in  brackets  refer  to  the  figures,  and 
are  not  significative  : — 

Barbara  celarent  Darii  ferioque  [prioris], 
Caesare  Camestris  Festino  Baroko  [secundae]. 
[Tertia]  darapti  disamis  datisi  felapton. 
Bokardo  ferisson  habei  [Quarta  insuper  addit]. 
Bramantip  camenes  dimaris  ferapton  fresison. 

In  these  lines  each  significative  word  has  three  vowels, 
the  first  vowel  refers  to  the  major  premiss,  and  gives  the 
mood  of  that  premiss,  "a"  signifying,  for  instance,  that 
the  major  mood  is  in  mood  a.  The  second  vowel  refers 
to  the  minor  premiss,  and  gives  its  mood.  The  third 
vowel  refers  to  the  conclusion,  and  gives  its  mood.  Thus 
(prioris) — of  the  first  figure — the  first  mnemonic  word  is 
"  barbara,"  and  this  gives  major  premiss,  mood  A  ;  minor 
premiss,  mood  A  ;  conclusion,  mood  A.  Accordingly  in  the 
first  of  our  four  cubes  we  mark  the  lowest  left-hand  front 
cube.  To  take  another  instance  in  the  third  figure  "  Tertia," 
the  word  "  ferisson  "  gives  us  major  premiss  mood  E — e.g., 
no  M  is  P,  minor  premiss  mood  I ;  some  M  is  s,  conclusion, 
mood  p  5  some  s  is  not  p.  The  region  to  be  marked  then 


102  THE   FOURTH   DIMENSION 

in  the  third  representative  cube  is  the  one  in  the  second 
wall  to  the  right  for  the  major  premiss,  the  third  wall 
from  the  front  for  the  minor  premiss,  and  the  top  layer 
for  the  conclusion. 

It  is  easily  seen  that  in  the  diagram  this  cube  is 
marked,  and  so  with  all  the  valid  conclusions.  The 
regions  marked  in  the  total  region  show  which  com- 
binations of  the  four  variables,  major  premiss,  minor 
premiss,  figure,  and  conclusion  exist. 

That  is  to  say,  we  objectify  all  possible  conclusions,  and 
build  up  an  ideal  manifold,  containing  all  possible  com- 
binations of  them  with  the  premisses,  and  then  out  of 
this  we  eliminate  all  that  do  not  satisfy  the  laws  of  logic. 
The  residue  is  the  syllogism,  considered  as  a  canon  of 
reasoning. 

Looking  at  the  shape  which  represents  the  totality 
of  the  valid  conclusions,  it  does  not  present  any  obvious 
symmetry,  or  easily  characterisable  nature.  A  striking 
configuration,  however,  is  obtained,  if  we  project  the  four- 
dimensional  figure  obtained  into  a  three-dimensional  one ; 
that  is,  if  we  take  in  the  base  cube  all  those  cubes  which 
have  a  marked  space  anywhere  in  the  series  of  four 
regions  which  start  from  that  cube. 

This  corresponds  to  making  abstraction  of  the  figures, 
giving  all  the  conclusions  which  are  valid  whatever  the 
figure  may  be. 

Proceeding  in  this  way  we  obtain  the  arrangement  of 
marked  cubes  shown  in  fig.  57.     We  see 
that  the    valid   conclusions    are    arranged 
almost  symmetrically  round  one  cube — the 
one  on  the  top  of  the  column  starting  from 
AAA.      There  is  one  breach  of  continuity 
however   in   this    scheme.      One   cube   is 
Fig.  57.         unmarked,  which   if    marked   would    give 
symmetry.     It  is  the  one  which  would  be  denoted  by  the 


THE    USE   OF   FOUR   DIMENSIONS   IN   THOUGHT 


103 


letters  I,  E,  o,  in  the  third  wall  to  the  right,  the  second 
wall  away,  the  topmost  layer.  Now  this  combination  of 
premisses  in  the  mood  IE,  with  a  conclusion  in  the  mood 
o,  is  not  noticed  in  any  book  on  logic  with  which  I  am 
familiar.  Let  us  look  at  it  for  ourselves,  as  it  seems 
that  there  must  be  something  curious  in  connection  with 
this  break  of  continuity  in  the  poiograph. 


M 


©0 


2nd  figure. 


0© 


8rd  figure. 


Fig.  58. 


4th  figure. 


The  propositions  I,  E,  in  the  various  figures  are  the 
following,  as  shown  in  the  accompanying  scheme,  fig.  58  : — 
First  figure :  some  M  is  p  ;  no  S  is  M.  Second  figure  : 
some  P  is  M ;  no  S  is  M.  Third  figure  :  some  M  is  p ;  no 
M  is  S.  Fourth  figure  :  some  p  is  M  ;  no  M  is  s. 

Examining  these  figures,  we  see,  taking  the  first,  that 
jf  some  M  is  P  and  no  S  is  M,  we  have  no  conclusion  of 


104  THE   FOURTH   DIMENSION 

the  form  s  is  p  in  the  various  moods.  It  is  quite  inde- 
terminate how  the  circle  representing  s  lies  with  regard 
to  the  circle  representing  p.  It  may  lie  inside,  outside, 
or  partly  inside  P.  The  same  is  true  in  the  other  figures 
2  and  3.  But  when  we  come  to  the  fourth  figure,  since 
M  and  s  lie  completely  outside  each  other,  there  cannot 
lie  inside  s  that  part  of  p  which  lies  inside  M.  Now 
we  know  by  the  major  premiss  that  some  of  P  does  lie 
in  M.  Hence  s  cannot  contain  the  whole  of  p.  In 
words,  some  P  is  M,  no  M  is  s,  therefore  s  does  not  contain 
the  whole  of  P.  If  we  take  P  as  the  subject,  this  gives 
us  a  conclusion  in  the  mood  0  about  p.  Some  P  is  not  s. 
But  it  does  not  give  us  conclusion  about  s  in  any  one 
of  the  four  forms  recognised  in  the  syllogism  and  called 
its  moods.  Hence  the  breach  of  the  continuity  in  the 
poiograph  has  enabled  us  to  detect  a  lack  of  complete- 
ness in  the  relations  which  are  considered  in  the  syllogism. 

To  take  an  instance : — Some  Americans  (p)  are  of 
African  stock  (M);  No  Aryans  (s)  are  of  African  stock 
(M)  ;  Aryans  (s)  do  not  include  all  of  Americans  (p). 

In  order  to  draw  a  conclusion  about  s  we  have  to  admit 
the  statement,  "  s  does  not  contain  the  whole  of  p,"  as 
a  valid  logical  form — it  is  a  statement  about  s  which  can 
be  made.  The  logic  which  gives  us  the  form,  "  some  p 
is  not  s,"  and  which  does  not  allow  us  to  give  the  exactly 
equivalent  and  equally  primary  form,  "  S  does  not  con- 
tain the  whole  of  P,"  is  artificial. 

And  I  wish  to  point  out  that  this  artificiality  leads 
to  an  error. 

If  one  trusted  to  the  mnemonic  lines  given  above,  one 
would  conclude  that  no  logical  conclusion  about  s  can 
be  drawn  from  the  statement,  "  some  P  are  M,  no  M  are  s." 

But  a  conclusion  can  be  drawn  :  s  does  not  contain 
the  whole  of  p. 

}t  is  not  that  thp  result  is  given  expressed  Jn  another 


THE    USE   OF   FOUR   DIMENSIONS   IN  THOUGHT         105 

form.  The  mnemonic  lines  deny  that  any  conclusion 
can  be  drawn  from  premisses  in  the  moods  I,  E,  respectively 

Thus  a  simple  four-dimensional  poiograph  has  enabled 
us  to  detect  a  mistake  in  the  mnemonic  lines  which  have 
been  handed  down  unchallenged  from  mediaeval  times. 
To  discuss  the  subject  of  these  lines  more  fully  a  logician 
defending  them  would  probably  say  that  a  particular 
statement  cannot  be  a  major  premiss;  and  so  deny  the 
existence  of  the  fourth  figure  in  the  combination  of  moods. 

To  take  our  instance :  some  Americans  are  of  African 
stock ;  no  Aryans  are  of  African  stock.  He  would  sav 
that  the  conclusion  is  some  Americans  are  not  Aryans  ; 
and  that  the  second  statement  is  the  major.  He  would 
refuse  to  say  anything  about  Aryans,  condemning  us  to 
an  eternal  silence  about  them,  as  far  as  these  premisses 
are  concerned !  But,  if  there  is  a  statement  involving 
the  relation  of  two  classes,  it  must  be  expressible  as  a 
statement  about  either  of  them. 

To  bar  the  conclusion,  "Aryans  do  not  include  the 
whole  of  Americans,"  is  purely  a  makeshift  in  favour  of 
a  false  classification. 

And  the  argument  drawn  from  the  universality  of  the 
major  premiss  cannot  be  consistently  maintained.  It 
would  preclude  such  combinations  as  major  o,  minor  A, 
conclusion  o — i.e.,  such  as  some  mountains  (M)  are  not 
permanent  (p);  all  mountains  (M)  are  scenery  (s)  ;  some 
scenery  (s)  is  not  permanent  (p). 

This  is  allowed  in  "  Jevon's  Logic,"  and  his  omission  to 
discuss  I,  E,  o,  in  the  fourth  figure,  is  inexplicable.  A 
satisfactory  poiograph  of  the  logical  scheme  can  be  made 
by  admitting  the  use  of  the  words  some,  none,  or  all, 
about  the  predicate  as  well  as  about  the  subject.  Then 
we  can  express  the  statement,  "  Aryans  do  not  include  the 
whole  of  Americans,"  clumsily,  but,  when  its  obscurity 
is  fathomed,  correctly,  as  "  Some  Aryans  are  not  all 


106  THE   FOURTH   DIMENSION 

Americans."     And    this   method   is   what    is   called    the 
"  quantification  of  the  predicate." 

The  laws  of  formal  logic  are  coincident  with  the  con- 
clusions which  can  be  drawn  about  regions  of  space,  which 
overlap  one  another  in  the  various  possible  ways.  It  is 
not  difficult  so  to  state  the  relations  or  to  obtain  a 
symmetrical  poiograph.  But  to  enter  into  this  branch  of 
geometry  is  beside  our  present  purpose,  which  is  to  show 
the  application  of  the  poiograph  in  a  finite  and  limited 
region,  without  any  of  those  complexities  which  attend  its 
use  in  regard  to  natural  objects. 

If  we  take  the  latter — plants,  for  instance — and,  without 
assuming  fixed  directions  in  space  as  representative  of 
definite  variations,  arrange  the  representative  points  in 
such  a  manner  as  to  correspond  to  the  similarities  of  the 
objects,  we  obtain  configuration  of  singular  interest ;  and 
perhaps  in  this  way,  in  the  making  of  shapes  of  shapes, 
bodies  with  bodies  omitted,  some  insight  into  the  structure 
of  the  species  and  genera  might  be  obtained. 


CHAPTER    IX 

APPLICATION    TO    KANT'S    THEORY    OP 
EXPERIENCE 

WHEN  we  observe  the  heavenly  bodies  we  become  aware 
that  they  all  participate  in  one  universal  motion — a 
diurnal  revolution  round  the  polar  axis. 

In  the  case  of  fixed  stars  this  is  most  unqualifiedly  true, 
but  in  the  case  of  the  sun,  and  the  planets  also,  the  single 
motion  of  revolution  can  be  discerned,  modified,  and 
slightly  altered  by  other  and  secondary  motions. 

Hence  the  universal  characteristic  of  the  celestial  bodies 
is  that  they  move  in  a  diurnal  circle. 

But  we  know  that  this  one  great  fact  which  is  true  of 
them  all  has  in  reality  nothing  to  do  with  them.  The 
diurnal  revolution  which  they  visibly  perform  is  the  result 
of  the  condition  of  the  observer.  It  is  because  the 
observer  is  on  a  rotating  earth  that  a  universal  statement 
can  be  made  about  all  the  celestial  bodies. 

The  universal  statement  which  is  valid  about  every  one 
of  the  celestial  bodies  is  that  which  does  not  concern 
them  at  all,  and  is  but  a  statement  of  the  condition  of 
the  observer. 

Now  there  are  universal  statements  of  other  kinds 
which  we  can  make.  We  can  say  that  all  objects  of 
experience  are  in  space  and  subject  to  the  laws  of 
geometry. 

wr 


108  THE   FOURTH    DIMENSION 

Does  this  mean  that  space  and  all  that  it  means  is  due 
to  a  condition  of  the  observer  ? 

If  a  universal  law  in  one  case  means  nothing  affecting 
the  objects  themselves,  but  only  a  condition  of  observa- 
tion, is  this  true  in  every  case?  There  is  shown  us  in 
astronomy  a  vera  causa  for  the  assertion  of  a  universal. 
Is  the  same  cause  to  be  traced  everywhere? 

Such  is  a  first  approximation  to  the  doctrine  of  Kant's 
critique. 

It  is  the  apprehension  of  a  relation  into  which,  on  the 
one  side  and  the  other,  perfectly  definite  constituents 
enter — the  human  observer  and  the  stars — and  a  trans- 
ference of  this  relation  to  a  region  in  which  the  con- 
stituents on  either  side  are  perfectly  unknown. 

If  spatiality  is  due  to  a  condition  of  the  observer,  the 
observer  cannot  be  this  bodily  self  of  ours — the  body,  like 
the  objects  around  it,  are  equally  in  space. 

This  conception  Kant  applied,  not  only  to  the  intuitions 
of  sense,  but  to  the  concepts  of  reason — wherever  a  universal 
statement  is  made  there  is  afforded  him  an  opportunity 
for  the  application  of  his  principle.  He  constructed  a 
system  in  which  one  hardly  knows  which  the  most  to 
admire,  the  architectonic  skill,  or  the  reticence  with  regard 
to  things  in  themselves,  and  the  observer  in  himself. 

His  system  can  be  compared  to  a  garden,  somewhat 
formal  perhaps,  but  with  the  charm  of  a  quality  more 
than  intellectual,  a  besonnenheit,  an  exquisite  moderation 
over  all.  And  from  the  ground  he  so  carefully  prepared 
with  that  buried  in  obscurity,  which  it  is  fitting  should 
be  obscure,  science  blossoms  and  the  tree  of  real  knowledge 
grows. 

The  critique  is  a  storehouse  of  ideas  of  profound  interest. 
The  one  of  which  I  have  given  a  partial  statement  leads, 
as  we  shall  see  on  studying  it  in  detail,  to  a  theory  of 
mathematics  suggestive  of  enquiries  in  many  direptjon?. 


APPLICATION   TO  KAJTr's   THEORY  OF  EXPERIENCE    109 

The  justification  for  my  treatment  will  be  found 
amongst  other  passages  in  that  part  of  the  transcendental 
analytic,  in  which  Kant  speaks  of  objects  of  experience 
subject  to  the  forms  of  sensibility,  not  subject  to  the 
concepts  of  reason. 

Kant  asserts  that  whenever  we  think  we  think  of 
objects  in  space  and  time,  but  he  denies  that  the  space 
and  time  exist  as  independent  entities.  He  goes  about 
to  explain  them,  and  their  universality,  not  by  assuming 
them,  as  most  other  philosophers  do,  but  by  postulating 
their  absence.  How  then  does  it  come  to  pass  that  the 
world  is  in  space  and  time  to  us  ? 

Kant  takes  the  same  position  with  regard  to  what  we 
call  nature — a  great  system  subject  to  law  and  order. 
"  How  do  you  explain  the  law  and  order  in  nature  ?  "  we 
ask  the  philosophers.  All  except  Kant  reply  by  assuming 
law  and  order  somewhere,  and  then  showing  how  we  can 
recognise  it. 

In  explaining  our  notions,  philosophers  from  ether  than 
the  Kantian  standpoint,  assume  the  notions  as  existing 
outside  us,  and  then  it  is  no  difficult  task  to  show  how 
they  come  to  us,  either  by  inspiration  or  by  observation. 

We  ask  "  Why  do  we  have  an  idea  of  law  in  nature  ?  " 
"  Because  natural  processes  go  according  to  law,"  we  are 
answered,  "  and  experience  inherited  or  acquired,  gives  us 
this  notion." 

But  when  we  speak  about  the  law  in  nature  we  are 
speaking  about  a  notion  of  our  own.  So  all  that  these 
expositors  do  is  to  explain  our  notion  by  an  assumption 
of  it. 

Kant  is  very  different.  He  supposes  nothing.  An  ex- 
perience such  as  ours  is  very  different  from  experience 
in  the  abstract.  Imagine  just  simply  experience,  suc- 
cession of  states,  of  consciousness !  Why,  there  would 
be  no  connecting  any  two  together,  there  would  be  no 


110  TftE    FOURTH    DIMENSION 

personal  identity,  no  memory.  It  is  out  of  a  general 
experience  such  as  this,  which,  in  respect  to  anything  we 
call  real,  is  less  than  a  dream,  that  Kant  shows  the 
genesis  of  an  experience  such  as  ours. 

Kant  takes  up  the  problem  of  the  explanation  of  space, 
time,  order,  and  so  quite  logically  does  not  presuppose 
them. 

But  how,  when  every  act  of  thought  is  of  things  in 
space,  and  time,  and  ordered,  shall  we  represent  to  our- 
selves that  perfectly  indefinite  somewhat  which  is  Kant's 
necessary  hypothesis — that  which  is  not  in  space  or  time 
and  is  not  ordered.  That  is  our  problem,  to  represent 
that  which  Kant  assumes  not  subject  to  any  of  our  forms 
of  thought,  and  then  show  some  function  which  working 
on  that  makes  it  into  a  "  nature "  subject  to  law  and 
order,  in  space  and  time.  Such  a  function  Kant  calls  the 
"Unity  of  Apperception";  i.e.,  that  which  makes  our  state 
of  consciousness  capable  of  being  woven  into  a  system 
with  a  self,  an  outer  world,  memory,  law,  cause,  and  order. 

The  difficulty  that  meets  us  in  discussing  Kant's 
hypothesis  is  that  everything  we  think  of  is  in  space 
and  time — how  then  shall  we  represent  in  space  an  exis- 
tence not  in  space,  and  in  time  an  existence  not  in  time  ? 
This  difficulty  is  still  more  evident  when  we  come  to 
construct  a  poiograph,  for  a  poiograph  is  essentially  a 
space  structure.  But  because  more  evident  the  difficulty 
is  nearer  a  solution.  If  we  always  think  in  space,  i.e. 
using  space  concepts,  the  first  condition  requisite  for 
adapting  them  to  the  representation  of  non-spatial  exis- 
tence, is  to  be  aware  of  the  limitation  of  our  thought, 
and  so  be  able  to  take  the  proper  steps  to  overcome  it. 
The  problem  before  us,  then,  is  to  represent  in  space  an 
existence  not  in  space. 

The  solution  is  an  easy  one.  It  is  provided  by  the 
conception  of  alternativity. 


APPLICATION  TO  KANT*S   THEORY  OF  EXPERIENCE    111 

To  get  our  ideas  clear  let  us  go  right  back  behind  the 
distinctions  of  an  inner  and  an  outer  world.  Both  of 
these,  Kant  says,  are  products.  Let  us  take  merely  states 
of  consciousness,  and  not  ask  the  question  whether  they  are 
produced  or  superinduced — to  ask  such  a  question  is  to 
have  got  too  far  on,  to  have  assumed  something  of  which 
we  have  not  traced  the  origin.  Of  these  states  let  us 
simply  say  that  they  occur.  Let  us  now  use  the  word 
a  "posit"  for  a  phase  of  consciousness  reduced  to  its 
last  possible  stage  of  evanescence ;  let  a  posit  be  that 
phase  of  consciousness  of  which  all  that  can  be  said  is 
that  it  occurs. 

Let  a,  b,  c,  be  three  such  posits.  We  cannot  represent 
them  in  space  without  placing  them  in  a  certain  order, 
as  a,  b,  c.  But  Kant  distinguishes  between  the  forms 
of  sensibility  and  the  concepts  of  reason.  A  dream  in 
which  everything  happens  at  haphazard  would  be  an 
experience  subject  to  the  form  of  sensibility  and  only 
partially  subject  to  the  concepts  of  reason.  It  is  par- 
tially subject  to  the  concepts  of  reason  because,  although 
there  is  no  order  of  sequence,  still  at  any  given  time 
there  is  order.  Perception  of  a  thing  as  in  space  is  a 
form  of  sensibility,  the  perception  of  an  order  is  a  concept 
of  reason. 

We  must,  therefore,  in  order  to  get  at  that  process 
which  Kant  supposes  to  be  constitutive  of  an  ordered 
experience  imagine  the  posits  as  in  space  without 
order. 

As  we  know  them  they  must  be  in  some  order,  abc, 
bca,  cab,  acb,  cba,  bac,  one  or  another. 

To  represent  them  as  having  no  order  conceive  all 
these  different  orders  as  equally  existing.  Introduce  the 
conception  of  alternativity — let  us  suppose  that  the  order 
abc,  and  bac,  for  example,  exist  equally,  so  that  we 
cannot  say  about  a  that  it  comes  before  or  after  b.  This 


112  THE   FOURTH   DIMENSION 

would  correspond  to  a  sudden  and  arbitrary  change  of  rt 
into  6  and  b  into  a,  so  that,  to  use  Kant's  words,  it  would 
be  possible  to  call  one  thing  by  one  name  at  one  time 
and  at  another  time  by  another  name. 

In  an  experience  of  this  kind  we  have  a  kind  of  chaos, 
in  which  no  order  exists;  it  is  a  manifold  not  subject  to 
the  concepts  of  reason. 

Now  is  there  any  process  by  which  order  can  be  intro- 
duced into  such  a  manifold — is  there  any  function  of 
consciousness  in  virtue  of  which  an  ordered  experience 
could  arise  ? 

In  the  precise  condition  in  which  the  posits  are,  as 
described  above,  it  does  not  seem  to  be  possible.  But 
if  we  imagine  a  duality  to  exist  in  the  manifold,  a 
function  of  consciousness  can  be  easily  discovered  which 
will  produce  order  out  of  no  order. 

Let  us  imagine  each  posit,  then,  as  having,  a  dual  aspect. 
Let  a  be  la  in  which  the  dual  aspect  is  represented  by  the 
combination  of  symbols.  And  similarly  let  b  be  26, 
c  be  3c,  in  which  2  and  b  represent  the  dual  aspects 
of  6,  3  and  c  those  of  c. 

Since  a  can  arbitrarily  change  into  6,  or  into  c,  and 
so  on,  the  particular  combinations  written  above  cannot 
be  kept.  We  have  to  assume  the  equally  possible  occur- 
rence of  form  such  as  2a,  2b,  and  so  on ;  and  in  order 
to  get  a  representation  of  all  those  combinations  out  of 
which  any  set  is  alternatively  possible,  we  must  take 
every  aspect  with  every  aspect.  We  must,  that  is,  have 
every  letter  with  every  number. 

Let  us  now  apply  the  method  of  space  represention. 

Note. — At  the  beginning  of  the  next  chapter  the  same 
structures  as  those  which  follow  are  exhibited  in 
more  detail  and  a  reference  to  them  will  remove 
any  obscurity  which  may  be  found  in  the  imme- 
diately following  passages.  They  are  there  carried 


APPLICATION  TO  KANT'S  THEORY  OF  EXPERIENCE   113 

on  to  a  greater  multiplicity  of  dimensions,  and  the 
significance  of  the  process  here  briefly  explained 
becomes  more  apparent. 

Take  three  mutually  rectangular  axes  in  space  1,  2,  3 
(fig.  59),  and  on  each  mark  three  points, 
the  common  meeting  point  being  the 
first  on  each  axis.  Then  by  means  of 
these  three  points  on  each  axis  we 
define  27  positions,  27  points  in  a 
cubical  cluster,  shown  in  fig.  60,  the 
same  method  of  co-ordination  being 
used  as  has  been  described  before. 
Each  of  these  positions  can  be  named  by  means  of  the 
axes  and  the  points  combined. 

Thus,  for  instance,  the  one  marked  by  an  asterisk  can 
<*k  be    called    Ic,  26,  3c,  because  it  is 

opposite  to  c  on  1,  to  6  on   2,   to 
c  on  3. 

Let  us  now  treat  of  the  states  of 
consciousness  corresponding  to  these 
positions.  Each  point  represents  a 
composite  of  posits,  and  the  mani- 
fold of  consciousness  corresponding 
Fig.  <;o.  ,,  r  ,  .  ,*_.. 

to  them  is  of  a  certain  complexity. 

Suppose  now  the  constituents,  the  points  on  the  axes, 
to  interchange  arbitrarily,  any  one  to  become  any  other, 
and  also  the  axes  1,  2,  and  3,  to  interchange  amongst 
themselves,  any  one  to  become  any  other,  and  to  be  sub- 
ject to  no  system  or  law,  that  is  to  say,  that  order  does 
not  exist,  and  that  the  points  which  run  abc  on  each  axis 
may  run  bac,  and  so  on. 

Then  any  one  of  the  states  of  consciousness  represented 
by  the  points  in  the  cluster  can  become  any  other.  We 
have  a  representation  of  a  random  consciousness  of  a 
certain  degree  of  complexity 

8 


114 


FOURTH  DIMENSION 


«c2a3c 


Now  let  us  examine  carefully  one  particular  case  of 
arbitrary  interchange  of  the  points,  a,  b,  c ;  as  one  such 
case,  carefully  considered,  makes  the  whole  clear. 

Consider  the  points  named  in  the  figure  Ic,  2a,  3c ; 

Ic,  2c,  3a  ;    la,  2c,   3c,   and 
examine   the   effect   on   them 
>32c3c    when  a  change  of  order  takes 
(  place.      Let    us    suppose,    for 

instance,  that  a  changes  into  6, 
and  let  us  call  the  two  sets  of 
points  we  get,  the  one  before 
and  the  one  after,  their  change 
conjugates. 


1c2c3a 
Fig.  01. 


Before  the  change     Ic  2a  Be    Ic  2c  3a     la  2c  3c 
After  the  change       Ic  2b  3f     Ic  2e  3b     Ib  2c  3o 


\  Conjugates. 


The  points  surrounded  by  rings  represent  the  conjugate 
points. 

It  is  evident  that  as  consciousness,  represented  first  by 
the  first  set  of  points  and  afterwards  by  the  second  set  of 
points,  would  have  nothing  in  common  in  its  two  phases. 
It  would  not  be  capable  of  giving  an  account  of  itself. 
There  would  be  no  identity. 

If,  however,  we  can  find  any  set  of  points  in  the 
cubical  cluster,  which,  when  any  arbitrary  change  takes 
place  in  the  points  on  the  axes,  or  in  the  axes  themselves, 
repeats  itself,  is  reproduced,  then  a  consciousness  repre- 
sented by  those  points  would  have  a  permanence.  It 
would  have  a  principle  of  identity.  Despite  the  no  law, 
the  no  order,  of  the  ultimate  constituents,  it  would  have 
an  order,  it  would  form  a  system,  the  condition  of  a 
personal  identity  would  be  fulfilled. 

The  question  comes  to  this,  then.  Can  we  find  a 
system  of  points  which  is  self-conjugate  which  is  such 
that  when  any  posit  on  the  axes  becomes  any  other,  or 


APPLICATION  TO  KANT'S  THEORY  OF  EXPERIENCE  115 


when  any  axis  becomes  any  other,  such  a  set  is  trans- 
formed into  itself,  its  identity 
is  not  submerged,  but  rises 
superior  to  the  chaos  of  its 
constituents? 

Such  a  set  can  be  found. 
Consider  the  set  represented 
in  fig.  62,  and  written  down  in 
the  first  of  the  two  lines — 


Fig.  62. 


Self-  flaZbSc  \b  la  3c  Ic  2a  3b  Ic  2b  3a  Ib  2c  3a  Ia2o3b 
conjugate. \lc2b3a  Ib  2c  3a  la  2c  3b  la  2b  3c  Ib  2d  3c  Ic  2a  3b 

If  now  a  change  into  c  and  c  into  a,  we  get  the  set  in 
the  second  line,  which  has  the  same  members  as  are  in  the 
upper  line.  Looking  at  the  diagram  we  see  that  it  would 
correspond  simply  to  the  turning  of  the  figures  as  a 
whole.*  Any  arbitrary  change  of  the  points  on  the  axes, 
or  of  the  axes  themselves,  reproduces  the  same  set. 

Thus,  a  function,  by  which  a  random,  an  unordered,  con- 
sciousness could  give  an  ordered  and  systematic  one,  can 
be  represented.  It  is  noteworthy  that  it  is  a  system  of 
selection.  If  out  of  all  the  alternative  forms  that  only  is 
attended  to  which  is  self-conjugate,  an  ordered  conscious- 
ness is  formed.  A  selection  gives  a  feature  of  permanence. 

Can  we  say  that  the  permanent  consciousness  is  this 
selection  ? 

An  analogy  between  Kant  and  Darwin  comes  into  light. 
That  which  is  swings  clear  of  the  fleeting,  in  virtue  of  its 
presenting  a  feature  of  permanence.  There  is  no  need 
to  suppose  any  function  of  "  attending  to."  A  con- 
sciousness capable  of  giving  au  account  of  itself  is  one 
which  is  characterised  by  this  combination.  All  com- 
binations exist — of  this  kind  is  the  consciousness  which 
can  give  an  account  of  itself.  And  the  very  duality  which 

*  These  figures  are  described  more  fully,  and  extended,  in  the  next 
chapter. 


116  THE   FOURTH   DIMENSION 

we  have  presupposed  may  be  regarded  as  originated  by 
a  process  of  selection. 

Darwin  set  himself  to  explain  the  origin  of  the  fauna 
and  flora  of  the  world.  He  denied  specific  tendencies. 
He  assumed  an  indefinite  variability — that  is,  chance — 
but  a  chance  confined  within  narrow  limits  as  regards  the 
magnitude  of  any  consecutive  variations.  He  showed  that 
organisms  possessing  features  of  permanence,  if  they 
occurred  would  be  preserved.  So  his  account  of  any 
structure  or  organised  being  was  that  it  possessed  features 
of  permanence. 

Kant,  undertaking  not  the  explanation  of  any  particular 
phenomena  but  of  that  which  we  call  nature  as  a  whole, 
had  an  origin  of  species  of  his  own,  an  account  of  the 
flora  and  fauna  of  consciousness.  He  denied  any  specific 
tendency  of  the  elements  of  consciousness,  but  taking  our 
own  consciousness,  pointed  out  that  in  which  it  resembled 
any  consciousness  which  could  survive,  which  could  give 
an  account  of  itself. 

He  assumes  a  chance  or  random  world,  and  as  great 
and  small  were  not  to  him  any  given  notions  of  which  he 
could  make  use,  he  did  not  limit  the  chance,  the  random- 
ness, in  any  way.  But  any  consciousness  which  is  per- 
manent must  possess  certain  features — those  attributes 
namely  which  give  it  permanence.  Any  consciousness 
like  our  own  is  simply  a  consciousness  which  possesses 
those  attributes.  The  main  thing  is  that  which  he  calls 
the  unity  of  apperception,  which  we  have  seen  above  is 
simply  the  statement  that  a  particular  set  of  phases  of 
consciousness  on  the  basis  of  complete  randomness  will  be 
self-conjugate,  and  so  permanent. 

As  with  Darwin  so  with  Kant,  the  reason  for  existence 
of  any  feature  comes  to  this — show  that  it  tends  to  the 
permanence  of  that  which  possesses  it. 

We  can  thus  regard  Kant  as  the  creator  of  the  first  of 


APPLICATION  TO   KANT's   THEORY   OP  EXPERIENCE    117 

the  modern  evolution  theories.  And,  as  is  so  often  the 
case,  the  first  effort  was  the  most  stupendous  in  its  scope. 
Kant  does  not  investigate  the  origin  of  any  special  part 
of  the  world,  such  as  its  organisms,  its  chemical  elements, 
its  social  communities  of  men.  He  simply  investigates 
the  origin  of  the  whole — of  all  that  is  included  in  con- 
sciousness, the  origin  of  that  "thought  thing"  whose 
progressive  realisation  is  the  knowable  universe. 

This  point  of  view  is  very  different  from  the  ordinary 
one,  in  which  a  man  is  supposed  to  be  placed  in  a  world 
like  that  which  he  has  come  to  think  of  it,  and  then  to 
learn  what  he  has  found  out  from  this  model  which  he 
himself  has  placed  on  the  scene. 

We  all  know  that  there  are  a  number  of  questions  in 
attempting  an  answer  to  which  such  an  assumption  is  not 
allowable. 

Mill,  for  instance,  explains  our  notion  of  "  law  "  by  an 
invariable  sequence  in  nature.  But  what  we  call  nature 
is  something  given  in  thought.  So  he  explains  a  thought 
"-f  law  and  order  by  a  thought  of  an  invariable  sequence. 
lie  leaves  the  problem  where  he  found  it. 

Kant's  theory  is  not  unique  and  alone.  It  is  one  of 
a  number  of  evolution  theories.  A  notion  of  its  import 
and  significance  can  be  obtained  by  a  comparison  of  it 
with  other  theories. 

Thus  in  Darwin's  theoretical  world  of  natural  selection 
a  certain  assumption  is  made,  the  assumption  of  indefinite 
variability — slight  variability  it  is  true,  over  any  appre- 
ciable lapse  of  time,  but  indefinite  in  the  postulated 
epochs  of  transformation — and  a  whole  chain  of  results 
is  shown  to  follow. 

This  element  of  chance  variation  is  not,  however,  an 
ultimate  resting  place.  It  is  a  preliminary  stage.  This 
supposing  the  all  is  a  preliminary  step  towards  finding 
out  what  is.  If  every  kind  of  organism  can  come  into 


118  THE   FOURTH   DIMENSION 

being,  those  that  do  survive  will  present  such  and  such 
characteristics.  This  is  the  necessary  beginning  for  ascer- 
taining what  kinds  of  organisms  do  come  into  existence. 
And  so  Kant's  hypothesis  of  a  random  consciousness  is 
the  necessary  beginning  for  the  rational  investigation 
of  consciousness  as  it  is.  His  assumption  supplies,  as 
it  were,  the  space  in  which  we  can  observe  the  pheno- 
mena. It  gives  the  general  laws  constitutive  of  any 
experience.  If,  on  the  assumption  of  absolute  random- 
ness in  the  constituents,  such  and  such  would  be 
characteristic  of  the  experience,  then,  whatever  the  con- 
stituents, these  characteristics  must  be  universally  valid. 

We  will  now  proceed  to  examine  more  carefully  the 
poiograph,  constructed  for  the  purpose  of  exhibiting  an 
illustration  of  Kant's  unity  of  apperception. 

In  order  to  show  the  derivation  order  out  of  non-order 
it  has  been  necessary  to  assume  a  principle  of  duality — 
we  have  had  the  axes  and  the  posits  on  the  axes — there 
are  two  sets  of  elements,  each  non-ordered,  and  it  is  in 
the  reciprocal  relation  of  them  that  the  order,  the  definite 
system,  originates. 

Is  there  anything  in  our  experience  of  the  nature  of  a 
duality  ? 

There  certainly  are  objects  in  our  experience  which 
have  order  and  those  which  are  incapable  of  order.  The 
two  roots  of  a  quadratic  equation  have  no  order.  No  one 
can  tell  which  comes  first.  If  a  body  rises  vertically  and 
then  goes  at  right  angles  to  its  former  course,  no  one  can 
assign  any  priority  to  the  direction  of  the  north  or  to  the 
east.  There  is  no  priority  in  directions  of  turning.  We 
associate  turnings  with  no  order  progressions  in  a  line 
with  order.  But  in  the  axes  and  points  we  have  assumed 
above  there  is  no  such  distinction.  It  is  the  same,  whether 
we  assume  an  order  among  the  turnings,  and  no  order 
among  the  points  on  the  axes,  or,  vice  versa,  an  order  in 


APPLICATION  TO  KANT'S   THEORY   OF  EXPERIENCE    119 

the  points  and  no  order  in  the  turnings.  A  being  with 
an  infinite  number  of  axes  mutually  at  right  angles, 
with  a  definite  sequence  between  them  and  no  sequence 
between  the  points  on  the  axes,  would  be  in  a  condition 
formally  indistinguishable  from  that  of  a  creature  who, 
according  to  an  assumption  more  natural  to  us,  had  on 
each  axis  an  infinite  number  of  ordered  points  and  no 
order  of  priority  amongst  the  axes,  A  being  in  such 
a  constituted  world  would  not  be  able  to  tell  which 
was  turning  and  which  was  length  along  an  axis,  in 
order  to  distinguish  between  them.  Thus  to  take  a  per- 
tinent illustration,  we  may  be  in  a  world  of  an  infinite 
number  of  dimensions,  with  three  arbitrary  points  on 
each — three  points  whose  order  is  indifferent,  or  in  a 
world  of  three  axes  of  arbitary  sequence  with  an  infinite 
number  of  ordered  points  on  each.  We  can't  tell  which 
is  which,  to  distinguish  it  from  the  other. 

Thus  it  appears  the  mode  of  illustration  which  we 
have  used  is  not  an  artificial  one.  There  really  exists 
in  nature  a  duality  of  the  kind  which  is  necessary  to 
explain  the  origin  of  order  out  of  no  order — the  duality, 
namely,  of  dimension  and  position.  Let  us  use  the  term 
group  for  that  system  of  points  which  remains  unchanged, 
whatever  arbitrary  change  of  its  constituents  takes  place. 
We  notice  that  a  group  involves  a  duality,  is  inconceivable 
without  a  duality. 

Thus,  according  to  Kant,  the  primary  element  of  ex- 
perience is  the  group,  and  the  theory  of  groups  would  be 
the  most  fundamental  branch  of  science.  Owing  to  an 
expression  in  the  critique  the  authority  of  Kant  is  some- 
times adduced  against  the  assumption  of  more  than  three 
dimensions  to  space.  It  seems  to  me,  however,  that  the 
whole  tendency  of  his  theory  lies  in  the  opposite  direction, 
and  points  to  a  perfect  duality  between  dimension  and 
position  in  a  dimension. 


120  THE   FOURTH  DIMENSION 

If  the  order  and  the  law  we  see  is  due  to  the  conditions 
of  conscious  experience,  we  must  conceive  nature  as 
spontaneous,  free,  subject  to  no  predication  that  we  can 
devise,  but,  however  apprehended,  subject  to  our  logic. 

And  our  logic  is  simply  spatiality  in  the  general  sense 
— that  resultant  of  a  selection  of  the  permanent  from  the 
unpermanent,  the  ordered  from  the  unordered,  by  the 
means  of  the  group  and  its  underlying  duality. 

We  can  predicate  nothing  about  nature,  only  about  the 
way  in  which  we  can  apprehend  nature.  All  that  we  can 
say  is  that  all  that  which  experience  gives  us  will  be  con- 
ditioned as  spatial,  subject  to  our  logic.  Thus,  in  exploring 
the  facts  of  geometry  from  the  simplest  logical  relations 
to  the  properties  of  space  of  any  number  of  dimensions, 
we  are  merely  observing  ourselves,  becoming  aware  of 
the  conditions  under  which  we  must  perceive.  Do  any 
phenomena  present  themselves  incapable  of  explanation 
under  the  assumption  of  the  space  we  are  dealing  with, 
then  we  must  habituate  ourselves  to  the  conception  of  a 
higher  space,  in  order  that  our  logic  may  be  equal  to  the 
task  before  us. 

We  gain  a  repetition  of  the  thought  that  came  before, 
experimentally  suggested.  If  the  laws  of  the  intellectual 
comprehension  of  nature  are  those  derived  from  con- 
sidering her  as  absolute  chance,  subject  to  no  law  save 
that  derived  from  a  process  of  selection,  then,  perhaps,  the 
order  of  nature  requires  different  faculties  from  the  in- 
tellectual to  apprehend  it.  The  source  and  origin  of 
ideas  may  have  to  be  sought  elsewhere  than  in  reasoning. 

The  total  outcome  of  the  critique  is  to  leave  the 
ordinary  man  just  where  he  is,  justified  in  his  practical 
attitude  towards  nature,  liberated  from  the  fetters  of  his 
own  mental  representations. 

The  truth  of  a  picture  lies  in  its  total  effect.  It  is  vain 
to  seek  information  about  the  landscape  from  an  examina- 


APPLICATION  TO   KANT'S   THEORY  OF   EXPERIENCE    121 

tion  of  the  pigments.  And  in  any  method  of  thought  it 
is  the  complexity  of  the  whole  that  brings  us  to  a  know- 
ledge of  nature.  Dimensions  are  artificial  enough,  but  in 
the  multiplicity  of  them  we  catch  some  breath  of  nature. 

We  must  therefore,  and  this  seems  to  me  the  practical 
conclusion  of  the  whole  matter,  proceed  to  form  means  of 
intellectual  apprehension  of  a  greater  and  greater  degree 
of  complexity,  both  dimensionally  and  in  extent  in  any 
dimension.  Such  means  of  representation  must  always 
be  artificial,  but  in  the  multiplicity  of  the  elements  with 
which  we  deal,  however  incipiently  arbitrary,  lies  our 
chance  of  apprehending  nature. 

And  as  a  concluding  chapter  to  this  part  of  the  book, 
I  will  extend  the  figures,  which  have  been  used  to  repre- 
sents Kant's  theory,  two  steps,  so  that  the  reader  may 
have  the  opportunity  of  looking  at  a  four-dimensional 
figure  which  can  be  delineated  without  any  of  the  special 
apparatus,  to  the  consideration  of  which  I  shall  subse- 
quently pass  on. 


CHAPTER    X 

A    FOUR-DIMENSIONAL    FIGURE 

THE  method  used  in  the  preceding  chapter  to  illustrate 
the  problem  of  Kant's  critique,  gives  a  singularly  easy 
and  direct  mode  of  constructing  a  series  of  important 
figures  in  any  number  of  dimensions. 

We  have  seen  that  to  represent  our  space  a  plane  being 
must  give  up  one  of  his  axes,  and  similarly  to  represent 
the  higher  shapes  we  must  give  up  one  amongst  our 
three  axes. 

But  there  is  another  kind  of  giving  up  which  reduces 
the  construction  of  higher  shapes  to  a  matter  of  the 
utmost  simplicity. 

Ordinarily  we  have  on  a  straight  line  any  number  of 
positions.  The  wealth  of  space  in  position  is  illimitable, 
while  there  are  only  three  dimensions. 

I  propose  to  give  up  this  wealth  of  positions,  and  to 
consider  the  figures  obtained  by  taking  just  as  many 
positions  as  dimensions. 

In  this  way  I  consider  dimensions  and  positions  as  two 
"  kinds,"  and  applying  the  simple  rule  of  selecting  every 
one  of  one  kind  with  every  other  of  every  other  kind, 
get  a  series  of  figures  which  are  noteworthy  because 
they  exactly  fill  space  of  any  number  of  dimensions 
(as  the  hexagon  fills  a  plane)  by  equal  repetitions  of 

themselves. 

122 


A   FOUR-DIMENSIONAL   FIGURE  123 

The  rule  will  be  made  more  evident  by  a  simple 
application. 

Let  us  consider  one  dimension  and  one  position.  I  will 
call  the  axis  i,  and  the  position  o. 


Here  the  figure  is  the  position  o  on  the  line  i.     Take 
now  two  dimensions  and  two  positions  on  each. 

We  have  the  two  positions  o  ;    1  on  i,  and  the  two 
positions  o,  1  on  j,  fig.  63.     These  give 
J    rise  to  a  certain  complexity.     I  will 
let  the  two  lines  i  and  j  meet  in  the 
position  I  call  o  on  each,  and  I  will 
consider  i  as  a  direction  starting  equally 
big.  63.  from    every   position    on  j,   and  j  as 


B 


it.oj 


starting   equally   from  every   position   on   i.      We  thus 
obtain  the  following  figure  :  —  A  is  both  oi  and  oj,  B  is  1  i 
A  _  C          ai]d  o},  and  so  on  as  shown  in  fig.  636. 
The  positions  on  AC  are  all  oi  positions. 
They  are,  if  we  like  to  consider  it  in 
that  way,  points  at  no  distance  in  the  i 
direction  from  the  line  AC.       We  can 
call  the  line  AC  the  oi  line.    Similarly 
the  points  on  AB  are  those  no  distance 
Fig.  63ft.  from  AB  in  thej  direction,  and  we  can 

call  them  oj  points  and  the  line  AB  the  oj  line.  Again, 
the  line  CD  can  be  called  the  Ij  line  because  the  points 
on  it  are  at  a  distance,  1  in  the  j  direction. 

We  have  then  four  positions  or  points  named  as  shown, 
and,  considering  directions  and  positions  as  "  kinds,"  we 
have  the  combination  of  two  kinds  with  two  kinds.  Now, 
selecting  every  one  of  one  kind  with  every  other  of  every 
other  kind  will  mean  that  we  take  1  of  the  kind  i  and 


134 


THE   FOURTH   DIMENSION 


with  it  o  of  the  kind  j  ;  and  then,  that  we  take  o  of  the 
kind  i  and  with  it  1  of  the  kind  j. 

Thus  we  get  a  pair  of  positions  lying  in  the  straight 
C  line  BC,  fig.  64.  We  can  call  this  pair  10 
and  01  if  we  adopt  the  plan  of  mentally, 
adding  an  i  to  the  first  and  a  j  to  the 
second  of  the  symbols  written  thus—  01 
is  a  short  expression  for  Oi,  Ij. 

Coining  now  to  our  space,  we  have  three 
dimensions,  so  we  take  three  positions  on  each.  These 
positions  I  will  suppose  to  be  at  equal  distances  along  each 


.  64. 


Fig.  65. 

axis.  The  three  axes  and  the  three  positions  on  each  are 
shown  in  the  accompanying  diagrams,  fig.  65,  of  which 
the  first  represents  a  cube  with  the  front  faces  visible,  the 
second  the  rear  faces  of  the  same  cube ;  the  positions  I 
will  call  0,  1,  2  ;  the  axes,  i,j,  k.  I  take  the  base  ABC  as 
the  starting  place,  from  which  to  determine  distances  in 
the  k  direction,  and  hence  every  point  in  the  base  ABC 
will  be  an  ok  position,  and  the  base  ABC  can  be  called  an 
ok  plane. 

In  the  same  way,  measuring  the  distances  from  the  face 
\DC,  we  see  that  every  position  in  the  face  ADC  is  a  oi 
position,  and  the  whole  plane  of  the  face  may  be  called  an 
oi  plane.  Thus  we  see  that  with  the  introduction  of  a 


A  FOUR-DIMENSIONAL   FIGURE 


125 


hew  dimension  the  signification  of  a  compound  symbol, 
such  as  "  oi,"  alters.  In  the  plane  it  meant  the  line  AC. 
In  space  it  means  the  whole  plane  ACD. 

Now,  it  is  evident  that  we  have  twenty-seven  positions, 
each  of  them  named.  If  the  reader  will  follow  this 
nomenclature  in  respect  of  the  positions  marked  in  the 
figures  he  will  have  no  difficulty  in  assigning  names  to 
each  one  of  the  twenty-seven  positions.  A  is  oi,  oj,  uk. 
It  is  at  the  distance  0  along  i,  0  along  j,  0  along  k,  and 
io  can  be  written  in  short  000,  where  the  ijk  symbols 
are  omitted. 

The  point  immediately  above  is  001,  for  it  is  no  dis- 
tance in  the  i  direction,  and  a  distance  of  1  in  the  k 
direction.  Again,  looking  at  B,  it  is  at  a  distance  of  2 
from  A,  or  from  the  plane  ADC,  in  the  i  direction,  0  in  the 
j  direction  from  the  plane  ABD,  and  0  in  the  k  direction, 
measured  from  the  plane  ABC.  Hence  it  is  200  written 
for  2i,  Oj,  Ok. 

Now,  out  of  these  twenty-seVen  "things  "  or  compounds 
of  position  and  dimension,  select  those  which  are  given  by 
the  rule,  every  one  of  one  kind  with  every  other  of  every 
other  kind. 

Take  2  of  the  i  kind.     With  this 
we  must  have  a   1   of  the  j  kind, 
and  then  by  the  rule  we  can  only 
have  a  0  of  the  k  kind,  for  if  we 
had   any  other  of  the   k   kind  we 
should  repeat  one  of  the  kinds  we 
already    had.      In   2i,    Ij,    Ik,    for 
instance,  1  is  repeated.     The  point 
we  obtain  is  that  marked  210,  fig.  66. 
Proceeding   in    this    way,    we    pick    out  the  following 
cluster   of  points,   fig.    67.      They   are  joined   by   lines, 
dotted  where  they  are  hidden  by  the  body  of  the  cube, 
and  we  see  that  they  form  a  figure— a  hexagon  which 


126 


FOURTft   DIMENSION 


could  be  taken  out  of  the  cube  and  placed  on  a  plane. 
It  is  a  figure  which  will  fill  a 
plane  by  equal  repetitions  of  itself. 
The  plane  being  representing  this 
construction  in  his  plane  would 
take  three  squares  to  represent  the 
cube.  Let  us  suppose  that  he 
takes  the  ij  axes  in  his  space  and 
k  represents  the  axis  running  out 
of  his  space,  fig.  68.  In  each  of 
the  three  squares  shown  here  as  drawn  separately  he 
could  select  the  points  given  by  the  rule,  and  he  would 


Fig.  Gi 


then  have  to  try  to  discover  the  figure  determined  by 
the  three  lines  drawn.  The  line  from  210  to  120  is 
given  in  the  figure,  but  the  line  from  201  to  102  or  GK 
is  not  given.  He  can  determine  GK  by  making  another 
set  of  drawing^  and  discovering  in  them  what  the  relation 
between  these  two  extremities  is. 


I02.X' 

"x-fc 


201 


C2I 


^B 

Fig.  69. 

Let  him  draw  the  i  and  k  axes  in  his  plane,  fig.  69. 
The  j  axis  then  runs  out  and  he  has  the  accompanying 
figure.  In  the  first  of  these  three  squares,  fig.  69,  he  can 


127 


pick  out  by  the  rule  the  two  points  201,  102 — u,  and  K. 
Here  they  occur  in  one  plane  and  he  can  measure  the 
distance  between  them.  In  his  first  representation  they 
occur  at  G  and  K  in  separate  figures. 

Thus  the  plane  being  would  find  that  the  ends  of  each 
of  the  lines  was  distant  by  the  diagonal  of  a  unit  square 
from  the  corresponding  end  of  the  last  and  he  could  then 
place  the  three  lines  in  their  right  relative  position. 
Joining  them  he  would  have  the  figure  of  a  hexagon. 
We  may  also  notice  that  the  plane  being  could  make 
a  representation  of  the  whole  cube 
simultaneously.  The  three  squares, 
shown  in  perspective  in  fig.  70,  all 
lie  in  one  plane,  and  on  these  the 
plane  being  could  pick  out  any 
selection  of  points  just  as  well  as  on 
three  separate  squares.  He  would 
obtain  a  hexagon  by  joining  the 
points  marked.  This  hexagon,  as 
drawn,  is  of  the  right  shape,  but  it  would  not  be  so  if 
actual  squares  were  used  instead  of  perspective,  because 
the  relation  between  the  separate  squares  as  they  lie  in 
the  plane  figure  is  not  their  real  relation.  The  figure, 
however,  as  thus  constructed,  would  give  him  an  idea  of 
the  correct  figure,  and  he  could  determine  it  accurately 
by  remembering  that  distances  in  each  square  were 
correct,  but  in  passing  from  one  square  to  another  their 
distance  in  the  third  dimension  had  to  be  taken  into 
account. 

Coming  now  to  the  figure  made  by  selecting  according 
to  our  rule  from  the  whole  mass  of  points  given  by  four 
axes  and  four  positions  in  each,  we  must  first  draw  a 
catalogue  figure  in  which  the  whole  assemllage  is  shown. 

We  can  represent  this  assemblage  of  points  by  four 
solid  figures.  The  first  giving  all  those  positions  which 


Fig.  70. 


128 


THE  FOURTH  DIMENSION 


are  at  a  distance  o  from  our  space  in  the  fourth  dimen* 
sion,  the  second  showing  all  those  that  are  at  a  distance  1, 
and  so  on. 

These  figures  will  each  be  cubes.  The  first  two  are 
drawn  showing  the  front  faces,  the  second  two  the  rear 
faces.  We  will  mark  the  points  0,  1,2,  3,  putting  points 
at  those  distances  along  each  of  these  axes,  and  suppose 


Fig.  71. 

all  the  points  thus  determined  to  be  contained  in  solid 
models  of  which  our  drawings  in  fig.  71  are  represen- 
tatives. Here  we  notice  that  as  on  the  plane  Oi  meant 
the  whole  line  from  which  the  distances  in  the  i  direction 
was  measured,  and  as  in  space  Oi  means  the  whole  plane 
from  which  distances  in  the  i  direction  are  measured,  so 
now  Oh  means  the  whole  space  in  which  the  first  cube 
stands — measuring  away  from  that  space  by  a  distance 
of  one  we  come  to  the  second  cuhe  represented. 


A   FOUR-DIMENSIONAL   FIGUEE 


129 


Now  selecting  according  to  the  rule  every  one  of  one 
kind  with  every  other  of  every  other  kind,  we  must  take, 
for  instance,  3i,  2j,  Ik,  Oh.  This  point  is  marked 
3210  at  the  lower  star  in  the  figure.  It  is  3  in  the 
i  direction,  2  in  the  j  direction,  1  in  the  k  direction, 
0  in  the  h  direction. 

With  3i  we  must  also  take  1^  2k,  Oh.  This  point 
is  shown  by  the  second  star  in  the  cube  Ohi, 


In  the  first  cube,  since  all  the  points  are  Oh  points, 
we  can  only  have  varieties  in  which  i,  j,  k,  are  accom- 
panied by  3,  2,  1. 

The  points  determined  are  marked  off  in  the  diagram 
fig.  72,  and  lines  are  drawn  joining  the  adjacent  pairs 
in  each  figure,  the  lines  being  dotted  when  they  pass 
within  the  substance  of  the  cube  in  the  first  two  diagrams. 

Opposite  each  point,  on  one  side  or  the  other  of  each 

9 


130 


THE   FOURTH   DIMENSION 


cube,  is  written  its  name.  It  will  be  noticed  that  the 
figures  are  symmetrical  right  and  left;  and  right  and 
left  the  first  two  numbers  are  simply  interchanged. 

Now  this  being  our  selection  of  points,  what  figure  do 
they  make  when  all  are  put  together  in  their  proper 
relative  positions  ? 

To  determine  this  we  must  find  the  distance  between 
corresponding  corners  of  the  separate  hexagons. 


2103 


0215 


5102 


2K 


Fig.  73. 


3K 


To  do  this  let  us  keep  the  axes  i,  j,  in  our  space,  and 
draw  h  instead  of  k,  letting  k  run  out  in  the  fourth 
dimension,  fig.  73. 

Here  we  have  four  cubes  again,  in  the  first  of  which  all 
the  points  are  Ok  points ;  that  is,  points  at  a  distance  zero 
in  the  k  direction  from  the  space  of  the  three  dimensions 
ijh.  We  have  all  the  points  selected  before,  and  some 
of  the  distances,  which  in  the  last  diagram  led  from  figure 
to  figure  are  shown  here  in  the  same  figure,  and  so  capable 


A    POUR-DIMENSIONAL    FlGURfc 


131 


of  measurement.  Take  for  instance  the  points  3120  to 
3021,  which  in  the  first  diagram  (fig.  72)  lie  in  the  first 
and  second  figures.  Their  actual  relation  is  shown  in 
fig.  73  in  the  cube  marked  2K,  where  the  points  in  ques- 
tion are  marked  with  a  *  in  fig.  73.  We  see  that  the 
distance  in  question  is  the  diagonal  of  a  unit  square.  In 
like  manner  we  find  that  the  distance  between  corres- 
ponding points  of  any  two  hexagonal  figures  is  the 
diagonal  of  a  unit  square.  The  total  figure  is  now  easily 
constructed.  An  idea  . 

of  it  may  be  gained  by  T  BX^XJ 
drawing  all  the  four 
cubes  in  the  catalogue 
figure  in  one  (fig.  74). 
These  cubes  are  exact 
repetitions  of  one 
another,  so  one  draw- 
ing will  serve  as  a 
representation  of  the 
whole  series,  if  we 
take  care  to  remember 
where  we  are,  whether 
in  a  Oh,  a  Ih,  a  2h, 
or  a  3h  figure,  when 
we  pick  out  the  points  required.  Fig.  74  is  a  represen- 
tation of  all  the  catalogue  cubes  put  in  one.  For  the 
sake  of  clearness  the  front  faces  and  the  back  faces  of 
this  cube  are  represented  separately. 

The  figure  determined  by  the  selected  points  is  shown 
below. 

In  putting  the  sections  together  some  of  the  outlines 
in  them  disappear.  The  line  TW  for  instance  is  not 
wanted. 

We  notice  that  PQTW  and  TWRS  are  each  the  half 
of  a  hexagon.  Now  QV  and  VR  lie  in  .one  straight  line. 


Fig.  74. 


132 


THE   FOURTH   DIMENSION 


Hence  these  two  hexagons  fit  together,  forming  one 
hexagon,  and  the  line  TVV  is  only  wanted  when  we  con- 
sider a  section  of  the  whole  figure,  we  thus  obtain  the 
solid  represented  in  the  lower  part  of  fig.  74.  Equal 
repetitions  of  this  figure,  called  a  tetrakaidecagon,  will 
fill  up  three-dimensional  space. 

To  make  the  corresponding  four-dimensional  figure  we 
have  to  take  five  axes  mutually  at  right  angles  with  five 
points  on  each.  A  catalogue  of  the  positions  determined 
in  five-dimensional  space  can  be  found  thus. 

Take  a  cube  with  five  points  on  each  of  its  axes,  the 
fifth  point  is  at  a  distance  of  four  units  of  length  from  the 
first  on  any  one  of  the  axes.  And  since  the  fourth  dimen- 
also  stretches  to  a  distance  of  four  we  shall  need  to 

represent  the  succes- 

BO 


-ion 


4L 


sive  sets  of  points  at 
distances  0,  1,  2,  3,4, 
in  the  fourth  dimen- 
sions, five  cubes.  Now 
all  of  these  extend  to 
no  distance  at  all  in 
the  fifth  dimension. 
To  represent  what 
lies  in  the  fifth  dimen 
sion  we  shall  have  to 
draw,  starting  from 
each  of  our  cubes,  five 
similar  cubes  to  re- 
present the  four  steps 
on  in  the  fifth  dimension.  By  this  assemblage  we  get  a 
catalogue  of  all  the  points  shown  in  fig.  75,  in  which 
L  represents  the  fifth  dimension. 

Now,  as  we  saw  before,  there  is  nothing  to  prevent  us 
from  putting  all  the  cubes  representing  the  different 
stages  in  the  fourth  dimension  in  one  figure,  if  we  take 


OH 


Fig.  7r,. 


A    FOUR-DIMENSIONAL   FIGURE  133 

note  when  we  look  at  it,  whether  we  consider  it  as  a  OA,  a 
\h,  a  2h,  etc.,  cube.  Putting  then  the  Oh,  Ih,  2h,  3h,  4/4 
cubes  of  each  row  in  one,  we  have  five  cubes  with  the  sides 
of  each  containing  five  positions,  the  first  of  these  five 
cubes  represents  the  01  points,  and  has  in  it  the  i  points 
from  0  to  4,  the  j  points  from  0  to  4,  the  k  points  from 
0  to  4,  while  we  have  to  specify  with  regard  to  any 
selection  we  make  from  it,  whether  we  regard  it  as  a  Oh, 
a  Ih,  a  2h,  a  3&,  or  a  4 h  figure.  In  fig.  76  each  cube 
is  represented  by  two  drawings,  one  of  the  front  part,  the 
other  of  the  rear  part. 

Let  then  our  five  cubes  be  arranged  before  us  and  our 
selection  be  made  according  to  the  rule.  Take  the  first 
figure  in  which  all  points  are  Ql  points.  We  cannot 
have  0  with  any  other  letter.  Then,  keeping  in  the  first 
figure,  which  is  that  of  the  Ql  positions,  take  first  of  all 
that  selection  which  always  contains  Ih.  We  suppose, 
therefore,  that  the  cube  is  a  Ih  cube,  and  in  it  we  take 
i,j,  k  in  combination  with  4,  3,  2  according  to  the  rule. 

The  figure  we  obtain  is  a  hexagon,  as  shown,  the  one 
in  front.  The  points  on  the  right  hand  have  the  same 
figures  as  those  on  the  left,  with  the  first  two  numerals 
interchanged.  Next  keeping  still  to  the  Ql  figure  let 
us  suppose  that  the  cube  before  us  represents  a  section 
at  a  distance  of  2  in  the  h  direction.  Let  all  the  points 
in  it  be  considered  as  2h  points.  We  then  have  a  01,  2h 
region,  and  have  the  sets  ijk  and  431  left  over.  We 
must  then  pick  out  in  accordance  with  our  rule  all  such 
points  as  4i,  3jf,  Ik. 

These  are  shown  in  the  figure  and  we  find  that  we  can 
draw  them  without  confusion,  forming  the  second  hexagon 
from  the  front.  Going  on  in  this  way  it  will  be  seen 
that  in  each  of  the  five  figures  a  set  of  hexagons  is  picked 
out,  which  put  together  form  a  three-space  figure  some- 
thing like  the 


A  FOUR-DIMENSIONAL   FIGURE  135 

These  separate  figures  are  the  successive  stages  in 
which  the  whole  four-dimensional  figure  in  which  they 
cohere  can  be  apprehended. 

The  first  figure  and  the  last  are  tetrakaidecagons. 
These  are  two  of  the  solid  boundaries  of  the  figure.  The 
other  solid  boundaries  can  be  traced  easily.  Some  of 
them  are  complete  from  one  face  in  the  figure  to  the 
corresponding  face  in  the  next,  as  for  instance  the  solid 
which  extends  from  the  hexagonal  base  of  the  first  figure 
to  the  equal  hexagonal  base  of  the  second  figure.  This 
kind  of  boundary  is  a  hexagonal  prism.  The  hexagonal 
prism  also  occurs  in  another  sectional  series,  as  for 
instance,  in  the  square  at  the  bottom  of  the  first  figure, 
the  oblong  at  the  base  of  the  second  and  the  square  at 
the  basii  of  the  third  figure. 

Other  solid  boundaries  can  be  traced  through  four  of 
the  five  sectional  figures.  Thus  taking  the  hexagon  at 
the  top  of  the  first  figure  we  find  in  the  next  a  hexagon 
also,  of  which  some  alternate  sides  are  elongated.  The 
top  of  the  third  figure  is  also  a  hexagon  with  the  other 
set  of  alternate  rules  elongated,  and  finally  we  come  in 
the  fourth  figure  to  a  regular  hexagon. 

These  four  sections  are  the  sections  of  a  tetrakaidecagon 
as  can  be  recognised  from  the  sections  of  this  figure 
which  we  have  had  previously.  Hence  the  boundaries 
are  of  two  kinds,  hexagonal  prisms  and  tetrakaidecagons. 

These  four-dimensional  figures  exactly  fill  four-dimen- 
sional space  by  equal  repetitions  of  themselves. 


CHAPTER   XI 


NOMENCLATURE    AND    ANALOGIES    PRELIM- 

INARY    TO    THE     STUDY      OF      FOUR  DIMEN- 

SIONAL FIGURES 


IN  the  following  pages  a  method  of  designating  different 
regions  of  space  by  a  systematic  colour  scheme  has  been 
adopted.  The  explanations  have  been  given  in  such  a 
manner  as  to  involve  no  reference  to  models,  the  diagrams 
will  be  found  sufficient.  But  to  facilitate  the  study  a 
description  of  a  set  of  models  is  given  in  an  appendix 
which  the  reader  can  either  make  for  himself  or  obtain. 
If  models  are  used  the  diagrams  in  Chapters  XI.  and  XII. 
will  form  a  guide  sufficient  to  indicate  their  use.  Cubes 
of  the  colours  designated  by  the  diagrams  should  be  picked 
out  and  used  to  reinforce  the  diagrams.  The  reader, 
in  the  following  description,  should 
suppose  that  a  board  or  wall 
stretches  away  from  him,  against 
which  the  figures  are  placed. 

Take  a  square,  one  of  those 
shown  in  Fig.  77  and  give  it  a 
neutral  colour,  let  this  colour  be 
called  "null,"  and  be  such  that  it 
makes  no  appreciable  difference 


c 


130 


NOMENCLATURE   AND   ANALOGIES 


to  any  colour  with  which  it  mixed.  If  there  is  no 
such  real  colour  let  us  imagine  such  a  colour,  and 
assign  to  it  the  properties  of  the  number  zero,  which 
makes  no  difference  in  any  number  to  which  it  is 
added. 

Above  this  square  place  a  red  square.  Thus  we  symbolise 
the  going  up  by  adding  red  to  null. 

Away  from  this  null  square  place  a  yellow  square,  and 
represent  going  away  by  adding  yellow  to  null. 

To  complete  the  figure  we  need  a  fourth  square. 
Colour  this  orange,  which  is  a  mixture  of  red  and 
yellow,  and  so  appropriately  represents  a  going  in  a 
direction  compounded  of  up  and  away.  We  have  thus 
a  colour  scheme  which  will  serve  to  name  the  set  of 
squares  drawn.  We  have  two  axes  of  colours — red  and 

yellow — and  they  may  oc- 
cupy as  in  the  figure  the 
direction  up  and  away,  or 
they  may  be  turned  about ; 
in  any  case  they  enable  us 
to  name  the  four  squares 
drawn  in  their  relation  to 
one  another. 

Now  take,  in  Fig.  78, 
nine  squares,  and  suppose 
that  at  the  end  of  the 
going  in  any  direction  the 


Fig.  78. 


colour  started  with  repeats  itself. 

We  obtain  a  square  named  as  shown. 

Let  us  now,  in  fig.  79,  suppose  the  number  of  squares  to 
be  increased,  keeping  still  to  the  principle  of  colouring 
already  used. 

Here  the  nulls  remain  four  in  number.  There 
are  three  reds  between  the  first  null  and  the  null 
fvboye  it,  three  yellows  between  the  first  null  apd  the 


138 


THE   FOURTH   DIMENSION 


null  beyond  it,  while  the  oranges  increase  in  a  doublo 
way. 


Red 


Null 


Orange 


""Yellow 
Fig.  80. 


Red 
Null 


Fig.  79. 

Suppose  this  process  of  enlarging  the  number  of  the 
Null  Yellow  Null  squares  to  be  indefinitely  pursued  and 
the  total  figure  obtained  to  be  reduced 
in  size,  we  should  obtain  a  square  of 
which  the  interior  was  all  orange, 
while  the  lines  round  it  were  red  and 
yellow,  and  merely  the  points  null 
colour,  as  in  fig.  80.  Thus  all  the  points,  lines,  and  the 
area  would  have  a  colour. 

We  can  consider  this  scheme  to  originate  thus : — Let 
a  null  point  move  in  a  yellow  direction  and  trace  out  a 
yellow  line  and  end  in  a  null  point.  Then  let  the  whole 
line  thus  traced  move  in  a  red  direction.  The  null  points 
at  the  ends  of  the  line  will  produce  red  lines,  and  end  in 


NOMENCLATURE  AND  ANALOGIES 


139 


null  points.     The  yellow  line  will  trace  out  a  yellow  and 
red,  or  orange  square. 

Now,  turning  back  to  fig.  78,  we  see  that  these  two 
ways  of  naming,  the  one  we  started  with  and  the  one  we 
arrived  at,  can  be  combined. 

By  its  position  in  the  group  of  four  squares,  in  fig.  77, 
the  null  square  has  a  relation  to  the  yellow  and  to  the  red 
directions.  We  can  speak  therefore  of  the  red  line  of  the 
null  square  without  confusion,  meaning  thereby  the  line 
AB,  fig.  81,  which  runs  up  from  the 
initial  null  point  A  in  the  figure  as 
drawn.  The  yellow  line  of  the  null 
square  is  its  lower  horizontal  line  AC 
as  it  is  situated  in  the  figure. 

If  we  wish  to  denote  the  upper 
yellow  line  BD,  fig.  81,  we  can  speak 
of  it  as  the  yellow  r  line,  meaning 
Flg*  8L  the  yellow  line  which  is  separated 

from  the  primary  yellow  line  by  the  red  movement. 

In  a  similar  way  each  of  the  other  squares  has  null 
points,  red  and  yellow  lines.  Although  the  yellow  square 
is  all  yellow,  its  line  CD,  for  instance,  can  be  referred  to  as 
its  red  line. 

This  nomenclature  can  be  extended. 
If  the  eight  cubes  drawn,  in  fig.  82,  are  put  close 
together,  as  on  the  right  hand  of  the  diagram,  they  form 
a  cube,  and  in  them,  as  thus  arranged,  a  going  up  is 
represented  by  adding  red  to  the  zero,  or  null  colour,  a 
going  away  by  adding  yellow,  a  going  to  the  right  by 
adding  white.  White  is  used  as  a  colour,  as  a  pigment, 
which  produces  a  colour  change  in  the  pigments  with  which 
it  is  mixed.  From  whatever  cube  of  the  lower  set  we 
start,  a  motion  up  brings  us  to  a  cube  showing  a  change 
to  red,  thus  light  yellow  becomes  light  yellow  red,  or 
light  orange,  which  is  called  ochre,  And  going  tq  the 


140 


THE   FOURTH   DIMENSION 


right  from  the  null  on  the  left  we  have  a  change  involving 
the  introduction  of  white,  while  the  yellow  change  runs 
from  front  to  back.  There  are  three  colour  axes — the  red, 


yellow 


Fig.  82. 


the  white,  the  yellow — and  these  run  in  the  position  the 
cubes  occupy  in  the  drawing — up,  to  the  right,  away — but 
they  could  be  turned  about  to  occupy  any  positions  in  space. 


/  Null  /White/  Null  / 

A',  ll.m/1'1^'  /Yellow/ 
/              /  yellow  /              / 

/      /'    , 

/ 

/ 
/ 

( 

/ 

/ 

/ 

/T3 
U 
M 

%  / 

£  / 
^3 

iz, 
*/ 

<u  f 

f 

^ 

\ 

•  •& 

4? 

* 

<? 

S 

^ 

/ 

X 

^ 

^ 

/  Null  /White/  Null 

7        /LiVht   /V  n 
ellow/    .°     /Yellow 
_  /yellow/ 

Null    /White/  Null 

/  Red    /Pink    / 
/  -  -/——  —  + 
/Orange  /Ochre  /Orange 

/Red    /Pink        Red 

Null 


/  Null    / 


White 


Null       White      Null 


Third 

,QT7 

layer. 


Second 
layer. 


Fig.  88. 


We   can   conveniently   represent  a  block  of  cubes   by 
three  sets  of  squares,  representing  each  the  base  of  a  cube, 
the  block,  fig.  83,  can   be   represented   by  the 


NOMENCLATURE   AND   ANALOGIES  141 

layers  on  the  right.  Here,  as  in  the  case  of  the  plane, 
the  initial  colours  repeat  themselves  at  the  end  of  the 
series. 

Proceeding  now  to  increase  the  number  of  the  cubes 

.  -  7  -  7  -  7  -  7  -  7  we     obtain     fig.     84, 
/n     /wh/wh/wh/   n    / 

5y'  - 


v  Otters  of  the  colours 

/  y-  /'.y-/'-y:/».y:/  y-  /  , 

/  y.  /'•y./'-y.A-y./y.  /  are  glven   mstead  of 

/  n./  wh./  wh,./  wh./  n.  /  their  full  names. 

/  -  -7  -  T  -  -7  -  -7  -  -7  Here    we    see    that 

/  -  7^  -  •/•  —  —f-  —  —/-  -  /  there    are    four    null 
4,  /  or.  /  oc.  /  oc.  /  oc.  /  or.  / 

/or./oc./oc./oc./or./  cubes    as    betore'    but 

/  or.  /  oc.  /  oc.  /  oc  .  /or/  the  senes  whlch  SP"ng 

/  r.  /  p.  /  p.  /  p.  /  r.  /  from  tbe  initial  corner 

i  -  7  -  7  -  7  -  7  -  7  will   tend    to   become 

/  T-    /     P-    /    P-     /    P-    /     '•    /  ,.                   ,           , 

3  /on  /oc.  /oc  /oc.  /or.  /  lines  of  CubeS'  as  al^° 

/  or./oc./oc./oc    /or./  the    Sets    °f     Cubes 

/  or./  oc.  /  oc.  /oc.  /  or.  /  parallel  to  them,  start- 

/  *•/  p-  /  P-  /  P-  /  r-  /  inS  from  otner  corners. 

ri          p,          p,         p          rv  ThuS'  from  the  initial 

a  line    of 


g/or./oc./  oc.oc./  or. 

/  or.  /  oc  .  /  oc.  /  oc.  /  or"/          red    cubes»    a    line    of 
/  or.  /  oc.  /  oc.  /  oc.  /  or>  /          white  cubes,  and  a  line 
/  r.  /  p.  /  p.  /  p./    n  /  of  yellow  cubes. 

/  n    /wh./  wh./whi./n.  7       If  the  number  of  the 
1  /  y.  /I.  y  /  I.  y./l.'y./  y.  /      cubes    is    ^rgely   in- 
/  y>  /  ]-  y-/l-  y*>/  1.  y./  y-  /       creased,  and  the  size 
/  y.  /l-'y./i-  y./l.  y./  y.  /          of  the  whole  cube  is 
/  TI.  /  wh./  wh  ./  wh../  n.  /  diminished,     we     get 

p.     84  a     cube     with      null 

points,  and  the  edges 
coloured    with    these    three    colours. 

The  light  yellow  cubes  increase  in  two  ways,  forming 
ultimately  a  sheet  of  cubes,  and  the  same  is  true  of 
the  orange  and  pink  sets.  Hence,  ultimately  the  cube 


142 


THE  FOURTH  DIMENSION 


Null 


Null 


Null 


thus  formed  would  have  red,  white,  and  yellow  lines 
surrounding  pink,  orange,  and  light  yellow  faces.  The 
ochre  cubes  increase  in  three  ways,  and  hence  ulti- 
mately the  whole  interior  of  the  cube  would  be  coloured 
ochre. 

We  have  thus  a  nomenclature  for  the  points,  lines, 
faces,  and  solid  content  of  a  cube,  and  it  can  be  named 
as  exhibited  in  fig.  85. 

We  can  consider  the  cube  to  be  produced  in  the 
following  way.  A  null  point 
moves  in  a  direction  to  which 
we  attach  the  colour  indication 
yellow ;  it  generates  a  yellow  line 
and  ends  in  a  null  point.  The 
yellow  line  thus  generated  moves 
in  a  direction  to  which  we  give 
the  colour  indication  red.  This 
lies  up  in  the  figure.  The  yellow 
line  traces  out  a  yellow,  red,  or 
orange  square,  and  each  of  its  null  points  trace  out  a 
red  line,  and  ends  in  a  null  point. 

This  orange  square  moves  in  a  direction  to  which 
we  attribute  the  colour  indication  white,  in  this  case 
the  direction  is  the  right.  The  square  traces  out  a 
cube  coloured  orange,  red,  or  ochre,  the  red  lines  trace 
out  red  to  white  or  pink  squares,  and  the  yellow 
lines  trace  out  light  yellow  squares,  each  line  ending 
in  a  line  of  its  own  colour.  While  the  points  each 
trace  out  a  null  +  white,  or  white  line  to  end  in.  a  null 
point. 

Now  returning  to  the  first  block  of  eight  cubes  we  can 
name  each  point,  line,  and  square  in  them  by  reference  to 
the  colour  scheme,  which  they  determine  by  their  relation 
to  each  other. 

Thus,  in  fig.  86,  the  null  cube  touches  the  red  cube  by 


Fig.  85. 


NOMENCLATURE   AND   ANALOGIES 


143 


a  light  yellow  square;  it  touches  the  yellow  cube  by  a 


pink  square,  and  touches 
the  white  cube  by  an 
orange  square. 

There  are  three  axes 
to  which  the  colours  red, 
yellow,  and  white,  are 
assigned,  the  faces  of 
each  cube  are  designated 

Taking  all  the  colours 


Fig.  86 

by  taking  these  colours  in  pairs, 
together  we  get  a  colour  name  for  the  solidity  of  a  cube. 
Let  us  now  ask  ourselves  how  the  cube  could  be  pre- 
sented to  the  plane  being.  Without  going  into  the  question 
of  how  he  could  have  a  real  experience  of  it,  let  us  see 
how,  if  we  could  turn  it  about  and  show  it  to  him,  he, 
under  his  limitations,  could  get  information  about  it. 
If  the  cube  were  placed  with  its  red  and  yellow  axes 
against  a  plane,  that  is  resting  against  it  by  its  orange 


White 


Null  White  Null  wH. 


kce  previously  perceived 


Fig.  87. 

face,  the  plane  being  would  observe  a  square  surrounded 
by  red  and  yellow  lines,  and  having  null  points.  See  the 
dotted  square,  fig.  87. 

We  could  turn  the  cube  about  the  red  line  so  that 
a  different  face  comes  into  juxtaposition  with  the  plane. 

Suppose  the  cube  turned  about  the  red  line.     As  it 


144 


THE 


DIMENSION 


is  turning  from  its  first  position  all  of  it  except  the  red 
line  leaves  the  plane — goes  absolutely  out  of  the  range 
of  the  plane  being's  apprehension.  But  when  the  yellow 
line  points  straight  out  from  the  plane  then  the  pink 
face  comes  into  contact  with  it.  Thus  the  same  red  line 
remaining  as  he  saw  it  at  first,  now  towards  him  comes 
a  face  surrounded  by  white  and  red  lines. 

If  we  call  the  direction  to  the  right  the  unknown 
direction,  then  the  line  he  saw  before,  the  yellow  line, 
goes  out  into  this  unknown  direction,  and  the  line  which 
before  went  into  the  unknown  direction,  comes  in.  It 
comes  in  in  the  opposite  direction  to  that  in  which  the 
yellow  line  ran  before ;  the  interior  of  the  face  now 

against  the  plane  is  pink.  It  is 
a  property  of  two  lines  at  right 
angles  that,  if  one  turns  out  of 
a  given  direction  and  stands  at 
right  angles  to  it,  then  the  other 
'B  of  the  two  lines  comes  in,  but 

runs  the  opposite  way  in  that 
given  direction,  as  in  fig.  88. 
Now  these  two  presentations  of  the  cube  would  seem, 
to    the    plane   creature   like  perfectly  different  material 
bodies,   with   only  that  line   in   common  in  which  they 
both  meet. 

Again  our  cube  can  be  turned  about  the  yellow  line. 
In  this  case  the  yellow  square  would  disappear  as  before, 
but  a  new  square  would  come  into  the  plane  after  the 
cube  had  rotated  by  an  angle  of  90°  about  this  line. 
The  bottom  square  of  the  cube  would  come  in  thus 
in  figure  89.  The  cube  supposed  in  contact  with  the 
plane  is  rotated  about  the  lower  yellow  line  and  then 
the  bottom  face  is  in  contact  with  the  plane. 

Here,  as  before,  the  red  line  going  out  into  the  un- 
known dimension,  the  white  line  which  before  ran  in  the 


Fig.  83. 


NOMENCLATURE  ANt>  ANALOGIES 


145 


unknown   dimension   would    come  in   downwards  in  the 
opposite  sense  to  that  in  which  the  red  line  ran  before. 

Now  if  we  use  i,  j,  k,  for  the  three  space  directions, 
i  left  to  right,  j  from  near  away,  k  from  below  up  ;  then, 
using  the  colour  names  for  the  axes,  we  have  that  first 
of  all  white  runs  i,  yellow  runs  j,  red  runs  k ;  then  after 


Null-y  *  Null   Wfcite  NuIJ 
^Yellow 


Fig.  89. 

the  first  turning  round  the  k  axis,  white  runs  negative  j, 
yellow  runs  i,  red  runs  k ;  thus  we  have  the  table  : — 


i 

j 

li 

1st  position 

white 

yellow 

red 

2nd  position 

yellow 

white  — 

red 

3rd  position 

red 

yellow 

white  — 

Here  white  with  a  negative  sign  after  it  in  the  column 
under  j  means  that  white  runs  in  the  negative  sense  of 
the  j  direction. 

We  may  express  the  fact  in  the  following  way : — 
In  the  plane  there  is  room  for  two  axes  while  the  body 
has  three.  Therefore  in  the  plane  we  can  represent  any 
two.  If  we  want  to  keep  the  axis  that  goes  in  the 
unknown  dimension  always  running  in  the  positive  sense, 
then  the  axis  which  originally  ran  in  the  unknown 

10 


146  THE   FOURTH   DIMENSION 

dimension  (the  white  axis)  must  come  in  in  the  negative 
sense  of  that  axis  which  goes  out  of  the  plane  into  the 
unknown  dimension. 

It  is  obvious  that  the  unknown  direction,  the  direction 
in  which  the  white  line  runs  at  first,  is  quite  distinct  from 
any  direction  which  the  plane  creature  knows.  The  white 
line  may  come  in  towards  him,  or  running  down.  If  he 
is  looking  at  a  square,  which  is  the  face  of  a  cube 
(looking  at  it  by  a  line),  then  any  one  of  the  bounding  lines 
remaining  unmoved,  another  face  of  the  cube  may  come 
in.  any  one  of  the  faces,  namely,  which  have  the  white  line 
in  them.  And  the  white  line  comes  sometimes  in  one 
of  the  space  directions  he  knows,  sometimes  in  another. 

Now  this  turning  which  leaves  a  line  unchanged  is 
something  quite  unlike  any  turning  he  knows  in  the 
plane.  In  the  plane  a  figure  turns  round  a  point.  The 
square  can  turn  round  the  null  point  in  his  plane,  and 
the  red  and  yellow  lines  change  places,  only  of  course,  as 
with  every  rotation  of  lines  at  right  angles,  if  red  goes 
where  yellow  went,  yellow  comes  in  negative  of  red's  old 
direction. 

This  turning,  as  the  plane  creature  conceives  it,  we 
should  call  turning  about  an  axis  perpendicular  to  the 
plane.  What  he  calls  turning  about  the  null  point  we 
call  turning  about  the  white  line  as  it  stands  out  from 
his  plane.  There  is  no  such  thing  as  turning  about  a 
point,  there  is  always  an  axis,  and  really  much  more  turns 
than  the  plane  being  is  aware  of. 

Taking  now  a  different  point  of  view,  let  us  suppose  the 
cubes  to  be  presented  to  the  plane  being  by  being  passed 
transverse  to  his  plane.  Let  us  suppose  the  sheet  of 
matter  over  which  the  plane  being  and  all  objects  in  his, 
world  slide,  to  be  of  such  a  nature  that  objects  can  pass 
through  it  without  breaking  it.  Let  us  suppose  it  to  be 
of  the  same  nature  as  the  film  of  a  soap  bubble,  so  that 


NOMENCLATIVE  AND   ANALOGIES 


147 


Null 


it  closes  around  objects  pushed  through  it,  and,  however 
the  object  alters  its  shape  as  it  passes  through  it,  let  us 
suppose  this  film  to  run  up  to  the  contour  of  the  object 
in  every  part,  maintaining  its  plane  surface  unbroken. 

Then  we  can  push  a  cube  or  any  object  through  the 
film  and  the  plane  being  who  slips  about  in  the  film 
will  know  the  contour  of  the  cube  just  and  exactly  where 
the  film  meets  it. 

Fig.  90  represents  a  cube  passing  through  a  plane  film. 
The  plane  being  now  comes  into 
contact  with  a  very  thin  slice 
of  the  cube  somewhere  between 
the  left  and  right  hand  faces. 
This  very  thin  slice  he  thinks 
of  as  having  no  thickness,  and 
consequently  his  idea  of  it  is 
what  we  call  a  section.  It  is 
bounded  by  him  by  pink  lines 
front  and  back,  coming  from 
the  part  of  the  pink  face  he  is 
in  contact  with,  and  above  and  below,  by  light  yellow 
lines.  Its  corners  are  not  null-coloured  points,  but  white 
points,  and  its  interior  is  ochre,  the  colour  of  the  interior 
of  the  cube,  tf 

If  now  we  suppose  the  cube  to  be  an  inch  in  each 
dimension,  and  to  pass  across,  from  right  to  left,  through 
the  plane,  then  we  should  explain  the  appearances  pre- 
sented to  the  plane  being  by  saying :  First  of  all  you 
have  the  face  of  a  cube,  this  lasts  only  a  moment ;  then 
you  have  a  figure  of  the  same  shape  but  differently 
coloured.  This,  which  appears  not  to  move  to  you  in  any 
direction  which  you  know  of,  is  really  moving  transverse 
to  your  plane  world.  Its  appearance  is  unaltered,  but 
each  moment  it  is  something  different — a  section  further 
on,  in  the  white,  the  unknown  dimension.  Finally,  at  the 


Fig.  90. 


148  THE   FOUfcTH   DIMENSION 

end  of  the  minute,  a  face  comes  in  exactly  like  the  face 
you  first  saw.  This  finishes  up  the  cube — it  is  the  further 
face  in  the  unknown  dimension. 

The  white  line,  which  extends  in  length  just  like  the 
red  or  the  yellow,  you  do  not  see  as  extensive ;  you  appre- 
hend it  simply  as  an  enduring  white  point.  The  null 
point,  under  the  condition  of  movement  of  the  cube, 
vanishes  in  a  moment,  the  lasting  white  point  is  really 
your  apprehension  of  a  white  line,  running  in  the  unknown 
dimension.  In  the  same  way  the  red  line  of  the  face  by 
which  the  cube  is  first  in  contact  with  the  plane  lasts  only 
a  moment,  it  is  succeeded  by  the  pink  line,  and  this  pink 
line  lasts  for  the  inside  of  a  minute.  This  lasting  pink 
line  in  your  apprehension  of  a  surface,  which  extends  in 
two  dimensions  just  like  the  orange  surface  extends,  as  you 
know  it,  when  the  cube  is  at  rest. 

But  the  plane  creature  might  answer,  "  This  orange 
object  is  substance,  solid  substance,  bounded  completely 
and  on  every  side." 

Here,  of  course,  the  difficulty  comes  in.  His  solid  is  our 
surface — his  notion  of  a  solid  is  our  notion  of  an  abstract 
surface  with  no  thickness  at  all. 

We  should  have  to  explain  to  him  that,  from  every  point 
of  what  he  called  a  solid,  a  new  dimension  runs  away. 
From  every  point  a  line  can  be  drawn  in  a  direction 
unknown  to  him,  and  there  is  a  solidity  of  a  kind  greater 
than  that  which  he  knows.  This  solidity  can  only  be 
realised  by  him  by  his  supposing  an  unknown  direction, 
by  motion  in  which  what  he  conceives  to  be  solid  matter 
instantly  disappears.  The  higher  solid,  however,  which 
extends  in  this  dimension  as  well  as  in  those  which  he 
knows,  lasts  when  a  motion  of  that  kind  takes  place, 
different  sections  of  it  come  consecutively  in  the  plane  of 
his  apprehension,  and  take  the  place  of  the  solid  which  he 
at  first  conceives  to  be  all.  Thus,  the  higher  solid — our 


NOMENCLATURE   AND   ANALOGIES  149 

solid  in  contradistinction  to  his  area  solid,  his  two- 
dimensional  solid,  must  be  conceived  by  him  as  something 
which  has  duration  in  it,  under  circumstances  in  which  his 
matter  disappears  out  of  his  world. 

We  may  put  the  matter  thus,  using  the  conception  of 
motion. 

A  null  point  moving  in  a  direction  away  generates  a 
yellow  line,  and  the  yellow  line  ends  in  a  null  point.  We 
suppose,  that  is,  a  point  to  move  and  mark  out  the 
products  of  this  motion  in  such  a  manner.  Now 
suppose  this  whole  line  as  thus  produced  to  move  in 
an  upward  direction;  it  traces  out  the  two-dimensional 
solid,  and  the  plane  being  gets  an  orange  square.  The 
null  point  moves  in  a  red  line  and  ends  in  a  null  point, 
the  yellow  line  moves  and  generates  an  orange  square  and 
ends  in  a  yellow  line,  the  farther  null  point  generates 
a  red  line  and  ends  in  a  null  point.  Thus,  by  move- 
ment in  two  successive  directions  known  to  him,  he 
can  imagine  his  two-dimensional  solid  produced  with  all 
its  boundaries. 

Now  we  tell  him  :  "  This  whole  two-dimensional  solid 
can  move  in  a  third  or  unknown  dimension  to  you.  The 
null  point  moving  in  this  dimension  out  of  your  world 
generates  a  white  line  and  ends  in  a  null  point.  The 
yellow  line  moving  generates  a  light  yellow  two- 
dimensional  solid  and  ends  in  a  yellow  line,  and  thus 
two-dimensional  solid,  lying  end  on  to  your  plane  world,  is 
bounded  on  the  far  side  by  the  other  yellow  line.  In 
the  same  way  each  of  the  lines  surrounding  your  square 
traces  out  an  area,  just  like  the  orange  area  you  know. 
But  there  is  something  new  produced,  something  which 
you  had  no  idea  of  before  ;  it  is  that  which  is  produced  by 
the  movement  of  the  orange  square.  That,  than  which 
you  can  imagine  nothing  more  solid,  itself  moves  in  a 
direction  open  to  it  and  produces  a  three-dimensional 


150  THE   FOURTH   DIMENSION 

solid.  Using  the  addition  of  white  to  symbolise  the 
products  of  this  motion  this  new  kind  of  solid  will  be  light 
orange  or  ochre,  and  it  will  be  bounded  on  the  far  side  by 
the  final  position  of  the  orange  square  which  traced  it 
out,  and  this  final  position  we  suppose  to  be  coloured  like 
the  square  in  its  first  position,  orange  with  yellow  and 
red  boundaries  and  null  corners." 

This  product  of  movement,  which  it  is  so  easy  for  us  to 
describe,  would  be  difficult  for  him  to  conceive.  But  this 
difficulty  is  connected  rather  with  its  totality  than  with 
any  particular  part  of  it. 

Any  line,  or  plane  of  this,  to  him  higher,  solid  we  could 
show  to  him,  and  put  in  his  sensible  world. 

We  have  already  seen  how  the  pink  square  could  be  put 
in  his  world  by  a  turning  of  the  cube  about  the  red  line. 
And  any  section  which  we  can  conceive  made  of  the  cube 
could  be  exhibited  to  him.  You  have  simply  to  turn  the 
cube  and  push  it  through,  so  that  the  plane  of  his  existence 
is  the  plane  which  cuts  out  the  given  section  of  the  cube, 
then  the  section  would  appear  to  him  as  a  solid.  In  his 
world  he  would  see  the  contour,  get  to  any  part  of  it  by 
digging  down  into  it. 


THE  PROCESS  BY  WHICH  A  PLANE  BEING  WOULD  GAIN 
A  NOTION  OF  A  SOLID. 

If  we  suppose  the  plane  being  to  have  a  general  idea  of 
the  existence  of  a  higher  solid — our  solid — we  must  next 
trace  out  in  detail  the  method,  the  discipline,  by  which 
he  would  acquire  a  working  familiarity  with  our  space 
existence.  The  process  begins  with  an  adequate  realisa- 
tion of  a  simple  solid  figure.  For  this  purpose  we  will 
suppose  eight  cubes  forming  a  larger  cube,  and  first  we 
will  suppose  each  cube  to  be  coloured  throughout  uniformly. 


NOMENCLATURE    AND    ANALOGIES 


151 


Let  the  cubes  in  fig.  91  be  the  eight  making  a  larger 
cube. 

Now,  although  each  cube  is  supposed  to  be  coloured 
entirely  through  with  the  colour,  the  name  of  which  is 
written  on  it,  still  we  can  speak  of  the  faces,  edges,  and 
corners  of  each  cube  as  if  the  colour  scheme  we  have 
investigated  held  for  it.  Thus,  on  the  null  cube  we  can 
speak  of  a  null  point,  a  red  line,  a  white  line,  a  pink  face,  and 
so  on.  These  colour  designations  are  shown  on  No.  1  of 
the  views  of  the  tesseract  in  the  plate.  Here  these  colour 


Fig.  91. 

names  are  used  simply  in  their  geometrical  significance. 
They  denote  what  the  particular  line,  etc.,  referred  to  would 
have  as  its  colour,  if  in  reference  to  the  particular  cube 
the  colour  scheme  described  previously  were  carried  out. 

If  such  a  block  of  cubes  were  put  against  the  plane  and 
then  passed  through  it  from  right  to  left,  at  the  rate  of  an 
inch  a  minute,  each  cube  being  an  inch  each  way,  the 
plane  being  would  have  the  following  appearances  : — 

First  of  all,  four  squares  null,  yellow,  red,  orange,  lasting 
each  a  minute;  and  secondly,  taking  the  exact  places 
of  these  four  squares,  four  others,  coloured  white,  light 
yellow,  pink,  ochre.  Thus,  to  make  a  catalogue  of  the 
solid  body,  he  would  have  to  put  side  by  side  in  his  world 
two  sets  of  four  squares  eacli,  as  in  fig.  92.  The  first 


152 


THE   FOURTH   DIMENSION 


are  supposed  to  last  a  minute,  and  then  the  others  to 

eome  in  in  place  of  them, 
and  also  last  a  minute. 

In  speaking  of  them 
he  would  have  to  denote 
what  part  of  the  respective 
cube  each  square  repre- 
sents. Thus,  at  the  begin- 
ning he  would  have  null 
cube  orange  face,  and  after 
the  motion  had  begun  he 
would  have  null  cube  ochre 
section.  As  he  could  get 
the  same  coloured  section  whichever  way  the  cube  passed 
through,  it  would  be  best  for  him  to  call  this  section  white 
section,  meaning  that  it  is  transverse  to  the  white  axis. 
These  colour-names,  of  course,  are  merely  used  as  names, 
and  do  not  imply  in  this  case  that  the  object  is  really 
coloured.  Finally,  after  a  minute,  as  the  first  cube  was 
passing  beyond  his  plane  he  would  have  null  cube  orange 
face  again. 

The  same  names  will  hold  for  each  of  the  other  cubes, 
describing  what  face  or  section  of  them  the  plane  being 
has  before  him  ;  and  the  second  wall  of  cubes  will  come 
on,  continue,  and  go  out  in  the  same  manner.  In  the 
area  he  thus  has  he  can  represent  any  movement  which 
we  carry  out  in  the  cubes,  as  long  as  it  does  not  involve 
a  motion  in  the  direction  of  the  white  axis.  The  relation 
of  parts  that  succeed  one  another  in  the  direction  of  the 
white  axis  is  realised  by  him  as  a  consecution  of  states. 

Now,  his  means  of  developing  his  space  apprehension 
lies  in  this,  that  that  which  is  represented  as  a  time 
sequence  in  one  position  of  the  cubes,  can  become  a  real 
co-existence,  if  something  that  has  a  real  co-existence 
becomes  a  time  sequence. 


NOMENCLATURE  AND  ANALOGIES 


153 


We  must  suppose  the  cubes  turned  round  each  of  the 
Axes,  the  red  line,  and  the  yellow  line,  then  something, 
which  was  given  as  time  before,  will  now  be  given  as  the 
plane  creature's  space ;  something,  which  was  given  as  space 
before,  will  now  be  given  as  a  time  series  as  the  cube  is 
passed  through  the  plane. 

The  three  positions  in  which  the  cubes  must  be  studied 
are  the  one  given  above  and  the  two  following  ones.  In 
each  case  the  original  null  point  which  was  nearest  to  us 
at  first  is  marked  by  an  asterisk.  In  figs.  93  and  94  the 


och, 


I.  y. 


Fig.  93. 

The  cu>e  swung  round  the  red  line,  so  that  the  white  line  points 
towards  us. 

point  marked  with  a  star  is  the  same  in  the  cubes  and  in 
the  plane  view. 

In  fig.  93  the  cube  is  swung  round  the  red  line  so  as  to 
point  towards  us,  and  consequently  the  pink  face  comes 
next  to  the  plane.  As  it  passes  through  there  are  two 
varieties  of  appearance  designated  by  the  figures  1  and  2 
in  the  plane.  These  appearances  are  named  in  the  figure, 
and  are  determined  by  the  order  in  which  the  cubes 


-154 


THE   FOURTH   DIMENSION 


come  in  the  motion  of  the  whole  block  through  the 
plane. 

With  regard  to  these  squares  severally,  however, 
different  names  must  be  used,  determined  by  their 
relations  in  the  block. 

Thus,  in  fig.  93,  when  the  cube  first  rests  against  the 
plane  the  null  cube  is  in  contact  by  its  pink  face ;  as  the 
block  passes  through  we  get  an  ochre  section  of  the  null 
cube,  but  this  is  better  called  a  yellow  section,  as  it  is 
made  by  a  p'ane  perpendicular  to  the  yellow  line.  When 


A 


\ 


\ 


Fig.  94. 


The  cube  swung  round  yellow  line,  with  red  line  running  from  left 
to  right,  and  white  line  running  down. 

the  null  cube  has  passed  through  the  plane,  as  it  is 
leaving  it,  we  get  again  a  pink  face. 

The  same  series  of  changes  take  place  with  the  cube 
appearances  which  follow  on  those  of  the  null  cube.  In 
this  motion  the  yellow  cube  follows  on  the  null  cube,  and 
the  square  marked  yellow  in  2  in  the  plane  will  be  first 
"  yellow  pink  face,"  then  "  yellow  yellow  section,"  then 
"  yellow  pink  face." 

In  fig.  94,  in  which  the  cube  is  turned  about  the  yellow 
line,  we  have  a  certain  difficulty,  for  the  plane  being  will 


NOMENCLATURE   AND   ANALOGIES 


155 


find  that  the  position  his  squares  are  to  be  placed  in  will 
lie  below  that  which  they  first  occupied.  They  will  come 
where  the  support  was  on  which  he  stood  his  first  set  of 
squares.  He  will  get  over  this  difficulty  by  moving  his 
support. 

Then,  since  the  cubes  come  upon  his  plane  by  the  light 
yellow  face,  he  will  have,  taking  the  null  cube  as  before  for 
an  example,  null,  light  yellow  face;  null,  red  section, 
because  the  section  is  perpendicular  to  the  red  line  ;  and 
finally,  as  the  null  cube  leaves  the  plane,  null,  light  yellow 
face.  Then,  in  this  case  red  following  on  null,  he  will 


Mill 


Null 

r.  y.  wh    s 


X 


XI 


Null 
r.  y,  wh« 


X 


ite 


Null 


3        4 


have  the  same  series  of  views  of  the  red  as  he  had  of  the 
null  cube. 

There  is  another  set  of  considerations  which  we  will 
briefly  allude  to. 

Suppose  there  is  a  hollow  cube,  and  a  string  is  stretched 
across  it  from  null  to  null,  r,  y,  w/i,  as  we  may  call  the 
far  diagonal  point,  how  will  this  string  appear  to  the 
plane  being  as  the  cube  moves  transverse  to  his  plane  ? 

Let  us  represent  the  cube  as  a  number  of  sections,  say 
5,  corresponding  to  4  equal  divisions  made  along  the  white 
line  perpendicular  to  it. 

We  number  these  sections  0,  1,  2,  3,  4,  corresponding 
to  the  distances  along  the  white  line  at  which  they  are 


156  THE   FOURTH    DIMENSION 

taken,  and  imagine  each  section  to  come  in  successively, 
taking  the  place  of  the  preceding  one. 

These  sections  appear  to  the  plane  being,  counting  from 
the  first,  to  exactly  coincide  each  with  the  preceding  one. 
But  the  section  of  the  string  occupies  a  different  place  in 
each  to  that  which  it  does  in  the  preceding  section.  The 
section  of  the  string  appears  in  the  position  marked  by 
the  dots.  Hence  the  slant  of  the  string  appears  as  a 
motion  in  the  frame  work  marked  out  by  the  cube  sides. 
If  we  suppose  the  motion  of  the  cube  not  to  be  recognised, 
then  the  string  appears  to  the  plane  being  as  a  moving 
point.  Hence  extension  on  the  unknown  dimension 
appears  as  duration.  Extension  sloping  in  the  unknown 
direction  appears  as  continuous  movement. 


CHAFFER  XII 

THE     SIMPLEST    FOUR-DIMENSIONAL    SOLID 

A  PLANE  being,  in  learning  to  apprehend  solid  existence, 
must  first  of  all  realise  that  there  is  a  sense  of  direction 
altogether  wanting  to  him.  That  which  we  call  right 
and  left  does  not  exist  in  his  perception.  He  must 
assume  a  movement  in  a  direction,  and  a  distinction  of 
positive  and  negative  in  that  direction,  which  has  no 
reality  corresponding  to  it  in  the  movements  he  can. 
make.  This  direction,  this  new  dimension,  he  can  only 
make  sensible  to  himself  by  bringing  in  time,  and  sup- 
posing that  changes,  which  take  place  in  time,  are  due  to 
objects  of  a  definite  configuration  in  three  dimensions 
passing  transverse  to  his  plane,  and  the  different  sections 
of  it  being  apprehended  as  changes  of  one  and  the  same 
plane  figure. 

He  must  also  acquire  a  distinct  notion  about  his  plane 
world,  he  must  no  longer  believe  that  it  is  the  all  of 
space,  but  that  space  extends  on  both  sides  of  it.  In 
order,  then,  to  prevent  his  moving  off  in  this  unknown 
direction,  he  must  assume  a  sheet,  an  extended  solid  sheet, 
in  two  dimensions,  against  which,  in  contact  with  which, 
all  his  movements  take  place. 

When  we  come  to  think  of  a  four-dimensional  solid, 
what  are  the  corresponding  assumptions  which  we  must 
make  ? 

We  must  suppose  a  sense  which  we  have  not,  a  sense 

157 


158  tttE  FOURT& 


of  direction  wanting  in  us,  something  which  a  being  in 
a  four-dimensional  world  has,  and  which  we  have  not.  It 
is  a  sense  corresponding  to  a  new  space  direction,  a 
direction  which  extends  positively  and  negatively  from 
every  point  of  our  space,  and  which  goes  right  away  from 
any  space  direction  we  know  of.  The  perpendicular  to  a 
plane  is  perpendicular,  not  only  to  two  lines  in  it,  but  to 
every  line,  and  so  we  must  conceive  this  fourth  dimension 
as  running  perpendicularly  to  each  and  every  line  we  can 
draw  in  our  space. 

And  as  the  plane  being  had  to  suppose  something 
which  prevented  his  moving  off  in  the  third,  the 
unknown  dimension  to  him,  so  we  have  to  suppose 
something  which  prevents  us  moving  off  in  the  direction 
unknown  to  us.  This  something,  since  we  must  be  in 
contact  with  it  in  every  one  of  our  movements,  must  not 
be  a  plane  surface,  but  a  solid  ;  it  must  be  a  solid,  which 
in  every  one  of  our  movements  we  are  against,  not  in.  It 
must  be  supposed  as  stretching  out  in  every  space  dimension 
that  we  know  ;  but  we  are  not  in  it,  we  are  against  it,  we 
are  next  to  it,  in  the  fourth  dimension. 

That  is,  as  the  plane  being  conceives  himself  as  having 
a  very  small  thickness  in  the  third  dimension,  of  which 
he  is  not  aware  in  his  sense  experience,  so  we  must 
suppose  ourselves  as  having  a  very  small  thickness  in 
the  fourth  dimension,  and,  being  thus  four-dimensional 
beings,  to  be  prevented  from  realising  that  we  are 
such  beings  by  a  constraint  which  keeps  us  always  in 
contact  with  a  vast  solid  sheet,  which  stretches  on  in 
every  direction.  We  are  against  that  sheet,  so  that,  if  we 
had  the  power  of  four-dimensional  movement,  we  should 
either  go  away  from  it  or  through  it  ;  all  our  space 
movements  as  we  know  them  being  such  that,  performing 
them,  we  keep  in  contact  with  this  solid  sheet. 

Now  consider  the  exposition  a  plane  being  would  make 


fttE   SIMPLEST   FOUfe-fciMENSiOKAL   SOLID 


159 


for  himself  as  to  the  question  of  the  enclosure  of  a  square, 

and  of  a  cube. 

He  would  say  the  square  A,  in  Fig.  96,  is  completely 
enclosed  by  the  four  squares,  A  far, 
A  near,  A  above,  A  below,  or  as  they 
are  written  ATI,  A/,  Aa,  Ab. 

If  now  he  conceives  the  square  A 
to  move  in  the,  to  him,  unknown 
dimension  it  will  trace  out  a  cube, 
and  the  bounding  squares  will  form 
cubes.  Will  these  completely  sur- 
round the  cube  generated  by  A  ?  No  ; 
there  will  be  two  faces  of  the  cube 
made  by  A  left  uncovered  ;  the  first, 
that  face  which  coincides  with  the 


Fig.  96. 


square  A  in  its  first  position  ;  the  next,  that  which  coincides 
with  the  square  A  in  its  final  position.  Against  these 
two  faces  cubes  must  be  placed  in  order  to  completely 
enclose  the  cube  A.  These  may  be  called  the  cubes  left 
and  right  or  Al  and  AT.  Thus  each  of  the  enclosing 
squares  of  the  square  A  becomes  a  cube  and  two  more 
cubes  are  wanted  to  enclose  the  cube  formed  by  the 
movement  of  A  in  the  third  dimension. 

The  plane  being  could  not  see  the  square  A  with  the 
squares  An,  A/,  etc.,  placed  about  it, 
because  they  completely  hide  it  from 
view ;  and  so  we,  in  the  analogous 
case  in  our  three-dimensional  world, 
cannot  see  a  cube  A  surrounded  by 
six  other  cubes.  These  cubes  we 
will  call  A  near  ATI,  A  far  A/,  A  above 
Aa,  A  below  Ab,  A  left  Al,  A  right  Ar, 
shown  in  fig.  97.  If  now  the  cube  A 




Aa 

7fAf 

/       / 

Al 

/ 

/ 

Ar 

An 

D 

J 

Al 

Pig.  97. 


moves  in  the  fourth  dimension  right  out  of  space,  it  traces 
out  a  higher   cube — a  tesseract,   as   it   may  be   called. 


160  THE   FOURTH   DlMENSlOK 

Each  of  the  six  surrounding  cubes  carried  on  in  the  satne 
motion  will  make  a  tesseract  also,  and  these  will  be 
grouped  around  the  tesseract  formed  by  A.  But  will  they 
enclose  it  completely  ? 

All  the  cubes  An,  A/,  etc.,  lie  in  our  space.  But  there  is 
nothing  between  the  cube  A  and  that  solid  sheet  in  contact 
with  which  every  particle  of  matter  is.  When  the  cube  A 
moves  in  the  fourth  direction  it  starts  from  its  position, 
say  Ak,  and  ends  in  a  final  position  An  (using  the  words 
"  ana  "  and  "  kata  "  for  up  and  down  in  the  fourth  dimen- 
sion). Now  the  movement  in  this  fourth  dimension  is 
not  bounded  by  any  of  the  cubes  An,  A/,  nor  by  what 
they  form  when  thus  moved.  The  tesseract  which  A 
becomes  is  bounded  in  the  positive  and  negative  ways  in 
this  new  direction  by  the  first  position  of  A  and  the  last 
position  of  A.  Or,  if  we  ask  how  many  tesseracts  lie 
around  the  tesseract  which  A  forms,  there  are  eight,  of 
which  one  meets  it  by  the  cube  A,  and  another  meets  it 
by  a  cube  like  A  at  the  end  of  its  motion. 

We  come  here  to  a  very  curious  thing.  The  whole 
solid  cube  A  is  to  be  looked  on  merely  as  a  boundary  of 
the  tesseract. 

Yet  this  is  exactly  analogous  to  what  the  plane  being 
would  come  to  in  his  study  of  the  solid  world.  The 
square  A  (fig.  96),  which  the  plane  being  looks  on  as  a 
solid  existence  in  his  plane  world,  is  merely  the  boundary 
of  the  cube  which  he  supposes  generated  by  its  motion. 

The  fact  is  that  we  have  to  recognise  that,  if  there  is 
another  dimension  of  space,  our  present  idea  of  a  solid 
body,  as  one  which  has  three  dimensions  only,  does  not 
correspond  to  anything  real,  but  is  the  abstract  idea  of  a 
three-dimensional  boundary  limiting  a  four-dimensional 
solid,  which  a  four-dimensional  being  would  form.  The 
plane  being's  thought  of  a  square  is  not  the  thought 
of  what  we  should  call  a  possibly  existing  real  square, 


THE   SIMPLEST  FOUH-DlMENSIONAL   SOLID 


161 


but  the  thought  of  an  abstract  boundary ,  the   face   of 
a  cube. 

Let  us  now  take  our  eight  coloured  cubes,  which  form 
ft  cube  in  space,  and  ask  what  additions  we  must  make 
to  them  to  represent  the  simplest  collection  of  four-dimen- 
sional bodies — namely,  a  group  of  them  of  the  same  extent 
in  every  direction.  In  plane  space  we  have  four  squares. 
In  solid  space  we  have  eight  cubes.  So  we  should  expect 
in  four-dimensional  space  to  have  sixteen  four-dimen- 
sional bodies — bodies  which  in  four-dimensional  space 
correspond  to  cubes  in  three-dimensional  space,  and  these 
bodies  we  call  tesseracts. 

Given  then  the  null,  white,  red,  yellow  cubes,  and 
those  which  make  up  the  block,  we 
notice  that  we  represent  perfectly 
well  the  extension  in  three  directions 
(fig.  98).  From  the  null  point  of 
the  null  cube,  travelling  one  inch,  we 
come  to  the  white  cube ;  travelling 
one  inch  away  we  come  to  the  yellow 
cube  ;  travelling  one  inch  up  we  come 
to  the  red  cube.  Now,  if  there  is 
a  fourth  dimension,  then  travelling 
from  the  same  null  point  for  one 


Red 

Pink 

N 
\ 
\ 

u 

\ 

Null 

X 

White 

\   \ 

\fellow\     §, 
\  vellov 

( Orange  hidden) 
Fig.  98. 

inch  in  that  direction,  we  must  come  to  the  body  lying 
beyond  the  null  region. 

I  say  null  region,  not  cube ;  for  with  the  introduction 
of  the  fourth  dimension  each  of  our  cubes  must  become 
something  different  from  cubes.  If  they  are  to  have 
existence  in  the  fourth  dimension,  they  must  be  "  filled 
up  from  "  in  this  fourth  dimension. 

Now  we  will  assume  that  as  we  get  a  transference  from 
null  to  white  going  in  one  way,  from  null  to  yellow  going 
in  another,  so  going  from  null  in  the  fourth  direction  we 
have  a  transference  from  null  to  blue,  using  thus  the 

11 


162 


THE  FOURTH  DIMENSION 


colours  white,  yellow,  red,  blue,  to  denote  transferences  in 
each  of  the  four  directions — right,  away,  up,  unknown  or 
fourth  dimension. 

Hence,  as  the  plane  being  must  represent  the  solid  re- 
gions, he  would  come  to  by  going  right,  as  four  squares  lying 

in  some  position  in 
his  plane,  arbitrarily 
chosen,  side  by  side 
with  his  original  four 
squares,  so  we  must 
represent  those  eight 
four-dimensional  re- 
gions, which  we 

Fig.  99.  should    come    to   by 

A  plane  being's  representation  of  a  block   going    in    the   fourth 
of  eight  cubes  by  two  sets  of  four  squares.       dimension  from  each 

of  our  eight  cubes,  by  eight  cubes  placed  in  some  arbitrary 
position  relative  to  our  first  eight  cubes. 


Red 

Pink 

\ 
\ 
\ 

CA 
o 

* 

Null 

X 

White 

\           \ 

\Yellow\L'gh<v 

(1) 

Orange    hidden 


(2) 
Brown     hidden 


Fig.  100. 


Our  representation  of  a  block  of  sixteen  tesse  acts  by 
two  blocks  of  eight  cubes.* 

Hence,  of  the  two  sets  of  eight  cubes,  each  one  will  serve 

*  The  eight  cubes  used  here  in  2  can  be  found  in  the  second  of  the 
model  blocks.    They  can  be  taken  out  and  used. 


THE   SIMPLEST   FOUR-DIMENSIONAL   SOLID  163 

us  as  a  representation  of  one  of  the  sixteen  tesseracts 
which  form  one  single  block  in  four-dimensional  space. 
Each  cube,  as  we  have  it,  is  a  tray,  as  it  were,  against 
which  the  real  four-dimensional  figure  rests — just  as  each 
of  the  squares  which  the  plane  being  has  is  a  tray,  so  to 
speak,  against  which  the  cube  it  represents  could  rest. 

If  we  suppose  the  cubes  to  be  one  inch  each  way,  then 
the  original  eight  cubes  will  give  eight  tesseracts  of  the 
same  colours,  or  the  cubes,  extending  each  one  inch  in  the 
fourth  dimension. 

But  after  these  there  come,  going  on  in  the  fourth  di- 
mension, eight  other  bodies,  eight  other  tesseracts.  These 
must-  be  there,  if  we  suppose  the  four-dimensional  body 
we  make  up  to  have  two  divisions,  one  inch  each  in  each 
of  four  directions. 

The  colour  we  choose  to  designate  the  transference  to 
this  second  region  in  the  fourth' dimension  is  blue.  Thus, 
starting  from  the  null  cube  and  going  in  the  fourth 
dimension,  we  first  go  through  one  inch  of  the  null 
tesseract,  then  we  come  to  a  blue  cube,  which  is  the 
beginning  of  a  blue  tesseract.  This  blue  tesseract  stretches 
one  inch  farther  on  in  the  fourth  dimension. 

Thus,  beyond  each  of  the  eight  tesseracts,  which  are  of 
the  same  colour  as  the  cubes  which  are  their  bases,  lie 
eight  tesseracts  whose  colours  are  derived  from  the  colours 
of  the  first  eight  by  adding  blue.     Thus — 
Null  gives  blue 

Yellow  „      green 

Red  „      purple 

Orange  „      brown 

White  „     light  blue 

Pink  „      light  purple 

Light  yellow  ,,      light  green 
Ochre  „      light  brown 

The  addition  of  blue  to  yellow  gives  green — this  is  a 


164  THE  FOURTH  DIMENSION 

natural  supposition  to  make.  It  is  also  natural  to  siippOse 
that  blue  added  to  red  makes  purple.  Orange  and  blue 
can  be  made  to  give  a  brown,  by  using  certain  shades  and 
proportions.  And  ochre  and  blue  can  be  made  to  give  a 
light  brown. 

But  the  scheme  of  colours  is  merely  used  for  getting 
a  definite  and  realisable  set  of  names  and  distinctions 
visible  to  the  eye.  Their  naturalness  is  apparent  to  any 
one  in  the  habit  of  using  colours,  and  may  be  assumed  to 
be  justifiable,  as  the  sole  purpose  is  to  devise  a  set  of 
names  which  are  easy  to  remember,  and  which  will  give 
us  a  set  of  colours  by  which  diagrams  may  be  made  easy 
of  comprehension.  No  scientific  classification  of  colours 
has  been  attempted. 

Starting,  then,  with  these  sixteen  colour  names,  we  hav^e 
a  catalogue  of  the  sixteen  tesseracts,  which  form  a  four- 
dimensional  block  analogous  to  the  cubic  block.  But 
the  cube  which  we  can  put  in  space  and  look  at  is  not  one 
of  the  constituent  tesseracts  ;  it  is  merely  the  beginning, 
the  solid  face,  the  side,  the  aspect,  of  a  tesseract. 

We  will  now  proceed  to  derive  a  name  for  each  region, 
point,  edge,  plane  face,  solid  and  a  face  of  the  tesseract. 

The  system  will  be  clear,  if  we  look  at  a  representation 
in  the  plane  of  a  tesseract  with  three,  and  one  with  four 
divisions  in  its  side. 

The  tesseract  made  up  of  three  tesseracts  each  way 
corresponds  to  the  cube  made  up  of  three  cubes  each  way, 
and  will  give  us  a  complete  nomenclature. 

In  this  diagram,  fig.  101,  1  represents  a  cube  of  27 
cubes,  each  of  which  is  the  beginning  of  a  tesseract. 
These  cubes  are  represented  simply  by  their  lowest  squares, 
the  solid  content  must  be  understood.  2  represents  the 
27  cubes  which  are  the  beginnings  of  the  27  tesseracts 
one  inch  on  in  the  fourth  dimension.  These  tesseracts 
are  represented  as  a  block  of  cubes  put  side  by  side  with. 


THE   SIMPLEST   FOUR-DIMENSIONAL   SOLID 


165 


the  first  block,  but  in  their  proper  positions  they  could 
not  be  in  space  with  the  first  set.     3  represents  27  cubes 

Fig.  101. 


Null 

White 

Null 

bellow 

Light 
yellow 

Yellow 

Null 

White 

Null 

Bhie 

Light 
blue 

Blue 

Green 

Light 
green 

Green 

Blue 

Light 
blue 

Blue 

Null 

White 

Null 

Yellow 

Light 
yellow 

Yellow 

Null 

White 

Null 

Red 

Pink 

Red 

)range 

Ochre 

Grange 

Red 

Pink 

Red 

Purple 

Light 
purple 

Purple 

Brown 

Light 
brown 

Brown 

Purple 

Light 
purple 

Purple 

Red 

Pink 

Red 

)range 

Ochre  i 

)  range 

Red 

Pink 

Red 

Null 

White 

Null 

Yellow 

Light 
yellow 

Yellow 

Null 

I 

White 

Null 

Eao'.i  cube  is  the  begin- 
ning of  the  first  tesseract 
going  in  the   fourth  di- 
mension. 


Bach  cube  is  the  begin- 
ning of  the  second 
tessenict. 


Null 

White 

Null 

Yellow 

Light 
yellow 

Yellow 

Null 

White 

Null 

3 

Each   cube  is  the  l>e<:i  li- 
ning of  the  thhd 
tesseract. 


(forming  a  larger  cube)  which  are  the  beginnings  of  the 
tesseraets,  which  begin  two  inches  in  the  fourth  direction 
from  our  space  and  coi}tinu.e  another  incft, 


166 


THE   FOURTH   DIMENSION 


In  fig.  102,  we  have  the  representation  of  a  block  of 
4x4x4x4  or   256   tesseracts.      They  are  given  in 

Fig.  102,* 


A  cube  of  64  cubes, 
each  1  in.  x  1  in. 
x  1  in.,  the  begin- 
ning of  a  tesseract. 


A  cube  of  04  tubes, 
each  1  in.  x  1  in. 
x  1  in.  the  begin- 
ning of  tesseracts 
1  in  from  our  space 
on  the  4th  dimen- 
sion. 


A  cube  or  04  cubes, 
each  1  in.  x  1  in. 
x  1  in.,  the  begin- 
ning of  tesseracts 
2  in.  from  our  space 
in  the  4th  dimen- 
sion. 


A  cube  of  64  cubes, 
each  1  in.  x  1  in. 
x  1  in.,  the  begin- 
ning of  tesseracts 
starting  3  in.  from 
our  space  in  the  4th 
dimension. 


four  consecutive  sections,  each  supposed  to  be  taken  one 
inch  apart  in  the  fourth  dimension,  and  so  giving  four 

*  The    coloured    plate,  figs.    1,  2,   3,  shovys    these    relations    more. 


THE    SIMPLEST   FOUR-DIMENSIONAL    SOLID  167 

blocks  of  cubes,  64  in  each  block.  Here  we  see,  com- 
paring it  with  the  figure  of  81  tesseracts,  that  the  number 
of  the  different  regions  show  a  different  tendency  of 
increase.  By  taking  five  blocks  of  five  divisions  each  way 
this  would  become  even  more  clear. 

We  see,  fig.  102,  that  starting  from  the  point  at  any 
corner,  the  white  coloured  regions  only  extend  out  in 
a  line.  The  same  is  true  for  the  yellow,  red,  and  blue. 
With  regard  to  the  latter  it  should  be  noticed  that  the 
line  of  blues  does  not  consist  in  regions  next  to  each 
other  in  the  drawing,  but  in  portions  which  come  in  in 
different  cubes.  The  portions  which  lie  next  to  one 
another  in  the  fourth  dimension  must  always  be  repre- 
sented so,  when  we  have  a  three-dimensional  representation. 
Again,  those  regions  such  as  the  pink  one,  go  on  increasing 
in  two  dimensions.  About  the  pink  region  this  is  seen 
without  going  out  of  the  cube  itself,  the  pink  regions 
increase  in  length  and  height,  but  in  no  other  dimension. 
In  examining  these  regions  it  is  sufficient  to  take  one  as 
a  sample. 

The  purple  increases  in  the  same  manner,  for  it  comes 
in  in  a  succession  from  below  to  above  in  block  2,  and  in 
a  succession  from  block  to  block  in  2  and  3.  Now,  a 
succession  from  below  to  above  represents  a  continuous 
extension  upwards,  and  a  succession  from  block  to  block 
represents  a  continuous  extension  in  the  fourth  dimension. 
Thus  the  purple  regions  increase  in  two  dimensions,  the 
upward  and  the  fourth,  so  when  we  take  a  very  great 
many  divisions,  and  let  each  become  very  small,  the 
purple  region  forms  a  two-dimensional  extension. 

In  the  same  way,  looking  at  the  regions  marked  1.  b.  or 
light  blue,  which  starts  nearest  a  corner,  we  see  that  the 
tesseracts  occupying  it  increase  in  length  from  left  to 
right,  forming  a  line,  and  that  there  are  as  many  lines  of 
light  blue  tesseracts  as  there  are  sections  between  the 


168  THE    FOURTH    DIMENSION 

first  and  last  section.  Hence  the  light  blue  tesseracts 
increase  in  number  in  two  ways — in  the  right  and  left, 
and  in  the  fourth  dimension.  They  ultimately  form 
what  we  may  call  a  plane  surface. 

Now  all  those  regions  which  contain  a  mixture  of  two 
simple  colours,  white,  yellow,  red,  blue,  increase  in  two 
ways.  On  the  other  hand,  those  which  contain  a  mixture 
of  three  colours  increase  in  three  ways.  Take,  for  instance, 
the  ochre  region;  this  has  three  colours,  white,  yellow, 
red ;  and  in  the  cube  itself  it  increases  in  three  ways. 

Now  regard  the  orange  region ;  if  we  add  blue  to  this 
we  get  a  brown.  The  region  of  the  brown  tesseracts 
extends  in  two  ways  on  the  left  of  the  second  block, 
No.  2  in  the  figure.  It  extends  also  from  left  to  right  in 
succession  from  one  section  to  another,  from  section  2 
to  section  3  in  our  figure. 

Hence  the  brown  tesseracts  increase  in  number  in  three 
dimensions  upwards,  to  and  fro,  fourth  dimension.  Hence 
they  form  a  cubic,  a  three-dimensional  region;  this  region 
extends  up  and  down,  near  and  far,  and  in  the  fourth 
direction,  but  is  thin  in  the  direction  from  left  to  right. 
It  is  a  cube  which,  when  the  complete  tesseract  is  repre- 
sented in  our  space,  appears  as  a  series  of  faces  on  the 
successive  cubic  sections  of  the  tesseract.  Compare  fig. 
103  in  which  the  middle  block,  2,  stands  as  representing  a 
great  number  of  sections  intermediate  between  1  and  3. 

In  a  similar  way  from  the  pink  region  by  addition  of 
blue  we  have  the  light  purple  region,  which  can  be  seen 
to  increase  in  three  ways  as  the  number  of  divisions 
becomes  greater.  The  three  ways  in  which  this  region  of 
tesseracts  extends  is  up  and  down,  right  and  left,  fourth 
dimension.  Finally,  therefore,  it  forms  a  cubic  mass  of 
very  small  tesseracts,  and  when  the  tesseract  is  given  in 
space  sections  it  appears  on  the  faces  containing  the 
upward  an4  the  right  and  left  dimensions, 


THE    SIMPLEST   FOUR-DIMENSIONAL   SOLID  169 

We  get  then  altogether,  as  three-dimensional  regions, 
ochre,  brown,  light  purple,  light  green. 

Finally,  there  i;>  the  region  which  corresponds  to  a 
mixture  of  all  the  colours ;  there  is  only  one  region  such 
as  this.  It  is  the  one  that  springs  from  ochre  by  the 
addition  of  blue — this  colour  we  call  light  brown. 

Looking  at  the  light  brown  region  we  see  that  it 
increases  in  four  ways.  Hence,  the  tesseracts  of  which  it 
is  composed  increase  in  number  in  each  of  four  dimen- 
sions, and  the  shape  they  form  does  not  remain  thin  in 
any  of  the  four  dimensions.  Consequently  this  region 
becomes  the  solid  content  of  the  block  of  tesseracts,  itself; 
it  is  the  real  four-dimensional  solid.  All  the  other  regions 
are  then  boundaries  of  this  light  brown  region.  If  we 
suppose  the  process  of  increasing  the  number  of  tesseracts 
and  diminishing  their  size  carried  on  indefinitely,  then 
the  light  brown  coloured  tesseracts  become  the  whole 
interior  mass,  the  three-coloured  tesseracts  become  three- 
dimensional  boundaries,  thin  in  one  dimension,  and  form 
the  ochre,  the  brown,  the  light  purple,  the  light  green. 
The  two-coloured  tesseracts  become  two-dimensional 
boundaries,  thin  in  two  dimensions,  e.g.,  the  pink,  the 
green,  the  purple,  the  orange,  the  light  blue,  the  light 
yellow.  The  one-coloured  tesseracts  become  bounding 
lines,  thin  in  three  dimensions,  and  the  null  points  become 
bounding  corners,  thin  in  four  dimensions.  From  these 
thin  real  boundaries  we  can  pass  in  thought  to  the 
abstractions — points,  lines,  faces,  solids — bounding  the 
four-dimensional  solid,  which  is  this  case  is  light  brown 
coloured,  and  under  this  supposition  the  light  brown 
coloured  region  is  the  only  real  one,  is  the  only  one  which 
is  not  an  abstraction. 

It  should  be  observed  that,  in  taking  a  square  as  the 
representation  of  a  cube  on  a  plane,  we  only  represent 
one  face,  or  the  section  between  two  faces.  The  squares, 


170  THE   FOURTH    DIMENSION 

as  drawn  by  a  plane  being,  are  not  the  cubes  themselves, 
but  represent  the  faces  or  the  sections  of  a  cube.  Thus 
in  the  plane  being's  diagram  a  cube  of  twenty-seven  cubes 
"  null "  represents  a  cube,  but  is  really,  in  the  normal 
position,  the  orange  square  of  a  null  cube,  and  may  be 
called  null,  orange  square. 

A  plane  being  would  save  himself  confusion  if  he  named 
his  representative  squares,  not  by  using  the  names  of  the 
cubes  simply,  but  by  adding  to  the  names  of  the  cubes  a 
word  to  show  what  part  of  a  cube  his  representative  square 
was. 

Thus  a  cube  null  standing  against  his  plane  touches  it 
by  null  orange  face,  passing  through  his  plane  it  has  in 
the  plane  a  square  as  trace,  which  is  null  white  section,  if 
we  use  the  phrase  white  section  to  mean  a  section  drawn 
perpendicular  to  the  white  line.  In  the  same  way  the 
cubes  which  we  take  as  representative  of  the  tesseract  are 
not  the  tesseract  itself,  but  definite  faces  or  sections  of  it. 
In  the  preceding  figures  we  should  say  then,  not  null,  but 
"  null  tesseract  ochre  cube,"  because  the  cube  we  actually 
have  is  the  one  determined  by  the  three  axes,  white,  red, 
yellow. 

There  is  another  way  in  which  we  can  regard  the  colour 
nomenclature  of  the  boundaries  of  a  tesseract. 

Consider  a  null  point  to  move  tracing  out  a  white  line 
one  inch  in  length,  and  terminating  in  a  null  point, 
see  fig.  103  or  in  the  coloured  plate. 

Then  consider  this  white  line  with  its  terminal  points 
itself  to  move  in  a  second  dimension,  each  of  the  points 
traces  out  a  line,  the  line  itself  traces  out  an  area,  and 
gives  two  lines  as  well,  its  initial  and  its  final  position. 

Thus,  if  we  call  "  a  region  "  any  element  of  the  figure, 
such  as  a  point,  or  a  line,  etc.,  every  "region"  in  moving 
traces  out  a  new  kind  of  region,  "  a  higher  region,"  and 
gives  two  regions  pf  its,  own  kind,  an  initial  a.nd  a  final 


THE   SIMPLEST   FOUR-DIMENSIONAL   SOLID  171 

position.  The  "  higher  region "  means  a  region  with 
another  dimension  in  it. 

Now  the  square  can  move  and  generate  a  cube.  The 
square  light  yellow  moves  and  traces  out  the  mass  of  the 
cube.  Letting  the  addition  of  red  denote  the  region 
made  by  the  motion  in  the  upward  direction  we  get  an 
ochre  solid.  The  light  yellow  face  in  its  initial  and 
terminal  positions  give  the  two  square  boundaries  of  the 
cube  above  and  below.  Then  each  of  the  four  lines  of  the 
light  yellow  square — white,  yellow,  and  the  white,  yellow 
opposite  them — trace  out  a  bounding  square.  So  there 
are  in  all  six  bounding  squares,  four  of  these  squares  being 
designated  in  colour  by  adding  red  to  the  colour  of  the 
generating  lines.  Finally,  each  point  moving  in  the  up 
direction  gives  rise  to  a  line  coloured  null  +  red,  or  red, 
and  then  there  are  the  initial  and  terminal  positions  of  the 
points  giving  eight  points.  The  number  of  the  lines  is 
evidently  twelve,  for  the  four  lines  of  this  light  yellow 
square  give  four  lines  in  their  initial,  four  lines  in  their 
final  position,  while  the  four  points  trace  out  four  lines, 
that  is  altogether  twelve  lines. 

Now  the  squares  are  each  of  them  separate  boundaries 
of  the  cube,  while  the  lines  belong,  each  of  them,  to  two 
squares,  thus  the  red  line  is  that  which  is  common  to  the 
orange  and  pink  squares. 

Now  suppose  that  there  is  a  direction,  the  fourth 
dimension,  which  is  perpendicular  alike  to  every  one 
of  the  space  dimensions  already  used — a  dimension 
perpendicular,  for  instance,  to  up  and  to  right  hand, 
so  that  the  pink  square  moving  in  this  direction  traces 
out  a  cube. 

A  dimension,  moreover,  perpendicular  to  the  up  and 
away  directions,  so  that  the  orange  square  moving  in  this 
direction  also  traces  out  a  cube,  and  the  light  yellow 
square,  too,  moving  jn  this  direction  traces  out  a  cub§. 


172  THE    FOURTH    DIMENSION 

Under  this  supposition,  the  whole  cube  moving  in  the 
unknown  dimension,  traces  out  something  new — a  new 
kind  of  volume,  a  higher  volume.  This  higher  volume 
is  a  four-dimensional  volume,  and  we  designate  it  in  colour 
by  adding  blue  to  the  colour  of  that  which  by  moving 
generates  it. 

It  is  generated  by  the  motion  of  the  ochre  solid,  and 
hence  it  is  of  the  colour  we  call  light  brown  (white,  yellow, 
red,  blue,  mixed  together).  It  is  represented  by  a  number 
of  sections  like  2  in  fig.  103. 

Now  this  light  brown  higher  solid  has  for  boundaries : 
first,  the  ochre  cube  in  its  initial  position,  second,  the 
same  cube  in  its  final  position,  1  and  3,  fig.  103.  Each 
of  the  squares  which  bound  the  cube,  moreover,  by  move- 
ment in  this  new  direction  traces  out  a  cube,  so  we  have 
from  the  front  pink  faces  of  the  cube,  third,  a  pink  blue  or 
light  purple  cube,  shown  as  a  light  purple  face  on  cube  2 
in  fig.  103,  this  cube  standing  for  any  number  of  inter- 
mediate sections ;  fourth,  a  similar  cube  from  the  opposite 
pink  face ;  fifth,  a  cube  traced  out  by  the  orange  face — 
this  is  coloured  brown  and  is  represented  by  the  brown 
face  of  the  section  cube  in  fig.  103  ;  sixth,  a  correspond- 
ing brown  cube  on  the  right  hand ;  seventh,  a  cube 
starting  from  the  light  yellow  square  below ;  the  unknown 
dimension  is  at  right  angles  to  this  also.  This  cube  is 
coloured  light  yellow  and  blue  or  light  green ;  and, 
finally,  eighth,  a  corresponding  cube  from  the  upper 
light  yellow  face,  shown  as  the  light  green  square  at  the 
top  of  the  section  cube. 

The  tesseract  has  thus  eight  cubic  boundaries.  These 
completely  enclose  it,  so  that  it  would  be  invisible  to  a 
four-dimensional  being.  Now,  as  to  the  other  boundaries, 
just  as  the  cube  has  squares,  lines,  points,  as  boundaries, 
so  the  tesseract  has  ciibes,  squares,  lines,  points,  a,$ 
boundaries, 


THE   SIMPLEST  FOUR- DIMENSIONAL   SOLID  173 

The  number  of  squares  is  found  thus — round  the  cube 
are  six  squares,  these  will  give  six  squares  in  their  initial 
and  six  in  their  final  positions.  Then  each  of  the  twelve 
lines  of  the  cube  trace  out  a  square  in  the  motion  in 
the  fourth  dimension.  Hence  there  will  be  altogether 
12  +  12  =  24  squares. 

If  we  look  at  any  one  of  these  squares  we  see  that  it 
is  the  meeting  surface  of  two  of  the  cubic  sides.  Thus, 
the  red  line  by  its  movement  in  the  fourth  dimension, 
traces  out  a  purple  square — this  is  common  to  two 
cubes,  one  of  which  is  traced  out  by  the  pink  square 
moving  in  the  fourth  dimension,  and  the  other  is 
traced  out  by  the  orange  square  moving  in  the  same 
way.  To  take  another  square,  the  light  yellow  one,  this 
is  common  to  the  ochre  cube  and  the  light  green  cube. 
The  ochre  cube  comes  from  the  light  yellow  square 
by  moving  it  in  the  up  direction,  the  light  green  cube 
is  made  from  the  light  yellow  square  by  moving  it  in 
the  fourth  dimension.  The  number  of  lines  is  thirty- 
two,  for  the  twelve  lines  of  the  cube  give  twelve  lines 
of  the  tesseract  in  their  initial  position,  and  twelve  in 
their  final  position,  making  twenty-four,  while  each  of 
the  eight  points  traces  out  a  line,  thus  forming  thirty- 
two  lines  altogether. 

The  lines  are  each  of  them  common  to  three  cubes,  or 
to  three  square  faces;  take,  for  instance,  the  red  line. 
This  is  common  to  the  orange  face,  the  pink  face,  and 
that  face  which  is  formed  by  moving  the  red  line  in  the 
sixth  dimension,  namely,  the  purple  face.  It  is  also 
common  to  the  ochre  cube,  the  pale  purple  cube,  and  the 
brown  cube. 

The  points  are  common  to  six  square  faces  and  to  four 
cubes  ;  thus,  the  null  point  from  which  we  start  is  common 
to  the  three  square  faces — pink,  light  yellow,  orange,  and 
to  the  three  square  faces  made  by  moving  the  three  lines 


174 


THE    FOURTH   DIMENSION 


white,  yellow,  red,  in  the  fourth  dimension,  namely,  the 
light  blue,  the  light  green,  the  purple  faces — that  is,  to 
six  faces  in  all.  The  four  cubes  which  meet  in  it  are  the 


£ 

\i 

*» 
-S-g 
?J 

f 

i      ^tdanj 

JO 

3      ajdjnj 

'"  S 

§  1 

s  -° 

^S  £ 

tuO     >>0 

j  ^3 

2    «- 

^      0 

CQ  '5 

4-» 
C 

J              #2 
<5>      "° 

£     ^      £ 

tn>      <j^              ta 

j  v455,       J 
aidjnj 

y^%° 

x>     $ 

v4*/ 

<fs. 

/    ** 

\i 

«w 

5P 

•5° 

^   «<X 
)     <oV 

S^ 

J9 
> 

1* 

5 

ajdjnj 

s^      I 

V    f 
•&      %  -g 

j      ,    *^H 

aidjnj 

\  X 

*  Y 

aidjnd      3 


1 


ochre  cube,  the  light  purple  cube,  the  brown  cube,  and 

the  light  green  cube.  :v  . 

A  complete  view  of  the  tesseract  in  its  various  space 


THE   SIMPLEST   FOUR-DIMENSIONAL   SOLID 


176 


f 


presentations  is  given  in  the  following  figures  or  catalogue 
cubes,   figp.    103-106.      The    first    cube   in    each   figure 


176 


THE   FOURTH   DIMENSION 


represents  the  view  of  a  tesseract  coloured  as  described  as 
it  begins  to  pass  transverse  to  our  space.  The  intermediate 
figure  represents  a  sectional  view  when  it  is  partly  through, 


/ 

P3H 

*-      V 

"5. 

u> 

•  /& 

ID                          <u 

4.3                                                  *J 

3 
0. 

eyxr  a 

"^          ^                '? 

•o 

-£-                  V^«                   "^ 

^       'V     ^ 

t 

£ 

rr- 

•$ 

p3H 

J 

*      U 
r*      ^s 

^x  \  ^  1 

and  the  final  figure  represents  the  far  end  as  it  is  just 
passing  out.  These  figures  will  be  explained  in  detail  in 
the  next  chapter. 


•THE   SIMPLEST   tOUR-DlMfcN SlONAL   SOtlt) 


177 


3[0jnj 

_*    4- 
•2    *<tf 

I     \       1 

i 

-    V 

1)                  ^ 

Oi     ,i|djnfj 

&k              <!/ 

f^\«f 

o 

I— I 
hb 


il 


We  have  thus  obtained  a  nomenclature  for  each  of  the 
regions  of  a  tesseract ;  we  can  speak  of  any  one  of  the 
eight  bounding  cubes,  the  twenty  square  faces,  the  thirty- 
two  lines,  the  sixteen  points. 


CHAPTER  XIII 
REMARKS   ON  THE  FIGURES 

AN  inspection  of  above  figures  will  give  an  answer  to 
many  questions  about  the  tesseract.  If  we  have  a 
tesseract  one  inch  each  way,  then  it  can  be  represented 
as  a  cube — a  cube  having  white,  yellow,  red  axes,  and 
from  this  cube  as  a  beginning,  a  volume  extending  into 
the  fourth  dimension.  Now  suppose  the  tesseract  to  pass 
transverse  to  our  space,  the  cube  of  the  red,  yellow,  white 
axes  disappears  at  once,  it  is  indefinitely  thin  in  the 
fourth  dimension.  Its  place  is  occupied  by  those  parts 
of  the  tesseract  which  lie  further  away  from  our  space 
in  the  fourth  dimension.  Each  one  of  these  sections 
will  last  only  for  one  moment,  but  the  whole  of  them 
will  take  up  some  appreciable  time  in  passing.  If  we 
take  the  rate  of  one  inch  a  minute  the  sections  will  take 
the  whole  of  the  minute  in  their  passage  across  our 
space,  they  will  take  the  whole  of  the  minute  except  the 
moment  which  the  beginning  cube  and  the  end  cube 
occupy  in  their  crossing  our  space.  In  each  one  of  the 
cubes,  the  section  cubes,  we  can  draw  lines  in  all  directions 
except  in  the  direction  occupied  by  the  blue  line,  the 
fourth  dimension  ;  lines  in  that  direction  are  represented 
by  the  transition  from  one  section  cube  to  another.  Thus 
to  give  ourselves  an  adequate  representation  of  the 
tesseract  we  ought  to  have  a  limitless  number  of  section 
cubes  intermediate  between  the  first  bounding  cube,  the 

178 


REMARKS   ON  THE  FIGURES 


170 


ochre  cube,  and  the  last  bounding  cube,  the  other  ochre 
cube.  Practically  three  intermediate  sectional  cubes  will 
be  found  sufficient  for  most  purposes.  We  will  take  then 
a  series  of  five  figures — two  terminal  cubes,  and  three 
intermediate  sections — and  show  how  the  different  regions 
appear  in  our  space  when  we  take  each  set  of  three  out 
of  the  four  axes  of  the  tesseract  as  lying  in  our  space. 

In  fig.  107  initial  letters  are  used  for  the  colours. 
A  reference  to  fig.  103  will  show  the  complete  nomen- 
clature, which  is  merely  indicated  here. 


'iw  I  •    \i  I     ^»  A-NJ       I      ^J/^NJ  I      -*\\L 

"•  wh.  n.     bl.  1.  bl.bl.   bl.  I.  bl.  bl.    bl  1.  bl.bl.    n.  wh.  n. 
interior       interior        interior         interior         interior 


Ochre     L.Brown      L.Brown      L.  Brown 
Fig.  107. 


Ochre 


In  this  figure  the  tesseract  is  shown  in  five  stages 
distant  from  our  space:  first,  zero  ;  second,  £  in.  5  third, 
f  in.  ;  fourth,  £  in.  ;  fifth,  1  in.;  which  are  called  60,  61, 
62,  63,  64,  because  they  are  sections  taken  at  distances 
0,  1,  2,  3,  4  quarter  inches  along  the  blue  line.  All  the 
regions  can  be  named  from  the  first  cube,  the  60  cube, 
as  before,  simply  by  remembering  that  transference  along 
the  6  axis  gives  the  addition  of  blue  to  the  colour  of 
(he  region  in  the  ochre,  the  60  cube.  In  the  final  cube 
64,  the  colouring  of  the  original  60  cube  is  repeated. 
Thus  the  red  line  moved  along  the  blue  axis  gives  a  red 
and  blue  or  purple  square.  This  purple  square  appears 
as  the  three  purple  lines  in  the  sections  61,  62,  63,  taken 
at  £,  £,  |  of  an  inch  in  the  fourth  dimension.  If  the 
tesseract  moves  transverse  to  our  space  we  have  then  in 
this  particular  region,  first  of  all  a  red  line  which  lasts 
for  a  moment,  secondly  a  purple  line  which  takes  its 


180  THE  FOURTH  DIMENSION 

place.  This  purple  line  lasts  for  a  minute — that  is,  all 
of  a  minute,  except  the  moment  taken  by  the  crossing 
our  space  of  the  initial  and  final  red  line.  The  purple 
line  having  lasted  for  this  period  is  succeeded  by  a  red 
line,  which  lasts  for  a  moment ;  then  this  goes  and  the 
tesseract  has  passed  across  our  space.  The  final  red  line 
we  call  red  bl.,  because  it  is  separated  from  the  initial 
red  line  by  a  distance  along  the  axis  for  which  we  use 
the  colour  blue.  Thus  a  line  that  lasts  represents  an 
area  duration ;  is  in  this  mode  of  presentation  equivalent 
to  a  dimension  of  space.  In  the  same  way  the  white 
line,  during  the  crossing  our  space  by  the  tesseract,  is 
succeeded  by  a  light  blue  line  which  lasts  for  the  inside 
of  a  minute,  and  as  the  tesseract  leaves  our  space,  having 
crossed  it,  the  white  bl.  line  appears  as  the  final 
termination. 

Take  now  the  pink  face.  Moved  in  the  blue  direction 
it  traces  out  a  light  purple  cube.  This  light  purple 
cube  is  shown  in  sections  in  &,,  62,  63,  and  the  farther 
face  of  this  cube  in  the  blue  direction  is  shown  in  64— 
a  pink  face,  called  pink  b  because  it  is  distant  from  the 
pink  face  we  began  with  in  the  blue  direction.  Thus 
the  cube  which  we  colour  light  purple  appears  as  a  lasting 
square.  The  square  face  itself,  the  pink  face,  vanishes 
instantly  the  tesseract  begins  to  move,  but  the  light 
purple  cube  appears  as  a  lasting  square.  Here  also 
duration  is  the  equivalent  of  a  dimension  of  space — a 
lasting  square  is  a  cube.  It  is  useful  to  connect  these 
diagrams  with  the  views  given  in  the  coloured  plate. 

Take  again  the  orange  face,  that  determined  by  the 
red  and  yellow  axes ;  from  it  goes  a  brown  cube  in  the 
blue  direction,  for  red  and  yellow  and  blue  are  supposed 
to  make  brown.  This  brown  cube  is  shown  in  three 
sections  in  the  faces  6t,  62,  63.  In  64  is  the  opposite 
orange  face  of  the  brown  cube,  the  face  called  orangp  b, 


BEMARKS   ON   THE   FIGURES 


181 


for  it  is  distant  in  the  blue  direction  from  the  orange 
face.  As  the  tesseract  passes  transverse  to  our  space, 
we  have  then  in  this  region  an  instantly  vanishing  orange 
square,  followed  by  a  lasting  brown  square,  and  finally 
an  orange  face  which  vanishes  instantly. 

Now,  as  any  three  axes  will  be  in  our  space,  let  us  send 
the  white  axis  out  into  the  unknown,  the  fourth  dimen- 
sion, and  take  the  blue  axis  into  our  known  space 
dimension.  Since  the  white  and  blue  axes  are  perpen- 
dicular to  each  other,  if  the  white  axis  goes  out  into 
the  fourth  dimension  in  the  positive  sense,  the  blue  axis 
will  come  into  the  direction  the  white  axis  occupied, 
in  the  negative  sense. 


wh/j 


\vh 


wh, 


-  bl.  n-         I.  bl.wh.       1.  bl.wh.     1.  bl.wh.    n.  bl.  n. 
Fig.  108. 

Hence,  not  to  complicate  matters  by  having  to  think 
of  two  senses  in  the  unknown  direction,  let  us  send  the 
white  line  into  the  positive  sense  of  the  fourth  dimen- 
sion, and  take  the  blue  one  as  running  in  the  negative 
sense  of  that  direction  which  the  white  line  has  left; 
let  the  blue  line,  that  is,  run  to  the  left.  We  have 
now  the  row  of  figures  in  fig.  108.  The  dotted  cube 
shows  where  we  had  a  cube  when  the  white  line  ran 
in  our  space — now  it  has  turned  out  of  our  space,  and 
another  solid  boundary,  another  cubic  face  of  the  tesseract 
comes  into  our  space.  This  cube  has  red  and  yellow 
axes  as  before ;  but  now,  instead  of  a  white  axis  running 
to  the  right,  there  is  a  blue  axis  running  to  the  left. 
Here  we  can  distinguish  the  regions  by  colours  in  a  per- 
fectly systematic  way.  The  red  line  traces  out  a  purple 


182  THE  FOURTH   DIMENSION 

square  in  the  transference  along  the  blue  axis  by  which 
this  cube  is  generated  from  the  orange  face.  This 
purple  square  made  by  the  motion  of  the  red  line  is 
the  same  purple  face  that  we  saw  before  as  a  series  of 
lines  in  the  sections  blt  62.  ^3-  Here,  since  both  red  and 
blue  axes  are  in  our  space,  we  have  no  need  of  duration 
to  represent  the  area  they  determine.  In  the  motion 
of  the  tesseract  across  space  this  purple  face  would 
instantly  disappear. 

From  the  orange  face,  which  is  common  to  the  initial 
cubes  in  fig.  107  and  fig.  108,  there  goes  in  the  blue 
direction  a  cube  coloured  brown.  This  brown  cube  is 
now  all  in  our  space,  because  each  of  its  three  axes  run 
in  space  directions,  up,  away,  to  the  left.  It  is  the  same 
brown  cube  which  appeared  as  the  successive  faces  on  the 
sections  6lf  62,  63.  Having  all  its  three  axes  in  our 
space,  it  is  given  in  extension  ;  no  part  of  it  needs  to 
be  represented  as  a  succession.  The  tesseract  is  now 
in  a  new  position  with  regard  to  our  space,  and  when 
it  moves  across  our  space  the  brown  cube  instantly 
disappears. 

In  order  to  exhibit  the  other  regions  of  the  tesseract 
we  must  remember  that  now  the  white  line  runs  in  the 
unknown  dimension.  Where  shall  we  put  the  sections 
at  distances  along  the  line  ?  Any  arbitrary  position  in 
our  space  will  do  :  there  is  no  way  by  which  we  can 
represent  their  real  position. 

However,  as  the  brown  cube  comes  off  from  the  orange 
face  to  the  left,  let  us  put  these  successive  sections  to 
the  left.  We  can  call  them  wh0,  ivhlt  wh.2,  wh3,  ivh±, 
because  they  are  sections  along  the  white  axis,  which 
now  runs  in  the  unknown  dimension. 

Eunning  from  the  purple  square  in  the  white  direction 
we  find  the  light  purplejmbe.  This  is  represented  in  the 
sections  whlt  wh2,  tdi*(t0A^fig.  108.  It  is  the  same  cube 


REMARKS   ON  THE   FIGURES  183 

that  is  represented  in  the  sections  61}  62,  b3  •  in  fig.  107 
the  red  and  white  axes  are  in  our  space,  the  blue  out  of 
it ;  in  the  other  case,  the  red  and  blue  are  in  our  space, 
the  white  out  of  it.  It  is  evident  that  the  face  pink  y, 
opposite  the  pink  face  in  fig.  107,  makes  a  cube  shown 
in  squares  in  61?  62,  63,  64,  on  the  opposite  side  to  the  I 
purple  squares.  Also  the  light  yellow  face  at  the  base 
of  the  cube  60,  makes  a  light  green  cube,  shown  as  a  series 
of  base  squares. 

The  same  light  green  cube  can  be  found  in  fig.  107. 
The  base  square  in  wh0  is  a  green  square,  for  it  is  enclosed 
by  blue  and  yellow  axes.  From  it  goes  a  cube  in  the 
white  direction,  this  is  then  a  light  green  cube  and  the 
same  as  the  one  just  mentioned  as  existing  in  the  sections 

60,  61,  &2>  &3>  bt. 

The  case  is,  however,  a  little  different  with  the  brown 
cube.  This  cube  we  have  altogether  in  space  in  the 
section  ivh6,  fig.  108,  while  it  exists  as  a  series  of  squares, 
the  left-hand  ones,  in  the  sections  &<>,  bu  b&  63,  64.  The 
brown  cube  exists  as  a  solid  in  our  space,  as  shown  in 
fig.  108.  In  the  mode  of  representation  of  the  tesseract 
exhibited  in  fig.  107,  the  same  brown  cube  appears  as  a 
succession  of  squares.  That  is,  as  the  tesseract  moves 
across  space,  the  brown  cube  would  actually  be  to  us  a 
square — it  would  be  merely  the  lasting  boundary  of  another 
solid.  It  would  have  no  thickness  at  all,  only  extension 
in  two  dimensions,  and  its  duration  would  show  its  solidity 
in  three  dimensions. 

It  is  obvious  that,  if  there  is  a  four-dimensional  space, 
matter  in  three  dimensions  only  is  a  mere  abstraction  ;  all 
material  objects  must  then  have  a  slight  four-dimensional 
thickness.  In  this  case  the  above  statement  will  undergo 
modification.  The  material  cube  which  is  used  as  the 
model  of  the  boundary  of  a  tesseract  will  have  a  slight 
thickness  in  the  fourth  dimension,  and  when  the  cube  is 


184  THE   FOURTH   DIMENSION 

presented  to  us  in  another  aspect,  it  would  not  be  a  mere 
surface.  But  it  is  most  convenient  to  regard  the  cubes 
we  use  as  having  no  extension  at  all  in  the  fourth 
dimension.  This  consideration  serves  to  bring  out  a  point 
alluded  to  before,  that,  if  there  is  a  fourth  dimension,  our 
conception  of  a  solid  is  the  conception  of  a  mere  abstraction, 
and  our  talking  about  real  three-dimensional  objects  would 
seem  to  a  four-dimensional  being  as  incorrect  as  a  two- 
dimensional  being's  telling  about  real  squares,  real 
triangles,  etc.,  would  seem  to  us. 

The  consideration  of  the  two  views  of  the  brown  cube 
shows  that  any  section  of  a  cube  can  be  looked  at  by  a 
presentation  of  the  cube  in  a  different  position  in  four- 
dimensional  space.  The  brown  faces  in  619  62>  b3,  are  the 
very  same  brown  sections  that  would  be  obtained  by 
cutting  the  brown  cube,  wh0,  across  at  the  right  distances 
along  the  blue  line,  as  shown  in  fig.  108.  But  as  these 
sections  are  placed  in  the  brown  cube,  wh0,  they  come 
behind  one  another  in  the  blue  direction.  Now,  in  the 
sections  ivhlt  wh2,  wh3,  we  are  looking  at  these  sections 
from  the  white  direction — the  blue  direction  does  not 
exist  in  these  figures.  So  we  see  them  in  a  direction  at 
right  angles  to  that  in  which  they  occur  behind  one 
another  in  wh0.  There  are  intermediate  vi^ews,  which 
would  come  in  the  rotation  of  a  tesseract.  These  brown 
squares  can  be  looked  at  from  directions  intermediate 
between  the  white  and  blue  axes.  It  must  be  remembered 
that  the  fourth  dimension  is  perpendicular  equally  to  all 
three  space  axes.  Hence  we  must  take  the  combinations 
of  the  blue  axis,  with  each  two  of  our  three  axes,  white, 
red,  yellow,  in  turn. 

In  fig.  109  we  take  red,  white,  and  blue  axes  in  space, 
sending  yellow  into  the  fourth  dimension.  If  it  goes  into 
the  positive  sense  of  the  fourth  dimension  the  blue  line 
will  come  in  the  opposite  direction  to  that  in  which  the 


REMARKS   ON   THE   FIGURES 


185 


yellow  line  ran  before.  Hence,  the  cube  determined  by 
the  white,  red,  blue  axes,  will  start  from  the  pink  plane 
and  run  towards  us.  The  dotted  cube  shows  where  the 
ochre  cube  was.  When  it  is  turned  out  of  space,  the  cube 
coming  towards  from  its  front  face  is  the  one  which  comes 
into  our  space  in  this  turning.  Since  the  yellow  line  now 
runs  in  the  unknown  dimension  we  call  the  sections 
2/o,  2/1,  2/2>  2/3?  2/4>  as  they  are  made  at  distances  0,  1,  2,  3,  4, 
quarter  inches  along  the  yellow  line.  We  suppose  these 
cubes  arranged  in  a  line  coming  towards  us — not  that 
that  is  any  more  natural  than  any  other  arbitrary  series 
of  positions,  but  it  agrees  with  the  plan  previously  adopted. 


Fig.  109. 

The  interior  of  the  first  cube,  2/0,  is  that  derived  from 
pink  by  adding  blue,  or,  as  we  call  it,  light  purple.  The 
faces  of  the  cube  are  light  blue,  purple,  pink.  As  drawn, 
we  can  only  see  the  face  nearest  to  us,  which  is  not  the 
one  from  which  the  cube  starts — but  the  face  on  the 
opposite  side  has  the  same  colour  name  as  the  face 
towards  us. 

The  successive  sections  of  the  series,  yoj  y^  y2,  etc.,  can 
be  considered  as  derived  from  sections  of  the  60  cube 
made  at  distances  along  the  yellow  axis.  What  is  distant 
a  quarter  inch  from  the  pink  face  in  the  yellow  direction  ? 
This  question  is  answered  by  taking  a  section  from  a  point 
a  quarter  inch  along  the  yellow  axis  in  the  cube  &<»  fig.  107. 
It  is  an  ochre  section  with  lines  orange  and  light  yellow. 
This  section  will  therefore  take  the  place  of  the  pink  face 


186 


THE   FOURTH   DIMENSION 


in  ^  when  we  go  on  in  the  yellow  direction.  Thus,  the 
first  section,  ylt  will  begin  from  an  ochre  face  with  light 
yellow  and  orange  lines.  The  colour  of  the  axis  which 
lies  in  space  towards  us  is  blue,  hence  the  regions  of  this 
section-cube  are  determined  in  nomenclature,  they  will  be 
found  in  full  in  fig.  105. 

There  remains  only  one  figure  to  be  drawn,  and  that  is 
the  one  in  which  the  red  axis  is  replaced  by  the  blue. 
Here,  as  before,  if  the  red  axis  goes  out  into  the  positive 
sense  of  the  fourth  dimension,  the  blue  line  must  come 
into  our  space  in  the  negative  sense  of  the  direction  which 
the  red  line  has  left.  Accordingly,  the  first  cube  will 


Fig.  110. 


come  in  beneath  the  position  of  our  ochre  cube,  the  one 
we  have  been  in  the  habit  of  starting  with. 

To  show  these  figures  we  must  suppose  the  ochre  cube 
to  be  on  a  movable  stand.  When  the  red  line  swings  out 
into  the  unknown  dimension,  and  the  blue  line  comes  in 
downwards,  a  cube  appears  below  the  place  occupied  by 
the  ochre  cube.  The  dotted  cube  shows  where  the  ochre 
cube  was.  That  cube  has  gone  and  a  different  cube  runs 
downwards  from  its  base.  This  cube  has  white,  yellow, 
and  blue  axes.  Its  top  is  a  light  yellow  square,  and  hence 
its  interior  is  light  yellow  +  blue  or  light  green.  Its  front 
face  is  formed  by  the  white  line  moving  along  the  blue 
axis,  and  is  therefore  light  blue,  the  left-hand  side  is 
formed  by  the  yellow  line  moving  along  the  blue  axis,  and 
therefore  green. 


REMARKS   ON   THE   FIGURES  187 

As  the  red  line  now  runs  in  the  fourth  dimension,  the 
imccessive  sections  can  be  called  TO,  r\,  r2,  r3,  r4,  these 
letters  indicating  that  at  distances  0,  J,  £ ,  f ,  1  inch  along 
the  red  axis  we  take  all  of  the  tesseract  that  can  be  found 
in  a  three-dimensional  space,  this  three-dimensional  space 
extending  not  at  all  in  the  fourth  dimension,  but  up  and 
down,  right  and  left,  far  and  near. 

We  can  see  what  should  replace  the  light  yellow  face  of 
ro,  when  the  section  r\  comes  in,  by  looking  at  the  cube 
60,  fig.  107.  What  is  distant  in  it  one-quarter  of  an  inch 
from  the  light  yellow  face  in  the  red  direction  ?  It  is  an 
ochre  section  with  orange  and  pink  lines  and  red  points  ; 
see  also  fig.  103. 

This  square  then  forms  the  top  square  of  r\.  Now  we 
can  determine  the  nomenclature  of  all  the  regions  of  r\  by 
considering  what  would  be  formed  by  the  motion  of  this 
square  along  a  blue  axis. 

But  we  can  adopt  another  plan.  Let  us  take  a  hori- 
zontal section  of  ro,  and  finding  that  section  in  the  figures, 
of  fig.  107  or  fig.  103,  from  them  determine  what  will 
replace  it,  going  on  in  the  red  direction. 

A  section  of  the  ro  cube  has  green,  light  blue,  green, 
light  blue  sides  and  blue  points. 

Now  this  square  occurs  on  the  base  of  each  of  the 
section  figures,  61,  b2,  etc.  In  them  we  see  that  £  inch  in 
the  red  direction  from  it  lies  a  section  with  brown  and 
light  purple  lines  and  purple  corners,  the  interior  being 
of  light  brown.  Hence  this  is  the  nomenclature  of  the 
section  which  in  n  replaces  the  section  of  r0  made  from  a 
point  along  the  blue  axis. 

Hence  the  colouring  as  given  can  be  derived. 

We  have  thus  obtained  a  perfectly  named  group  of 
tesseract s.  We  can  take  a  group  of  eighty-one  of  them 
3x3x3x3,  in  four  dimensions,  and  each  tesseract  will 
have  its  name  null,  red,  white,  yellow,  blue,  etc.,  and 


188 


THE   FOURTH   DIMENSION 


whatever  cubic  view  we  take 
what  sides  of  the  tesseracts 
they  touch  each  other.* 

Thus,  for  instance,  if  we 
shown  below,  we  can  ask  how 

In  the  arrangement  given 
white,  red,  yellow,  in  space, 
dimension.      Hence  we  have 
Imagine  now  the  tesseractic 
our  space — we  have  first  of 


of  them  we  can  say  exactly 
we  are  handling,  and  how 

have  the  sixteen   tesseracts 
does  null  touch  blue, 
in  fig.  Ill  we  have  the  axes 
blue  running  in  the  fourth 

the  ochre   cubes  as  bases, 
group  to  pass  transverse  to 

all  null  ochre  cube,  white 


•O'J 
<l>    (J 

os  £ 


cj 

\°c/JrX 

White 

\ 
\ 

\ 

\        \ 

Red 

Pink 

Null 

X 

White 

r. 
a 

\    -\Ligl 

it  brown 
White 

\ 
\ 
\ 

\        \ 

Purple 

Light 
purple 

Blue 

Light 
blue 

direction 


yellow  hidden 
A,b0 


Light  green 
!B 


Fig.  111. 


ochre  cube,  etc.;  these  instantly  vanish,  and  we  get  the 
section  shown  in  the  middle  cube  in  fig.  103,  and  finally, 
just  when  the  tesseract  block  has  moved  one  inch  trans- 
verse to  our  space,  we  have  null  ochre  cube,  and  then 
immediately  afterwards  the  ochre  cube  of  blue  comes  in. 
Hence  the  tesseract  null  touches  the  tesseract  blue  by  its 
ochre  cube,  which  is  in  contact,  each  and  every  point 
of  it,  with  the  ochre  cube  of  blue.  / 

How  does  null  touch  white,  we  may  ask  ?     Looking  at 
the   beginning   A,   fig.    Ill,   where   we   have  the   ochre 

*  At  this  point  the  reader  will  find  it  advantageous,  if  he  has  the 
models,  to  go  through  the  manipulations  described  in  the  appendix. 


REMARKS  ON  THE  FIGURES 


180 


Cubes,  we  see  that  null  ochre  touches  white  ochre  by  an 
orange  face.  Now  let  us  generate  the  null  and  white 
tesseracts  by  a  motion  in  the  blue  direction  of  each  of 
these  cubes.  Each  of  them  generates  the  corresponding 
tesseract,  and  the  plane  of  contact  of  the  cubes  generates 
the  cube  by  which  the  tesseracts  are  in  contact.  Now  an 
orange  plane  carried  along  a  blue  axis  generates  a  brown 
cube.  Hence  null  touches  white  by  a  brown  cube. 

If  we  ask  again  how  red  touches  light  blue  tesseract, 
let    us  rearrange  our  group,  fig.   112,  or  rather  turn  it 


White 

\ 

e. 

\ 

'o 

\Pfnk\ 

•3 

1 

« 

X     \ 

Light 
purple 

Light 
blue 

axis 

^,        White 
"Xp  p       hidden 

•o 

V 

OH 

c..v 

0 

4) 

E 

'•5 

§r 

-^ 

§ 
4 

^> 

r> 

^ 

1 

\P^X 

s 

1 
J 

~S   x 

Light 
Brown 

(^> 

-^ 

5 

Light 
green 

White 
direction 


Light  yellow 
Bidden 


Fig.  112. 


about  so  that  we  have  a  different  space  view  of  it ;  let 
the  red  axis  and  the  white  axis  run  up  and  right,  and  let 
the  blue  axis  come  in  space  towards  us,  then  the  yellow 
axis  runs  in  the  fourth  dimension.  We  have  then  two 
blocks  in  which  the  bounding  cubes  of  the  tesseracts  are 
given,  differently  arranged  with  regard  to  us — the  arrange- 
ment is  really  the  same,  but  it  appears  different  to  us. 
Starting  from  the  plane  of  the  red  and  white  axes  we 
have  the  four  squares  of  the  null,  white,  red,  pink  tesseracts 
as  shown  in  A,  on  the  red,  white  plane,  unaltered,  only 
from  them  now  comes  out  towards  us  the  blue  axis. 


190  THE  FOURTH 

Hence  we  have  null,  white,  red,  pink  tesseracts  in  contact 
with  our  space  by  their  cubes  which  have  the  red,  white, 
blue  axis  in  them,  that  is  by  the  light  purple  cubes. 
Following  on  these  four  tesseracts  we  have  that  which 
comes  next  to  them  in  the  blue  direction,  that  is  the 
four  blue,  light  blue,  purple,  light  purple.  These  are 
likewise  in  contact  with  our  space  by  their  light  purple 
cubes,  so  we  see  a  block  as  named  in  the  figure,  of  which 
each  cube  is  the  one  determined  by  the  red,  white,  blue, 
axes. 

The  yellow  line  now  runs  out  of  space ;  accordingly  one 
inch  on  in  the  fourth  dimension  we  come  to  the  tesseracts 
which  follow  on  the  eight  named  in  C,  fig.  112,  in  the 
yellow  direction. 

These  are  shown  in  C.yu  fig.  112.  Between  figure  C 
and  C.yi  is  that  four-dimensional  mass  which  is  formed 
by  moving  each  of  the  cubes  in  C  one  inch  in  the  fourth 
dimension — that  is,  along  a  yellow  axis ;  for  the  yellow 
axis  now  runs  in  the  fourth  dimension. 

In  the  block  C  we  observe  that  red  (light  purple 
cube)  touches  light  blue  (light  purple  cube)  by  a  point. 
Now  these  two  cubes  moving  together  remain  in  contact 
during  the  period  in  which  they  trace  out  the  tesseracts 
red  and  light  blue.  This  motion  is  along  the  yellow 
axis,  consequently  red  and  light  blue  touch  by  a  yellow 
line. 

We  have  seen  that  the  pink  face  moved  in  a  yellow 
direction  traces  out  a  cube ;  moved  in  the  blue  direction  it 
also  traces  out  a  cube.  Let  us  ask  what  the  pink  face 
will  trace  out  if  it  is  moved  in  a  direction  within  the 
tesseract  lying  equally  between  the  yellow  and  blue 
directions.  What  section  of  the  tesseract  will  it  make  ? 

We  will  first  consider  the  red  line  alone.  Let  us  take 
a  cube  with  the  red  line  in  it  and  the  yellow  and  blue 
axes. 


ON   THE   FIGURES 


191 


Red 


Yellow 


Blue     Null 


The  cube  with  the  yellow,  red,  blue  axes  is  shown  in 
fig.  113.  If  the  red  line  is 
moved  equally  in  the  yellow  and 
in  the  blue  direction  by  four 
equal  motions  of  £  inch  each,  it 
takes  the  positions  11,  22,  33, 
and  ends  as  a  red  line. 

Now,  the  whole  of  this   red, 
yellow,  blue,  or  brown  cube  ap- 
j'ig  liy  pears  as  a  series  of  faces  on  the 

successive  sections  of  the  tes- 
seract starting  from  the  ochre  cube  and  letting  the  blue 
axis  run  in  the  fourth  dimension.  Hence  the  plane 
traced  out  by  the  red  line  appears  as  a  series  of  lines  in 
the  successive  sections,  in  our  ordinary  way  of  representing 
the  tesseract;  these  lines  are  in  different  places  in  each 
successive  section. 


Yello 


Nufo  White   ' 
bo 


Fig.  114. 


Thus  drawing  our  initial  cube  and  the  successive 
sections,  calling  them  b0,  61,  63?  63,  &4,  fig-  115,  we  have 
the  red  line  subject  to  this  movement  appearing  in  the 
positions  indicated. 

We  will  now  investigate  what  positions  in  the  tesseract 
another  line  in  the  pink  face  assumes  when  it  is  moved  in 
a  similar  manner. 

Take  a  section  of  tha  original  cube  containing  a  vertical 
line,  4,  in  the  pink  plane,  fig.  115.  We  have,  in  the 
section,  the  yellow  direction,  but  not  the  blue. 


192 


FOtfRTft 


From  this  section  a  cube  goes  off  in  the  fourth  dimen- 
sion, which  is  formed  by  moving  each  point  of  the  section 
in  the  blue  direction. 


Yellow 


Null     White 
FiR.  115. 


Light  blue    White 
Fig.  11H. 


Drawing  this  cube  we  have  fig.  116. 

Now  this  cube  occurs  as  a  series  of  sections  in  our 
original  representation  of  the  tesseract.  Taking  four  steps 
as  before  this  cube  appears  as  the  sections  drawn  in  60,  b\, 
b-2,  b3,  64,  fig.  117,  and  if  the  line  4  is  subjected  to  a 
movement  equal  in  the  blue  and  yellow  directions,  it  will 
occupy  the  positions  designated  by  4,  4^  42,  43,  44. 


Fig.  117. 

Hence,  reasoning  in  a  similar  manner  about  every  line, 
it  is  evident  that,  moved  equally  in  the  blue  and  yellow 
directions,  the  pink  plane  will  trace  out  a  space  which  is 
shown  by  the  series  of  section  planes  represented  in  the 
diagram. 

Thus  the  space  traced  out  by  the  pink  face,  if  it  is 
moved  equally  in  the  yellow  and  blue  directions,  is  repre- 
sented by  the  set  of  planes  delineated  in  Fig.  118,  pink 


REMARKS   ON   TfiE   FIGURES 


fade  or  0,  then  1,  2,  3,  and  finally  pink  face  or  4.  This 
solid  is  a  diagonal  solid  of  the  tesseract,  running  from  a 
pink  face  to  a  pink  face.  Its  length  is  the  length  of  the 
diagonal  of  a  square,  its  side  is  a  square. 

Let  us  now  consider  the  unlimited  space  which  springs 
from  the  pink  face  extended. 

This  space,  if  it  goes  off  in  the  yellow  direction,  gives 
us  in  it  the  ochre  cube  of  the  tesseract.  Thus,  if  we  have 
the  pink  face  given  and  a  point  in  the  ochre  cube,  we 
have  determined  this  particular  space. 

Similarly  going  off  from  the  pink  face  in  the  blue 
direction  is  another  space,  which  gives  us  the  light  purple 
cube  of  the  tesseract  in  it.  And  any  point  being  taken  in 


Null  b0 


the  light  purple  cube,  this  space  going  off  from  the  pink 
face  is  fixed. 

The  space  we  are  speaking  of  can  be  conceived  as 
swinging  round  the  pink  face,  and  in  each  of  its  positions 
it  cuts  out  a  solid  figure  from  the  tesseract,  one  of  which 
we  have  seen  represented  in  fig.  118. 

Each  of  these  solid  figures  is  given  by  one  position  of 
the  swinging  space,  and  by  one  only.  Hence  in  each  of 
them,  if  one  point  is  taken,  the  particular  one  of  the 
slanting  spaces  is  fixed.  Thus  we  see  that  given  a  plane 
and  a  point  out  of  it  a  space  is  determined. 

Now,  two  points  determine  a  line. 

Again,  think  of  a  line  and  a  point  outside  it.  Imagine 
a  plane  rotating  round  the  line.  At  some  time  in  its 
rotation  it  passes  through  the  point.  Thus  a  line  and  a 

13 


194 


THE   FOURTH   DIMENSION 


point,  or  three  points,  determine  a  plane*  And  finally 
four  points  determine  a  space.  We  have  seen  that  a 
plane  and  a  point  determine  a  space,  and  that  three 
points  determine  a  plane ;  so  four  points  will  determine 
a  space. 

These  four  points  may  be  any  points,  and  we  can  take, 
for  instance,  the  four  points  at  the  extremities  of  the  red, 
white,  yellow,  blue  axes,  in  the  tesseract.  These  will 
determine  a  space  slanting  with  regard  to  the  section 
spaces  we  have  been  previously  considering.  This  space 
will  cut  the  tesseract  in  a  certain  figure. 

One  of  the  simplest  sections  of  a  cube  by  a  plane  is 
that  in  which  the  plane  passes  through  the  extremities 
of  the  three  edges  which  meet  in  a  point.  We  see  at 
once  that  this  plane  would  cut  the  cube  in  a  triangle,  but 
we  will  go  through  the  process  by  which  a  plane  being 
would  most  conveniently  treat  the  problem  of  the  deter- 
mination of  this  shape,  in  order  that  we  may  apply  the 
method  to  the  determination  of  the  figure  in  which  a 
space  cuts  a  tesseract  when  it  passes  through  the  4 
points  at  unit  distance  from  a  corner. 

We  know  that  two  points  determine  a  line,  three  points 
determine  a  plane,  and  given  any  two  points  in  a  plane 
the  line  between  them  lies  wholly  in  the  plane. 

Let  now  the  plane  being  study  the  section  made  by 

a  plane  parsing  through  the 
null  r,  null  wh,  and  null  y 
points,  fig.  119.  Looking  at 
the  orange  square,  which,  as 
usual,  we  suppose  to  be  ini- 
tially in  his  plane,  he  seen 
that  the  line  from  null  r  to 
null  y,  which  is  a  line  in  the 


Nully. 


Null-wh. 


Null       A 
Fig.  119. 

section    plane,   the   plane,   namely,  through    the    three 
extremities  of  the  edges  meeting  in  null,  cuts  the  orange 


fcEMAfcKS  ON   THE   fIGUfcES  196 

face  in  an  orange  line  with  null  points.  This  then  is  one 
of  the  boundaries  of  the  section  figure. 

Let  now  the  cube  be  so  turned  that  the  pink  face 
comes  in  his  plane.  The  points  null  r  and  null  ivh 
are  now  visible.  The  line  between  them  is  pink 
with  null  points,  and  since  this  line  is  common  to 
the  surface  of  the  cube  and  the  cutting  plane,  it  is 
a  boundary  of  the  figure  in  which  the  plane  cuts  the 
cube. 

Again,  suppose  the  cube  turned  so  that  the  light 
yellow  face  is  in  contact  with  the  plane  being's  plane. 
He  sees  two  points,  the  null  ivh  and  the  null  y.  The 
line  between  these  lies  in  the  cutting  plane.  Hence, 
since  the  three  cutting  lines  meet  and  enclose  a  portion 
of  the  cube  between  them,  he  has  determined  the 
figure  he  sought.  It  is  a  triangle  with  orange,  pink, 
and  light  yellow  sides,  all  equal,  and  enclosing  an 
ochre  area. 

Let  us  now  determine  in  what  figure  the  space, 
determined  by  the  four  points,  null  r,  null  y,  null 
wh,  null  b,  cuts  the  tesseract.  We  can  see  three 
of  these  points  in  the  primary  position  of  the  tesseract 
resting  against  our  solid  sheet  by  the  ochre  cube. 
These  three  points  determine  a  plane  which  lies  in 
the  space  we  are  considering,  and  this  plane  cuts 
the  ochre  cube  in  a  triangle,  the  interior  of  which 
is  ochre  (fig.  119  will  serve  for  this  view),  with  pink, 
light  yellow  and  orange  sides,  and  null  points.  Going 
in  the  fourth  direction,  in  one  sense,  from  this  plane 
we  pass  into  the  tesseract,  in  the  other  sense  we  pass 
away  from  it.  The  whole  area  inside  the  triangle  is 
common  to  the  cutting  plane  we  see,  and  a  boundary 
of  the  tesseract.  Hence  we  conclude  that  the  triangle 
drawn  is  common  to  the  tesseract  and  the  cutting 
space. 


196 


FOURTH  DIMENSION 


Red 


Now  let  the  ochre  cube  turn  out  and  the  brown  cube 

come  in.  The  dotted  lines 
show  the  position  the  ochre 
cube  has  left  (fig.  120). 

Here   we    see    three    out 
j    of  the  four  points  through 
1*J    which    the     cutting     plane 
passes,   null  r,  null  y,  and 
null    b.     The    plane    they 
cutting    space,    and    this    plane 


*\ 

gr- 

or. 

pur. 

A 

ellow 


Null- b.  Blue     Null 
Fig.  120. 

determine  lies  in  the 
cuts  out  of  the  brown  cube  a  triangle  with  orange, 
purple  and  green  sides,  and  null  points.  The  orange 
line  of  this  figure  is  the  same  as  the  orange  line  in 
the  last  figure. 

Now  let  the  light  purple  cube  swing  into  our  space, 
towards  us,  fig.  121.' 

The  cutting  space  which  passes  through  the  four  points, 

null  r,  y,  wh,  b,  passes  through 
the  null  r,  wh,  b,  and  there- 
fore the  plane  these  determine 
lies  in  the  cutting  space. 

This  triangle  lies  before  us. 
It  has  a  light  purple  interior 
and  pink,  light  blue,  and 
purple  edges  with  null  points. 
This,  since  it  is  all  of  the 
plane  that  is  common  to  it,  and  this  bounding  of  the 
tesseract,  gives  us  one  of  the  bounding  faces  of  our  sec- 
tional figure.  The  pink  line  in  it  is  the  same  as  the 
pink  line  we  found  in  the  first  figure — that  of  the  ochre 
cube. 

Finally,  let  the  tesseract  swing  about  the  light  yellow 
plane,  so  that  the  light  green  cube  comes  into  our  space. 
It  will  point  downwards. 

The  three  points,  n.y,  n.wh,  n.b,  are  in  the  cutting 


Nullb. 
Fig.  121. 


REMARKS   ON   THE   FIGURES 


197 


Null 


Null-wb. 


pur-n.r.pur. 


space,  and   the   triangle   they  determine  is   common  to 

the  tesseract  and  the  cut- 
ting space.  Hence  this 
boundary  is  a  triangle  hav- 
ing a  light  yellow  line, 
which  is  the  same  as  the 
light  yellow  line  of  the  first 
figure,  a  light  blue  line  and 
a  green  line. 

We  have  now  traced  the 
cutting  space  between  every 

NHll'b:  set   of   three    that    can   be 

made  out  of  the  four  points 

in  which  it  cuts  the  tesseract,  and  have  got  four  faces 
which  all  join  on  to  each  other  by  fines. 

The  triangles  are  shown  in  fig.  123  as  they  join  on  to 
the  triangle  in  the  ochre  cube.  But 
they  join  on  each  to  the  other  in  an 
exactly  similar  manner;  their  edges 
are  all  identical  two  and  two.  They 
form  a  closed  figure,  a  tetrahedron, 
enclosing  a  light  brown  portion  which 
is  the  portion  of  the  cutting  space 
which  lies  inside  the  tesseract. 

We  cannot  expect  to  see  this  light  brown  portion,  any 
more  than  a  plane  being  could  expect  to  see  the  inside 
of  a  cube  if  an  angle  of  it  were  pushed  through  his 
plane.  All  he  can  do  is  to  come  upon  the  boundaries 
of  it  in  a  different  way  to  that  in  which  he  would  if  it 
passed  straight  through  his  plane. 

Thus  in  this  solid  section  ;  the  whole  interior  lies  per- 
fectly open  in  the  fourth  dimension.  G-O  round  it  as 
we  may  we  are  simply  looking  at  the  boundaries  of  the 
tesseract  which  penetrates  through  our  solid  sheet.  If 
the  tess^rapt  were  not  to  pass  across  so  far,  tl^e  triangle 


198 


THE    FOURTH    DIMENSION 


Null-y. 


P.: 


Null 
Fig.  124. 


Null-wh. 


would  be  smaller ;  if  it  were  to  pass  farther,  we  should 
have  a  different  figure,  the  outlines  of  which  can  be 
determined  in  a  similar  manner. 

The  preceding  method  is  open  to  the  objection  that 
it  depends  rather  on  our  inferring  what  must  be,  than 
our  seeing  what  is.  Let  us  therefore  consider  our 
sectional  space  as  consisting  of  a  number  of  planes,  each 
very  close  to  the  last,  and  observe  what  is  to  be  found 
in-each  plane. 

The  corresponding  method  in  the  case  of  two  dimen- 
sions is  as  follows  : — The  plane 
being  can  see  that  line  of  the 
sectional  plane  through  null  y, 
null  iv,  null  r,  which  lies  in 
the  orange  plane.  Let  him 
now  suppose  the  cube  and  the 
section  plane  to  pass  half  way 
through  his  plane.  Replacing 
the  red  and  yellow  axes  are  lines  parallel  to  them,  sections 
of  the  pink  and  light  .yellow  faces. 

Where  will  the  section  plane  cut  these  parallels  to 
the  red  and  yellow  axes? 

Let  him  suppose  the  cube,  in  the  position  of  the 
drawing,  fig.  124,  turned  so  that  the  pink  face  lies 
against  his  plane.  He  can  see  the  line  from  the  null  r 
point  to  the  null  ivh  point,  and  can  see  (compare  fig.  119) 
that  it  cuts  A&  a  parallel  to  his  red  axis,  drawn  at  a  point 
half  way  along  the  white  line,  in  a  point  B,  half  way  up. 
I  shall  speak  of  the  axis  as  having  the  length  of  an  edge 
of  the  cube.  Similarly,  by  letting  the  cube  turn  so  that 
the  light  yellow  square  swings  against  his  plane,  he  can 
see  (compare  fig.  119)  that  a  parallel  to  his  yellow  axis 
drawn  from  a  point  half-way  along  the  white  axis,  is  cut 
at  half  its  length  by  the  trace  pf  the  section  plane  in  the 
light  yellow  facer 


REMARKS    ON   THE    felGURES  199 

Hence  when  the  cube  had  passed  half-way  through  he 
would  have — instead  of  the  orange  line  with  null  points, 
which  he  had  at  first — an  ochre  line  of  half  its  length, 
with  pink  and  light  yellow  points.  Thus,  as  the  cube 
passed  slowly  through  his  plane,  he  would  have  a  suc- 
cession of  lines  gradually  diminishing  in  length  and 
forming  an  equilateral  triangle.  The  whole  interior  would 
be  ochre,  the  line  from  which  it  started  would  be  orange. 
The  succession  of  points  at  the  ends  of  the  succeeding 
lines  would  form  pink  and  light  yellow  lines  and  the 
final  point  would  be  null.  Thus  looking  at  the  successive 
lines  in  the  section  plane  as  it  and  the  cube  passed  across 
his  plane  he  would  determine  the  figure  cut  out  bit 
by  bit. 

Coming  now  to  the  section  of  the  tesseract,  let  us 
imagine  that  the  "tesseract  and  its  cutting  space  pass 
slowly  across  our  space ;  we  can  examine  portions  of  it, 
and  their  relation  to  portions  of  tlie  cutting  space.  Take 
the  section  space  which  passes  through  the  four  points, 
null  r,  wh,  y,b;  we  can  see  in  the  ochre  cube  (fig.  119) 
the  plane  belonging  to  this  section  space,  which  passes 
through  the  three  extremities  of  the  red,  white,  yellow 
axes. 

Now  let  the  tesseract  pass  half  way  through  our  space. 
Instead  of  our  original  axes  we  have  parallels  to  them, 
purple,  light  blue,  and  green,  each  of  the  same  length  as 
the  first  axes,  for  the  section  of  the  tesseract  is  of  exactly 
the  same  shape  as  its  ochre  cube. 

But  the  sectional  space  seen  at  this  stage  of  the  trans- 
ference would  not  cut  the  section  of  the  tesseract  in  a 
plane  disposed  as  at  first. 

To  see  where  the  sectional  space  would  cut  these 
parallels  to  the  original  axes  let  the  tesseract  swing  so 
that,  the  orange  face  remaining  stationary,  the  blue  line 
cpmes  in  to  the  left. 


200 


THE    FOURTH   DIMENSION 


Null-b.  Blue 
Fi 


. _ 


Here  (fig.  125)  we  have  the  null  r,  y,  b  points,  and  of 

the  sectional  space  all  we 
see  is  the  plane  through  these 
three  points  in  it. 

In  this  figure  we  can  draw 
the  parallels  to  the  red  and 
yellow  axes  and  see  that,  if 
they  started  at  a  point  half 
way  along  the  blue  axis,  they 

would  each  be  cut  at  a  point  so  as  to  be  half  of  their 

previous  length. 

Swinging  the  tesseract  into  our  space  about  the  pink 

face  of  the  ochre  cube  we  likewise  find  that  the  parallel 

to  the  white  axis  is  cut  at  half  its  length  by  the  sectional 

space. 

Hence  in  a  section  made  when  the  tesseract  had  passed 

half  across  our  space  the  parallels  to  the  red,  white,  yellow 

axes,  which  are  now  in  our 
space,  are  cut  by  the  section 
space,  each  of  them  half  way 
along,  and  for  this  stage  of 
the  traversing  motion  we 
should  have  fig.  126.  The 


Blue    L.blue       bl. 
Section  bg  interior  Light  brown 
Fig.  126. 


section  made  of  this  cube  by 
the  plane  in  which  the  sec- 
tional space  cuts  it,  is  an 
equilateral  triangle  with  purple,  1.  blue,  green  points,  and 
1.  purple,  brown,  1.  green  lines. 

Thus  the  original  ochre  triangle,  with  null  points  and 
pink,  orange,  light  yellow  lines,  would  be  succeeded  by  a 
triangle  coloured  in  manner  just  described. 

This  triangle  would  initially  be  only  a  very  little  smaller 
than  the  original  triangle,  it  would  gradually  diminish, 
until  it  ended  in  a  point,  a  null  point.  Each  of  its 
edges  would  be  of  the  same  length.  Thus  the  successive 


REMARKS   ON   THE   FIGURES 


201 


sections  of  the  successive  planes  into  which  we  analyse  the 
cutting  space  would  be  a  tetrahedron  of  the  description 
shown  (fig.  123),  and  the  whole  interior  of  the  tetrahedron 
would  be  light  brown. 


Front  view. 


The  rear  faces. 


Fig.  127. 


In  fig.  127  the  tetrahedron  is  represented  by  means  of 
its  faces  as  two  triangles  which  meet  in  the  p.  line,  and 
two  rear  triangles  which  join  on  to  them,  the  diagonal 
of  the  pink  face  being  supposed  to  run  vertically 
upward. 

We  have  now  reached  a  natural  termination.  The 
reader  may  pursue  the  subject  in  further  detail,  but  will 
find  no  essential  novelty.  I  conclude  with  an  indication 
as  to  the  manner  in  which  figures  previously  given  may 
be  used  in  determining  sections  by  the  method  developed 
above. 

Applying  this  method  to  the  tesseract,  as  represented 
in  Chapter  IX.,  sections  made  by  a  space  cutting  the  axes 
equidistantly  at  any  distance  can  be  drawn,  and  also  the 
sections  of  tesseracts  arranged  in  a  block. 

If  we  draw  a  plane,  cutting  all  four  axes  at  a  point 
six  units  distance  from  null,  we  have  a  slanting  space. 
Jhjs  space  cuts  the  red,  white,  yellow  axes  in  the 


-          :5-    .-- 


nd  the 

to  our  spaee,  a 
11  a  section  of  the 
to  the 

"of  fire 

pninti  in 

the  cubical  wtioo  of  the 
predrawn.  In  2* (fig. 72)  the 
to  the  axes  at  a  distance 
iathe  wetTO 
br  it.  FinaOj  when  3& 
fjiy  jtxes  at  a  distance 
tnsngle,  of  which 
the  hfii&iiB  drawn  is  a  truncated  portion.  After  ttia 
the  teaKnet,  whidi  I'lTiin1!  rnfy  three  mnU  in  each  of 

of  ov  ffiacevaad  there  if  no  more  of  it  to  be  eat.    Hence, 
we  hare  the  section  determined  by  the  j«tkoiar  slanting 


CHAPTER  XIV.* 


07 


-i- 


(fee  it  O» 


:ir 


tlte. 


-  -    -          -      ^      -      - 

•dk    VBfr  ••m^BBB^t  *fc 


204  THE    FOURTH    DIMENSION 

visibly  moves.  Every  material  body  must  slip  or  slide 
along  this  sheet,  not  deviating  from  contact  with  it  in 
any  motion  which  we  can  observe. 

The  necessity  for  this  assumption  is  clearly  apparent,  if 
we  consider  the  analogous  case  of  a  suppositionary  plane 
world.  If  there  were  any  creatures  whose  experiences 
were  confined  to  a  plane,  we  must  account  for  their 
limitation.  If  they  were  free  to  move  in  every  space 
direction,  they  would  have  a  three-dimensional  motion ; 
hence  they  must  be  physically  limited,  and  the  only  way 
in  which  we  can  conceive  such  a  limitation  to  exist  is  by 
means  of  a  material  surface  against  which  they  slide. 
The  existence  of  this  surface  could  only  be  known  to 
them  indirectly.  It  does  not  lie  in  any  direction  from 
them  in  which  the  kinds  of  motion  they  know  of  leads 
them.  If  it  were  perfectly  smooth  and  always  in  contact 
with  every  material  object,  there  would  be  no  difference  in 
their  relations  to  it  which  would  direct  their  attention  to  it. 

But  if  this  surface  were  curved — if  it  were,  say,  in  the 
form  of  a  vast  sphere — the  triangles  they  drew  would 
really  be  triangles  of  a  sphere,  and  when  these  triangles 
are  large  enough  the  angles  diverge  from  the  magnitudes 
they  would  have  for  the  same  lengths  of  sides  if  the 
surface  were  plane.  Hence  by  the  measurement  of 
triangles  of  very  great  magnitude  a  plane  being  might 
detect  a  difference  from  the  laws  of  a  plane  world  in  his 
physical  world,  and  so  be  led  to  the  conclusion  that  there 
was  in  reality  another  dimension  to  space — a  third 
dimension — as  well  as  the  two  which  his  ordinary  experi- 
ence made  him  familiar  with. 

Now,  astronomers  have  thought  it  worth  while  to 
examine  the  measurements  of  vast  triangles  drawn  from 
one  celestial  body  to  another  with  a  view  to  determine  if 
there  is  anything  like  a  curvature  in  our  space — that  is  to 
say,  they  have  tried  astronomical  measurements  to  find 


RECAPITULATION   AND   EXTENSION  205 

out  if  the  vast  solid  sheet  against  which,  on  the  sup- 
position of  a  fourth  dimension,  everything  slides  is 
curved  or  not.  These  results  have  been  negative.  The 
solid  sheet,  if  it  exists,  is  not  curved  or,  being  curved,  has 
not  a  sufficient  curvature  to  cause  any  observable  deviation 
from  the  theoretical  value  of  the  angles  calculated. 

Hence  the  examination  of  the  infinitely  great  leads  to 
no  decisive  criterion.  If  it  did  we  should  have  to  decide 
between  the  present  theory  and  that  of  metageometry. 

Coming  now  to  the  prosecution  of  the  inquiry  in  the 
direction  of  the  infinitely  small,  we  have  to  state  the 
question  thus  :  Our  laws  of  movement  are  derived  from 
the  examination  of  bodies  which  move  in  three-dimensional 
space.  All  our  conceptions  are  founded  on  the  sup- 
position of  a  space  which  is  represented  analytically  by 
three  independent  axes  and  variations  along  them — that 
is,  it  is  a  space  in  which  there  are  three  independent 
movements.  Any  motion  possible  in  it  can  be  compounded 
out  of  these  three  movements,  which  we  may  call :  up, 
right,  away. 

To  examine  the  actions  of  the  very  small  portions  of 
matter  with  the  view  of  ascertaining  if  there  is  any 
evidence  in  the  phenomena  for  the  supposition  of  a  fourth 
dimension  of  space,  we  must  commence  by  clearly  defining 
what  the  laws  of  mechanics  would  be  on  the  supposition 
of  a  fourth  dimension.  It  is  of  no  use  asking  if  the 
phenomena  of  the  smallest  particles  of  matter  are  like — 
we  do  not  know  what.  We  must  have  a  definite  con- 
ception of  what  the  laws  of  motion  would  be  on  the 
supposition  of  the  fourth  dimension,  and  then  inquire  if 
the  phenomena  of  the  activity  of  the  smaller  particles  of 
matter  resemble  the  conceptions  which  we  have  elaborated. 
Now,  the  task  of  forming  these  conceptions  is  by  no 
means  one  to  be  lightly  dismissed.  Movement  in  space 
has  many  features  which  differ  entirely  from  movement 


206  THE  FOURTH   DIMENSION 

on  a  plane;  and  when  we  set  about  to  form  the  Con- 
ception of  motion  in  four  dimensions,  we  find  that  there 
is  at  least  as  great  a  step  as  from  the  plane  to  three- 
dimensional  space. 

I  do  not  say  that  the  step  is  difficult,  but  I  Want  to 
point  out  that  it  must  be  taken.  When  we  have  formed 
the  conception  of  four-dimensional  motion,  we  can  ask  a 
rational  question  of  Nature.  Before  we  have  elaborated 
our  conceptions  we  are  asking  if  an  unknown  is  like  an 
unknown — a  futile  inquiry. 

As  a  matter  of  fact,  four-dimensional  movements  are  in 
every  way  simple  and  more  easy  to  calculate  than  three- 
dimensional  movements,  for  four-dimensional  movements 
are  simply  two  sets  of  plane  movements  put  together. 

Without  the  formation  of  an  experience  of  four- 
dimensional  bodies,  their  shapes  and  motions,  the  subject 
can  be  but  formal — logically  conclusive,  not  intuitively 
evident.  It  is  to  this  logical  apprehension  that  I  must 
appeal. 

It  is  perfectly  simple  to  form  an  experiential  familiarity 
with  the  facts  of  four-dimensional  movement.  The 
method  is  analogous  to  that  which  a  plane  being  would 
have  to  adopt  to  form  an  experiential  familiarity  with 
three-dimensional  movements,  and  may  be  briefly 
summed  up  as  the  formation  of  a  compound  sense  by 
means  of  which  duration  is  regarded  as  equivalent  to 
extension. 

Consider  a  being  confined  to  a  plane.  A  square  enclosed 
by  four  lines  will  be  to  him  a  solid,  the  interior  of  which 
can  only  by  examined  by  breaking  through  the  lines. 
If  such  a  square  were  to  pass  transverse  to  his  plane,  it 
would  immediately  disappear.  It  would  vanish,  going  in 
no  direction  to  which  he  could  point. 

If,  now,  a  cube  be  placed  in  contact  with  his  plane,  its 
surface  of  contact  would  appear  like  the  square  which  we 


RECAPITULATION  AND  EXTENSION        207 

have  just  mentioned.  But  if  it  were  to  pass  transverse  to 
his  plane,  breaking  through  it,  it  would  appear  as  a  lasting 
square.  The  three-dimensional  matter  will  give  a  lasting 
appearance  in  circumstances  under  which  two-dimensional 
matter  will  at  once  disappear. 

Similarly,  a  four-dimensional  cube,  or,  as  we  may  call 
it,  a  tesseract,  which  is  generated  from  a  cube  by  a 
movement  of  every  part  of  the  cube  in  a  fourth  direction 
at  right  angles  to  each  of  the  three  visible  directions  in 
the  cube,  if  it  moved  transverse  to  our  space,  would 
appear  as  a  lasting  cube. 

A  cube  of  three-dimensional  matter,  since  it  extends  to 
no  distance  at  all  in  the  fourth  dimension,  would  instantly 
disappear,  if  subjected  to  a  motion  transverse  to  our  space. 
It  would  disappear  and  be  gone,  without  it  being  possible 
to  point  to  any  direction  in  which  it  had  moved. 

All  attempts  to  visualise  a  fourth  dimension  are  futile.  It 
must  be  connected  with  a  time  experience  in  three  space. 

The  most  difficult  notion  for  a  plane  being  to  acquire 
would  be  that  of  rotation  about  a  line.  Consider  a  plane 
being  facing  a  square.  If  he  were  told  that  rotation 
about  a  line  were  possible,  he  would  move  his  square  this 
way  and  that.  A  square  in  a  plane  can  rotate  about  a 
point,  but  to  rotate  about  a  line  would  seem  to  the  plane 
being  perfectly  impossible.  How  could  those  parts  of  his 
square  which  were  on  one  side  of  an  edge  come  to  the 
other  side  without  the  edge  moving  ?  He  could  under- 
stand their  reflection  in  the  edge.  He  could  form  an 
idea  of  the  looking-glass  image  of  his  square  lying  on  the 
opposite  side  of  the  line  of  an  edge,  but  by  no  motion 
that  he  knows  of  can  he  make  the  actual  square  assume 
that  position.  The  result  of  the  rotation  would  be  like 
reflection  in  the  edge,  but  it  would  be  a  physical  im- 
possibility to  produce  it  in  the  plane. 

The  demonstration  of  rotation  about  a  line  must  be  to 


208  THE   FOURTH   DIMENSION 

him  purely  formal.  If  he  conceived  the  notion  of  a  cube 
stretching  out  in  an  unknown  direction  away  from  his 
plane,  then  he  can  see  the  base  of  it,  his  square  in  the 
plane,  rotating  round  a  point.  He  can  likewise  apprehend 
that  every  parallel  section  taken  at  successive  intervals  in 
the  unknown  direction  rotates  in  like  manner  round  a 
point.  Thus  he  would  come  to  conclude  that  the  whole 
body  rotates  round  a  line  —  the  line  consisting  of  the 
succession  of  points  round  which  the  plane  sections  rotate. 
Thus,  given  three  axes,  x,  y,  z,  if  x  rotates  to  take  the 
the  place  of  y,  and  y  turns  so  as  to  point  to  negative  x, 
then  the  third  axis  remaining  unaffected  by  this  turning 
is  the  axis  about  which  the  rotation  takes  place.  This, 
then,  would  have  to  be  his  criterion  of  the  axis  of  a 
rotation  —  that  which  remains  unchanged  when  a  rotation 
of  every  plane  section  of  a  body  takes  place. 

There  is  another  way  in  which  a  plane  being  can  think 
about  three-dimensional  movements  ;  and,  as  it  affords 
the  type  by  which  we  can  most  conveniently  think  about 
four-dimensional  movements,  it  will  be  no  loss  of  time  to 
consider  it  in  detail. 

We  can  represent  the  plane  being  and  his  object  by 
figures  cut  out  of  paper,  which  slip  on  a  smooth  surface. 
The  thickness  of  these  bodies  must  be  taken  as  so  minute 
y  that  their  extension  in  the  third  di- 

mension escapes  the  observation  of  the 
plane  being,  and  he  thinks  about  them 
as  if  they  were  mathematical  plane 
5'  figures  in  a  plane  instead  of  being 
material  bodies  capable  of  moving  on 


A  B       x    a  plane  surface.      Let  Ax,  Ay  be  two 

Fig.  1  (129;.         axes   and   ABCD  a   square.      As  far  as 

movements  in  the  plane  are  concerned,  the  square  can 

rotate  about  a  point  A,  for  example.      It  cannot  rotate 

about  a  side,  such  as  AC. 


RECAPITULATION   AND   EXTENSION  209 

But  if  the  plane  being  is  aware  of  the  existence  of  a 
third  dimension  he  can  study  the  movements  possible  in 
the  ample  space,  taking  his  figure  portion  by  portion. 

His  plane  can  only  hold  two  axes.  But,  since  it  can 
hold  two,  he  is  able  to  represent  a  turning  into  the  third 
dimension  if  he  neglect  one  of  his  axes  and  represent  the 
third  axis  as  lying  in  his  plane.  He  can  make  a  drawing 
in  his  plane  of  what  stands  up  perpendicularly  from  his 
plane.  Let  AZ  be  the  axis,  which 
stands  perpendicular  to  his  plane  at 
A.  He  can  draw  in  his  plane  two 
lines  to  represent  the  two  axes,  Ax 
and  AZ.  Let  Fig.  2  be  this  draw- 
ing. Here  the  z  axis  has  taken 


^  8  the  place  of  the  y  axis,  and  the 

Fig.  2  (130).  plane  of  AX  AZ  is  represented  in  his 

plane.  In  this  figure  all  that  exists  of  the  square  ABCD 
will  be  the  line  AB. 

The  square  extends  from  this  line  in  the  y  direction, 
but  more  of  that  direction  is  represented  in  Fig.  2.  The 
plane  being  can  study  the  turning  of  the  line  AB  in  this 
diagram.  It  is  simply  a  case  of  plane  turning  around  the 
point  A.  The  line  AB  occupies  intermediate  portions  like  AB: 
and  after  half  a  revolution  will  lie  on  AX  produced  through  A. 

Now,  in  the  same  way,  the  plane  being  can  take 
another  point,  A',  and  another  line,  A'B',  in  his  square. 
He  can  make  the  drawing  of  the  two  directions  at  A',  one 
along  A'B',  the  other  perpendicular  to  his  plane.  He 
will  obtain  a  figure  precisely  similar  to  Fig.  2,  and  will 
see  that,  as  AB  can  turn  around  A,  so  A'(f  around  A. 

In  this  turning  AB  and  A'B'  would  not  interfere  with 
each  other,  as  they  would  if  they  moved  in  the  plane 
around  the  separate  points  A  and  A'. 

Hence  the  plane  being  would  conclude  that  a  rotation 
round  a  line  was  possible.  He  could  see  his  square  as  it 

14 


21 0  TttE   FOURTH   DIMENSION 

began  to  make  this  turning.  He  could  see  it  half  way 
round  when  it  came  to  lie  on  the  opposite  side  of  the  line 
AC.  But  in  intermediate  portions  he  could  not  see  it, 
for  it  runs  out  of  the  plane. 

Coming  now  to  the  question  of  a  four-dimensional  body, 
let  us  conceive  of  it  as  a  series  of  cubic  sections,  the  first 
in  our  space,  the  rest  at  intervals,  stretching  away  from 
our  space  in  the  unknown  direction. 

We  must  not  think  of  a  four-dimensional  body  as 
formed  by  moving  a  three-dimensional  body  in  any 
direction  which  we  can  see. 

Refer  for  a  moment  to  Fig.  3.  The  point  A,  moving  to 
the  right,  traces  out  the  line  AC.  The  line  AC,  moving 
away  in  a  new  direction,  traces  out  the  square  ACEG  at 
the  base  of  the  cube.  The  square  AEGC,  moving  in  a 
new  direction,  will  trace  out  the  cube  ACEGHDIIF.  The 
vertical  direction  of  this  last  motion  is  not  identical  with 
any  motion  possible  in  the  plane  of  the  base  of  the  cube. 
It  is  an  entirely  new  direction,  at  right  angles  to  every 
line  that  can  be  drawn  in  the  base.  To  trace  out  a 
tesseract  the  cube  must  move  in  a  new  direction — a 
direction  at  right  angles  to  any  and  every  line  that  can 
be  drawn  in  the  space  of  the  cube. 

The  cubic  sections  of  the  tesseract  are  related  to  the 
cube  we  see,  as  the  square  sections  of  the  cube  are  related 
to  the  square  of  its  base  which  a  plane  being  sees. 

Let  us  imagine  the  cube  in  our  space,  which  is  the  base 
of  a  tesseract,  to  turn  about  one  of  its  edges.  The  rotation 
will  carry  the  whole  body  with  it,  and  each  of  the  cubic 
sections  will  rotate.  The  axis  we  see  in  our  space  will 
remain  unchanged,  and  likewise  the  series  of  axes  parallel 
to  it  about  which  each  of  the  parallel  cubic  sections 
rotates.  The  assemblage  of  all  of  these  is  a  plane. 

Hence  in  four  dimensions  a  body  rotates  about  a  plane. 
There  is  no  such  thing  as  rotation  round  an  axis. 


RECAPITULATION  AND   EXTENSION 


211 


We  may  regard  the  rotation  from  a  different  point  of 
view.  Consider  four  independent  axes  each  at  right 
angles  to  all  the  others,  drawn  in  a  four-dimensional  body. 
Of  these,  four  axes  we  can  see  any  three.  The  fourth 
extends  normal  to  our  space. 

Rotation  is  the  turning  of  one  axis  into  a  second,  and 
the  second  turning  to  take  the  place  of  the  negative  of 
the  first.  It  involves  two  axes.  Thus,  in  this  rotation  of 
a  four-dimensional  body,  two  axes  change  and  two  remain 
at  rest.  Four-dimensional  rotation  is  therefore  a  turning 
about  a  plane. 

As  in  the  case  of  a  plane  being,  the  result  of  rotation 
about  a  line  would  appear  as  the  production  of  a  looking- 
glass  image  of  the  original  object  on  the  other  side  of  the 
line,  so  to  us  the  result  of  a  four-dimensional  rotation 
would  appear  like  the  production  of  a  looking-glass  image 
of  a  body  on  the  other  side  of  a  plane.  The  plane  -would 
be  the  axis  of  the  rotation,  and  the  path  of  the  body 
between  its  two  appearances  would  be  unimaginable  in 
three-dimensional  space. 

Let  us  now  apply  the  method  by  which  a  plane  being 
could  examine  the  nature  of  rota- 
tion about  a  line  in  our  examination 
of  rotation  about  a  plane.  Fig.  3 
represents  a  cube  in  our  space,  the 
three  axes  x,  y,  z  denoting  its 
three  dimensions.  Let  w  represent 
the  fourth  dimension.  Now,  since 
in  our  space  we  can  represent  any 
three  dimensions,  we  can,  if  we 


H 


B 


A  C 

Fig.  3  (131). 

choose,  make  a  representation  of  what  is  in  the  space 
determined  by  the  three  axes  x,  z,  w.  This  is  a  three- 
dimensional  space  determined  by  two  of  the  axes  we  have 
drawn,  x  and  z,  and  in  place  of  y  the  fourth  axis,  w.  We 
cannot,  keeping  x  and  z,  have  both  y  and  w  in  our  space ; 


B 


212  THE  FOURTH  DIMENSION 

so  we  will  let  y  go  and  draw  w  in  its  place.  What  will  be 
our  view  of  the  cube  ? 

Evidently  we  shall  have  simply  the  square  that  is  in 
the  plane  of  xz,  the  square  ACDB. 
The  rest  of  the  cube  stretches  in 
the  y  direction,  and,  as  we  have 
none  of  the  space  so  determined, 
we  have  only  the  face  of  the  cube. 
This  is  represented  in  fig.  4. 

Now,  suppose  the  whole  cube  to 
,  Fig.  4  (132).  be    turned   from  the  x  to  the  w 

direction.  Conformably  with  our  method,  we  will  not 
take  the  whole  of  the  cube  into  consideration  at  once,  but 
will  begin  with  the  face  ABCD. 

Let  this  face  begin  to  turn.  Fig.  5 
represents  one  of  the  positions  it  will 
occupy ;  the  line  AB  remains  on  the 
z  axis.  The  rest  of  the  face  extends 
between  the  x  and  the  w  direction. 

Now,  since  we  can  take  any  three 
axes,   let    us    look   at    what    lies   in 
Fig.  5  (133).  the  space  of  zyiv,  and  examine  the 

turning  there.  We  must  now  let  the  z  axis  disappear 
and  let  the  iv  axis  run  in  the  direction  in  which  the  z  ran. 
Making  this  representation,  what 
do  we  see  of  the  cube  ?  Obviously 

\we  bee  only  the  lower  face.    The  rest 
^  of  the  cube  lies  in  the  space  of  xyz. 

\  In  the  space  of  xyz  we  have  merely 


A  C          the  base  of  the  cube  lying  in  the 

Fig.  6  (134).  plane  of  xy,  as  shown  in  fig.  6. 

Now  let  the  x  to  w  turning  take  place.  The  square 
ACEG  will  turn  about  the  line  AE.  This  edge  will 
remain  along  the  y  axis  and  will  be  stationary,  however 
far  the  square  turns. 


BECAPITULAT10N   AND   EXTENSION 


213 


Thus,  if  the  cube  be  turned  by  an  x  to  w  turning,  both 
the  edge  AB  and  the  edge  AC  remain 
stationary  ;  hence  the  whole  face 
ABEF  in  the  yz  plane  remains  fixed. 
The  turning  has  taken  place  about 
the  face  ABEF. 

Suppose  this  turning  to   continue 
*  till    AC   runs    to    the    left    from    A. 
Fig.  7  (135).          r  occupy   the   position 


H 


shown  in  fig.  8.     This  is  the  looking-glass  image  of  the 
cube  in  fig.  3.     By  no  rotation  in  three-dimensional  .space 

can  the  cube  be  brought  from 
the  position  in  fig.  3  to  that 
shown  in  fig.  8. 

We  can  think  of  this  turning 
as  a  turning  of  the  face  ABCD 
about  AB,  and  a  turning  of  each 
section  parallel  to  ABCD  round 
the  vertical  line  in  which  it 
intersects  the  face  ABEF,  the 


2"?pQSiftcn .   Imposition 
Fig.  8  (13G> 


space  in  which  the  turning  takes  place  being  a  different 
one  from  that  in  which  the  cube  lies. 

One  of  the  conditions,  then,  of  our  inquiry  in  the 
direction  of  the  infinitely  small  is  that  we  form  the  con- 
ception of  a  rotation  about  a  plane.  The  production  of  a 
body  in  a  state  in  which  it  presents  the  appearance  of  a 
looking-glass  image  of  its  former  state  is  the  criterion 
for  a  four-dimensional  rotation. 

There  is  some  evidence  for  the  occurrence  of  such  trans- 
formations of  bodies  in  the  change  of  bodies  from  those 
which  produce  a  right>handed  polarisation  of  light  to 
those  which  produce  a  left-handed  polarisation;  but  this 
is  not  a  point  to  which  any  very  great  importance  can 
be  attached. 

Still,  in  this  connection,  let  me  quote  a  remark  from 


214  THE   FOURTH  DIMENSION 

Prof.  John  G.  McKendrick's  address  on  Physiology  before 
the  British  Association  at  Glasgow.  Discussing  the 
possibility  of  the  hereditary  production  of  characteristics 
through  the  material  structure  of  the  ovum,  he  estimates 
that  in  it  there  exist  12,000,000,000  biophors,  or  ultimate 
particles  of  living  matter,  a  sufficient  number  to  account 
for  hereditary  transmission,  and  observes :  "  Thus  it  is 
conceivable  that  vital  activities  may  also  be  determined 
by  the  kind  of  motion  that  takes  place  in  the  molecules 
of  that  which  we  speak  of  as  living  matter.  It  may  be 
different  in  kind  from  some  of  the  motions  known  to 
physicists,  and  it  is  conceivable  that  life  may  be  the 
transmission  to  dead  matter,  the  molecules  of  which  have 
already  a  special  kind  of  motion,  of  a  form  of  motion 
sui  generis." 

Now,  in  the  realm  of  organic  beings  symmetrical  struc- 
tures— those  with  a  right  and  left  symmetry — are  every- 
where in  evidence.  Granted  that  four  dimensions  exist, 
the  simplest  turning  produces  the  image  form,  and  by  a 
folding-over  structures  could  be  produced,  duplicated 
right  and  left,  just  as  is  the  case  of  symmetry  in  a 
plane. 

Thus  one  very  general  characteristic  of  the  forms  of 
organisms  could  be  accounted  for  by  the  supposition  that 
a  four-dimensional  motion  was  involved  in  the  process  of 
life. 

But  whether  four-dimensional  motions  correspond  in 
other  respects  to  the  physiologist's  demand  for  a  special 
kind  of  motion,  or  not,  I  do  not  know.  Our  business  is 
with  the  evidence  for  their  existence  in  physics.  For 
this  purpose  it  is  necessary  to  examine  into  the  signifi- 
cance of  rotation  round  a  plane  in  the  case  of  extensible 
and  of  fluid  matter. 

Let  us  dwell  a  moment  longer  on  the  rotation  of  a  rigid 
body.  Looking  at  the  cube  in  fig.  3,  which  turns  about 


RECAPITULATION  AND   EXTENSION 


215 


the  face  of  ABFE,  we  see  that  any  line  in  the  face  can 
take  the  place  of  the  vertical  and  horizontal  lines  we  have 
examined.  Take  the  diagonal  line  AF  and  the  section 
through  it  to  GH.  The  portions  of  matter  which  were  on 
one  side  of  AF  in  this  section  in  fig.  3  are  on  the 
opposite  side  of  it  in  fig.  8.  They  have  gone  round  the 
line  AF.  Thus  the  rotation  round  a  face  can  be  considered 
as  a  number  of  rotations  of  sections  round  parallel  lines 
in  it. 

The  turning  about  two  different  lines  is  impossible  in 
three-dimensional  space.  To  take  another  illustration, 
suppose  A  and  B  are  two  parallel  lines  in  the  xy  plane, 
and  let  CD  and  EF  be  two  rods  crossing  them.  Now,  in 
the  space  of  xyz  if  the  rods  turn  round  the  lines  A  and  B 

in  the  same  direction  they 
will  make  two  independent 
circles. 

When  the  end  F  is  goin^ 
down  the  end  c  will  be  coming 
up.  They  will  meet  and  con- 
flict. 

But  if  we  rotate  the  rods 
about  the  plane  of  AB  by  the 
z  to  w  rotation  these  move- 
ments will  not  conflict.  Sup- 
pose all  the  figure  removed 


B 


Fig.  9  (137). 


with  the  exception  of  the  plane  xz,  and  from  this  plane 
draw  the  axis  of  w,  so  that  we  are  looking  at  the  space 
of  xzw. 

Here,  fig.  10,  we  cannot  see  the  lines  A  and  B.  We 
see  the  points  G  and  H,  in  which  A  and  B  intercept 
the  x  axis,  but  we  cannot  see  the  lines  themselves,  for 
they  run  in  the  y  direction,  and  that  is  not  in  our 
drawing. 

Now,  if  the  rods  move  with  the  z  to  w  rotation  they  will 


216  THE   FOURTH   DIMENSION 

turn  in  parallel  planes,  keeping  their  relative  positions. 

The  point  D,  for  instance,  will 
describe  a  circle.  At  one  time 
it  will  be  above  the  line  A,  at 
another  time  below  it.  Hence 
it  rotates  round  A. 

Not  only   two  rods  but   any 

number   of    rods    crossing    the 
plane  will  move  round  it  har- 
moniously.    We   can   think  of 
w* 

this  rotation  by  supposing  the 

rods  standing  up  from  one  line 

to  move  round  that  line  and  remembering  that  it  is 
not  inconsistent  with  this  rotation  for  the  rods  standing 
up  along  another  line  also  to  move  round  it,  the  relative 
positions  of  all  the  rods  being  preserved.  Now,  if  the 
rods  are  thick  together,  they  may  represent  a  disk  of 
matter,  and  we  see  that  a  disk  of  matter  can  rotate 
round  a  central  plane. 

Rotation  round  a  plane  is  exactly  analogous  to  rotation 
round  an  axis  in  three  dimensions.  If  we  want  a  rod  to 
turn  round,  the  ends  must  be  free ;  so  if  we  want  a  disk 
of  matter  to  turn  round  its  central  plane  by  a  four-dimen- 
sional turning,  all  the  contour  must  be  free.  The  whole 
contour  corresponds  to  the  ends  of  the  rod.  Each  point 
of  the  contour  can  be  looked  on  as  the  extremity  of  an 
axis  in  the  body,  round  each  point  of  which  there  is  a 
rotation  of  the  matter  in  the  disk. 

If  the  one  end  of  a  rod  be  clamped,  we  can  twist  the 
rod,  but  not  turn  it  round ;  so  if  any  part  of  the  contour 
of  a  disk  is  clamped  we  can  impart  a  twist  to  the  disk, 
but  not  turn  it  round  its  central  plane.  In  the  case  of 
extensible  materials  a  long,  thin  rod  will  twist  round  its 
axis,  even  when  the  axis  is  curved,  as,  for  instance,  in  the 
case  of  a  ring  of  India  rubber. 


RECAPITULATION   AND   EXTENSION 


217 


In  an  analogous  manner,  in  four  dimensions  we  can  have 
rotation  round  a  curved  plane,  if  I  may  use  the  expression. 
A  sphere  can  be  turned  inside  out  in  four  dimensions. 

Let  fig.  11  represent  a 
spherical  surface,  on  each 
side  of  which  a  layer  of 
matter  exists.  The  thick- 
ness of  the  matter  is  rep- 
resented by  the  rods  CD  and 
EF,  extending  equally  with- 
'  out  and  within. 

Now,  take  the  section  of 
the  sphere  by  the  yz  plane 
we  have  a  circle — fig.  12. 
Now,  let  the  w  axis  be  drawn 
in  place  of  the  x  axis  so  that 


Fig.  ll  (139). 


we  have  the  space  of  yzw 

represented.     In  this  space  all  that  there  will  be  seen  of 
the  sphere  is  the  circle  drawn. 

Here  we  see  that  there  is  no  obstacle  to  prevent  the 

rods  turning  round.  If 
the  matter  is  so  elastic 
that  it  will  give  enough 
for  the  particles  at  E  and 
c  to  be  separated  as  they 
are  at  F  and  D,  they 
can  rotate  round  to  the 
position  D  and  F,  and  a 
similar  motion  is  possible 
for  all  other  particles. 
There  is  no  matter  or 
obstacle  to  prevent  them 
Fig.  12  (140).  from  moving  out  in  the 

w  direction,  and  then  on  round  the  circumference  as  an 
axis.     Now,  what  will  hold  for  one  section  will  hold  for 


218  THE   FOURTH  DIMENSION 

all,  as  the  fourth  dimension  is  at  right  angles  to  all  the 
sections  which  can  be  made  of  the  sphere. 

We  have  supposed  the  matter  of  which  the  sphere  is 
composed  to  be  three-dimensional.  If  the  matter  had  a 
small  thickness  in  the  fourth  dimension,  there  would  be 
a  slight  thickness  in  fig.  12  above  the  plane  of  the  paper 
— a  thickness  equal  to  the  thickness  of  the  matter  in  the 
fourth  dimension.  The  rods  would  have  to  be  replaced 
by  thin  slabs.  But  this  would  make  no  difference  as  to 
the  possibility  of  the  rotation.  This  motion  is  discussed 
by  Newcomb  in  the  first  volume  of  the  American  Journal 
of  Mathematics. 

Let  us  now  consider,  not  a  merely  extensible  body,  but 
a  liquid  one.  A  mass  of  rotating  liquid,  a  whirl,  eddy, 
or  vortex,  has  many  remarkable  properties.  On  first 
consideration  we  should  expect  the  rotating  mass  of 
liquid  immediately  to  spread  off  and  lose  itself  in  the 
surrounding  liquid.  The  water  flies  off  a  wheel  whirled 
round,  and  we  should  expect  the  rotating  liquid  to  be 
dispersed.  But  see  the  eddies  in  a  river  strangely  per- 
sistent. The  rings  that  occur  in  puffs  of  smoke  and  last 
so  long  are  whirls  or  vortices  curved  round  so  that  their 
opposite  ends  join  together.  A  cyclone  will  travel  over 
great  distances. 

Helmholtz  was  the  first  to  investigate  the  properties  of 
vortices.  He  studied  them  as  they  would  occur  in  a  perfect 
fluid — that  is,  one  without  friction  of  one  moving  portion 
or  another.  In  such  a  medium  vortices  would  be  inde- 
structible. They  would  go  on  for  ever,  altering  their 
shape,  but  consisting  always  of  the  same  portion  of  the 
fluid.  But  a  straight  vortex  could  not  exist  surrounded 
entirely  by  the  fluid.  The  ends  of  a  vortex  must  reach  to 
some  boundary  inside  or  outside  the  fluid. 

A  vortex  which  is  bent  round  so  that  its  opposite  ends 
join  is  capable  of  existing,  but  no  vortex  has  a  free  end  in 


RECAPITULATION   AND   EXTENSION  219 

the  fluid.     The  fluid  round  the  vortex  is  always  in  motion, 
and  one  produces  a  definite  movement  in  another. 

Lord  Kelvin  has  proposed  the  hypothesis  that  portions 
of  a  fluid  segregated  in  vortices  account  for  the  origin  of 
matter.  The  properties  of  the  ether  in  respect  of  its 
capacity  of  propagating  disturbances  can  be  explained 
by  the  assumption  of  vortices  in  it  instead  of  by  a  pro- 
perty of  rigidity.  It  is  difficult  to  conceive,  however, 
of  any  arrangement  of  the  vortex  rings  and  endless  vortex 
filaments  in  the  ether. 

Now,  the  further  consideration  of  four-dimensional 
rotations  shows  the  existence  of  a  kind  of  vortex  which 
would  make  an  ether  filled  with  a  homogeneous  vortex 
motion  easily  thinkable. 

To  understand  the  nature  of  this  vortex,  we  must  go 
on  and  take  a  step  by  which  we  accept  the  full  signifi- 
cance of  the  four-dimensional  hypothesis.  Granted  four- 
dimensional  axes,  we  have  seen  that  a  rotation  of  one  into 
another  leaves  two  unaltered,  and  these  two  form  the 
axial  plane  about  which  the  rotation  takes  place.  But 
what  about  these  two  ?  Do  they  necessarily  remain 
motionless  ?  There  is  nothing  to  prevent  a  rotation  of 
these  two,  one  into  the  other,  taking  place  concurrently 
with  the  first  rotation.  This  possibility  of  a  double 
rotation  deserves  the  most  careful  attention,  for  it  is  the 
kind  of  movement  which  is  distinctly  typical  of  four 
dimensions. 

Rotation  round  a  plane  is  analogous  to  rotation  round 
an  axis.  But  in  three-dimensional  space  there  is  no 
motion  analogous  to  the  double  rotation,  in  which,  while 
axis  1  changes  into  axis  2,  axis  3  changes  into  axis  4. 

Consider  a  four-dimensional  body,  with  four  independent 
axes,  x,  y,  z,  w.  A  point  in  it  can  move  in  only  one 
direction  at  a  given  moment.  If  the  body  has  a  velocity 
of  rotation  by  which  the  x  axis  changes  into  the  y  axis 


220  THE   FOURTH   DIMENSION 

and  all  parallel  sections  move  in  a  similar  manner,  then 
the  point  will  describe  a  circle.  If,  now,  in  addition  to 
the  rotation  by  which  the  x  axis  changes  into  the  y  axis  the 
body  has  a  rotation  by  which  the  z  axis  turns  into  the 
w  axis,  the  point  in  question  will  have  a  double  motion 
in  consequence  of  the  two  turnings.  The  motions  will 
compound,  and  the  point  will  describe  a  circle,  but  not 
the  same  circle  which  it  would  describe  in  virtue  of  either 
rotation  separately. 

We  know  that  if  a  body  in  three-dimensional  space  is 
given  two  movements  of  rotation  they  will  combine  into  a 
single  movement  of  rotation  round  a  definite  axis.  It  is 
in  no  different  condition  from  that  in  which  it  is  sub- 
jected to  one  movement  of  rotation.  The  direction  of 
the  axis  changes  ;  that  is  all.  The  same  is  not  true  about 
a  four-dimensional  body.  The  two  rotations,  x  to  y  and 
z  to  w,  are  independent.  A  body  subject  to  the  two  is  in 
a  totally  different  condition  to  that  which  it  is  in  wheiv 
subject  to  one  only.  When  subject  to  a  rotation  such  a* 
that  of  x  to  y,  a  whole  plane  in  the  body,  as  we  have 
seen,  is  stationary.  When  subject  to  the  double  rotatioh 
no  part  of  the  body  is  stationary  except  the  point  common 
to  the  two  planes  of  rotation. 

If  the  two  rotations  are  equal  in  velocity,  every  point 
in  the  body  describes  a  circle.  All  points  equally  distant 
from  the  stationary  point  describe  circles  of  equal  size. 

We  can  represent  a  four-dimensional  sphere  by  means 
of  two  diagrams,  in  one  of  which  we  take  the  three  axes, 
x,  y,  z;  in  the  other  the  axes  x,  w,  and  z.  In  fig.  13  we 
have  the  view  of  a  four-dimensional  sphere  in  the  space  of 
xyz.  Fig.  13  shows  all  that  we  can  see  of  the  four 
sphere  in  the  space  of  xyz,  for  it  represents  all  the 
points  in  that  space,  which  are  at  an  equal  distance  from 
the  centre. 

Let  us  now  take  the  xz  section,  and  let  the  axis  of  w 


RECAPITULATION  AND  EXTENSION 


221 


take  the  place  of  the  y  axis.  Here,  in  fig.  14,  we  have 
the  space  of  xzw.  In  this  space  we  have  to  take  all  the 
points  which  are  at  the  same  distance  from  the  centre, 
consequently  we  have  another  sphere.  If  we  had  a  three- 
dimensional  sphere,  as  has  been  shown  before,  we  should 
have  merely  a  circle  in  the  xzw  space,  the  xz  circle  seen 
in  the  space  of  xzw.  But  now,  taking  the  view  in  the 
space  of  xzw.  we  have  a  sphere  in  that  space  also.  In  a 
similar  manner,  whichever  set  of  three  axes  we  take,  we 
obtain  a  sphere. 


p'  Showing  axes 
xyz 

y 


Fig.  13  (141). 


Fig.  14  (142). 


In  fig.  13,  let  us  imagine  the  rotation  in  the  direction 
xy  to  be  taking  place.  The  point  x  will  turn  to  y-,  and  p 
to  p'.  The  axis  zz  remains  stationary,  and  this  axis  is  all 
of  the  plane  zw  which  we  can  see  in  the  space  section 
exhibited  in  the  figure. 

In  fig.  14,  imagine  the  rotation  from  z  to  w  to  be  taking 
place.  The  lu  axis  now  occupies  the  position  previously 
occupied  by  the  y  axis.  This  does  not  mean  that  the 
w  axis  can  coincide  with  the  y  axis.  It  indicates  that  we 
are  looking  at  the  four-dimensional  sphere  from  a  different 
point  of  view.  Any  three-space  view  will  show  us  three 
axes,  and  in  fig.  14  we  are  looking  at  xzw. 

The  only  part  that  is  identical  in  the  two  diagrams  is 
the  circle  of  the  x  and  z  axes,  which  axes  are  contained 
in  both  diagrams.  Thus  the  plane  zxz'  is  the  same  in 
both,  and  the  point  p  represents  the  same  point  in  both 


222  ME  FOURTH 

diagrams.  Now,  in  fig.  14  let  the  zw  rotation  take  place, 
the  z  axis  will  turn  toward  the  point  iv  of  the  w  axis,  and 
the  point  p  will  move  in  a  circle  about  the  point  x. 

Thus  in  fig.  13  the  point  p  moves  in  a  circle  parallel  to 
the  xy  plane ;  in  fig.  14  it  moves  in  a  circle  parallel  to  the 
zw  plane,  indicated  by  the  arrow. 

Now,  suppose  both  of  these  independent  rotations  com- 
pounded, the  point  p  will  move  in  a  circle,  but  this  circle 
will  coincide  with  neither  of  the  circles  in  which  either 
one  of  the  rotations  will  take  it.  The  circle  the  point  p 
will  move  in  will  depend  on  its  position  on  the  surface  of 
the  four  sphere. 

In  this  double  rotation,  possible  in  four-dimensional 
space,  there  is  a  kind  of  movement  totally  unlike  any 
with  which  we  are  familiar  in  three-dimensional  space. 
It  is  a  requisite  preliminary  to  the  discussion  of  the 
behaviour  of  the  small  particles  of  matter,  with  a  view  to 
determining  whether  they  show  the  characteristics  of  four- 
dimensional  movements,  to  become  familiar  with  the  main 
characteristics  of  this  double  rotation.  And  here  I  must 
rely  on  a  formal  and  logical  assent  rather  than  on  the 
intuitive  apprehension,  which  can  only  be  obtained  by  a 
more  detailed  study. 

In  the  first  place  this  double  rotation  consists  in  two 
varieties  or  kinds,  which  we  will  call  the  A  and  B  kinds. 
Consider  four  axes,  x,  y,  z,  iv.  The  rotation  of  x  to  y  can 
be  accompanied  with  the  rotation  of  z  to  iv.  Call  this 
the  A  kind. 

But  also  the  rotation  of  x  to  y  can  be  accompanied  by 
the  rotation,  of  not  z  to  w,  but  w  to  z.  Call  this  the 
B  kind. 

They  differ  in  only  one  of  the  component  rotations.  One 
is  not  the  negative  of  the  other.  It  is  the  semi-negative. 
The  opposite  of  an  x  to  y,  z  to  w  rotation  would  be  y  to  x, 
w  to  z.  The  semi-negative  is  x  to  y  and  w  to  z. 


feECAPlTtTLAflON  AND  EXTENSION  223 

If  four  dimensions  exist  and  we  cannot  perceive  them, 
because  the  extension  of  matter  is  so  small  in  the  fourth 
dimension  that  all  movements  are  withheld  from  direct 
observation  except  those  which  are  three-dimensional,  we 
should  not  observe  these  double  rotations,  but  only  the 
effects  of  them  in  three-dimensional  movements  of  the 
type  with  which  we  are  familiar. 

If  matter  in  its  small  particles  is  four-dimensional, 
we  should  expect  this  double  rotation  to  be  a  universal 
characteristic  of  the  atoms  and  molecules,  for  no  portion 
of  matter  is  at  rest.  The  consequences  of  this  corpus- 
cular motion  can  be  perceived,  but  only  under  the  form 
of  ordinary  rotation  or  displacement.  Thus,  if  the  theory 
of  four  dimensions  is  true,  we  have  in  the  corpuscles  of 
matter  a  whole  world  of  movement,  which  we  can  never 
study  directly,  but  only  by  means  of  inference. 

The  rotation  A,  as  I  have  defined  it,  consists  of  two 
equal  rotations — one  about  the  plane  of  zw,  the  other 
about  the  plane  of  xy.  It  is  evident  that  these  rotations 
are  not  necessarily  equal.  A  body  may  be  moving  with  a 
double  rotation,  in  which  these  two  independent  com- 
ponents are  not  equal ;  but  in  such  a  case  we  can  consider 
the  body  to  be  moving  with  a  composite  rotation — a 
rotation  of  the  A  or  B  kind  and,  in  addition,  a  rotation 
about  a  plane. 

If  we  combine  an  A  and  a  B  movement,  we  obtain  a 
rotation  about  a  plane;  for,  the  first  being  x  to  y  and 
0  to  iv,  and  the  second  being  x  to  y  and  iv  to  z,  when  (hey 
are  put  together  the  z  to  w  and  w  to  z  rotations  neutralise 
each  other,  and  we  obtain  an  x  to  y  rotation  only,  which 
is  a  rotation  about  the  plane  of  zw.  Similarly,  if  we 
take  a  B  rotation,  y  to  x  and  z  to  w,  we  get,  on  combining 
this  with  the  A  rotation,  a  rotation  of  z  to  w  about  the 
xy  plane.  In  this  case  the  plane  of  rotation  is  in  the 
three-dimensional  space  of  xyz,  and  we  have — what  has 


224  THE   FOURTH   DIMENSION 

been  described  before— a  twisting  about  a  plane  in  our 
space. 

Consider  now  a  portion  of  a  perfect  liquid  having  an  A 
motion.  It  can  be  proved  that  it  possesses  the  properties 
of  a  vortex.  It  forms  a  permanent  individuality — a 
separated-out  portion  of  the  liquid — accompanied  by  a 
motion  of  the  surrounding  liquid.  It  has  properties 
analogous  to  those  of  a  vortex  filament.  But  it  is  not 
necessary  for  its  existence  that  its  ends  should  reach  the 
boundary  of  the  liquid.  It  is  self-contained  and,  unless 
disturbed,  is  circular  in  every  section. 

If  we  suppose  the  ether  to  have  its  properties  of  trans- 
mitting vibration  given  it  by  such  vortices,  we  must 
inquire  how  they  lie  together  in  four-dimensional  space. 
Placing  a  circular  disk  on  a  plane  and  surrounding  it  by 
six  others,  we  find  that  if  the  central  one  is  given  a  motion 
of  rotation,  it  imparts  to  the  others  a  rotation  which  is 

antagonistic  in  every  two  ad- 
jacent ones.  If  A  goes  round, 
as  shown  by  the  arrow,  B  and 
C  will  be  moving  in  opposite 
ways,  and  each  tends  to  de- 
stroy the  motion  of  the  other. 
Now,  if  we  suppose  spheres 
to  be  arranged  in  a  corre- 
sponding manner  in  three- 
dimensional  space,  they  will 
Fig.  15(143).  ,  .  1  '  J  .  . 

be  grouped  m  figures  which 

are  for  three-dimensional  space  what  hexagons  are  for 
plane  space.  If  a  number  of  spheres  of  soft  clay  be 
pressed  together,  so  as  to  fill  up  the  interstices,  each  will 
assume  the  form  of  a  fourteen-sided  figure  called  a 
tetrakai  decagon. 

Now,  assuming  space  to  be  filled  with  such  tetrakai- 
decagons,  and  placing  a  sphere  in  each,  it  will  be  found. 


EECAPITULATION   AND   EXTENSION  225 

that  one  sphere  is  touched  by  eight  others.  The  re- 
maining six  spheres  of  the  fourteen  which  surround  the 
central  one  will  not  touch  it,  but  will  touch  three  of 
those  in  contact  with  it.  Hence,  if  the  central  sphere 
rotates,  it  will  not  necessarily  drive  those  around  it  so 
that  their  motions  will  be  antagonistic  to  each  other, 
but  the  velocities  will  not  arrange  themselves  in  a 
systematic  manner. 

In  four-dimensional  space  the  figure  which  forms  the 
next  term  of  the  series  hexagon,  tetrakaidecagon,  is  a 
thirty-sided  figure.  It  has  for  its  faces  ten  solid  tetra- 
kaidecagons  and  twenty  hexagonal  prisms.  Such  figures 
will  exactly  fill  four-dimensional  space,  five  of  them  meet- 
ing at  every  point.  If,  now,  in  each  of  these  figures  we 
suppose  a  solid  four-dimensional  sphere  to  be  placed,  any 
one  sphere  is  surrounded  by  thirty  others.  Of  these  it 
touches  ten,  and,  if  it  rotates,  it  drives  the  rest  by  means 
of  these.  Now,  if  we  imagine  the  central  sphere  to  be 
given  an  A  or  a  B  rotation,  it  will  turn  the  whole  mass  of 
sphere  round  in  a  systematic  manner.  Suppose  four- 
dimensional  space  to  be  filled  with  such  spheres,  each 
rotating  with  a  double  rotation,  the  whole  mass  would 
form  one  consistent  system  of  motion,  in  which  each  one 
drove  every  other  one,  with  no  friction  or  lagging  behind. 

Every  sphere  would  have  the  same  kind  of  rotation.  In 
three-dimensional  space,  if  one  body  drives  another  round 
the  second  body  rotates  with  the  opposite  kind  of  rotation  ; 
but  in  four-dimensional  space  these  four-dimensional 
spheres  would  each  have  the  double  negative  of  the  rotation 
of  the  one  next  it,  and  we  have  seen  that  the  double 
negative  of  an  A  or  B  rotation  is  still  an  A  or  B  rotation. 
Thus  fpur-dimensional  space  could  be  filled  with  a  system 
of  self-preservative  living  energy.  If  we  imagine  the 
four-dimensional  spheres  to  be  of  liquid  and  not  of  solid 
matter,  then,  even  if  the  liquid  were  not  quite  perfect  and 

15 


226  THE   FOURTH   DIMENSION 

there  were   a  slight   retarding   effect   of  one  vortex   on 
another,  the  system  would  still  maintain  itself. 

In  this  hypothesis  we  must  look  on  the  ether  as 
possessing  energy,  and  its  transmission  of  vibrations,  not 
as  the  conveying  of  a  motion  imparted  from  without,  but 
as  a  modification  of  its  own  motion. 

We  are  now  in  possession  of  pome  of  the  conceptions  of 
four-dimensional  mechanics,  and  will  turn  aside  from  the 
line  of  their  development  to  inquire  if  there  is  any 
evidence  of  their  applicability  to  the  processes  of  nature. 

Is  there  any  mode  of  motion  in  the  region  of  the 
minute  which,  giving  three-dimensional  movements  for 
its  effect,  still  in  itself  escapes  the  grasp  of  our  mechanical 
theories?  I  would  point  to  electricity.  Through  the 
labours  of  Faraday  and  Maxwell  we  are  convinced  that  the 
phenomena  of  electricity  are  of  the  nature  of  the  stress 
and  strain  of  a  medium ;  but  there  is  still  a  gap  to  be 
bridged  over  in  their  explanation — the  laws  of  elasticity, 
which  Maxwell  assumes,  are  not  those  of  ordinary  matter. 
And,  to  take  another  instance :  a  magnetic  pole  in  the 
neighbourhood  of  a  current  tends  to  move.  Maxwell  has 
shown  that  the  pressures  on  it  are  analogous  to  the 
velocities  in  a  liquid  which  would  exist  if  a  vortex  took 
the  place  of  the  electric  current ;  but  we  cannot  point  out 
the  definite  mechanical  explanation  of  these  pressures. 
There  must  be  some  mode  of  motion  of  a  body  or  of  the 
medium  in  virtue  of  which  a  body  is  said  to  be 
electrified. 

Take  the  ions  which  convey  charges  of  electricity  500 
times  greater  in  proportion  to  their  mass  than  are  carried 
by  the  molecules  of  hydrogen  in  electrolysis.  In  respect 
of  what  motion  can  these  ions  be  said  to  be  electrified  ? 
It  can  be  shown  that  the  energy  they  possess  is  not 
energy  of  rotation.  Think  of  a  short  rod  rotating.  If  it 
is  turned  over  it  is  found  to  be  rotating  in  the  opposite 


RECAPITULATION   AND   EXTENSION  221 

direction.  Now,  if  rotation  in  one  direction  corresponds  to 
positive  electricity,  rotation  in  the  opposite  direction  cor- 
responds to  negative  electricity,  and  the  smallest  electrified 
particles  would  have  their  charges  reversed  by  being 
turned  over — an  absurd  supposition. 

If  we  fix  on  a  mode  of  motion  as  a  definition  of 
electricity,  we  must  have  two  varieties  of  it,  one  for 
positive  and  one  for  negative  ;  and  a  body  possessing  the 
one  kind  must  not  become  possessed  of  the  other  by  any 
change  in  its  position. 

Ah1  three-dimensional  motions  are  compounded  of  rota- 
tions and  translations,  and  none  of  them  satisfy  this  first 
condition  for  serving  as  a  definition  of  electricity. 

But  consider  the  double  rotation  of  the  A  and  B  kinds. 
A  body  rotating  with  the  A  motion  cannot  have  its 
motion  transformed  into  the  B  kind  by  being  turned  over 
in  any  way.  Suppose  a  body  has  the  rotation  x  to  y  and 
0  to  w.  Turning  it  about  the  xy  plane,  we  reverse  the 
direction  of  the  motion  x  to  y.  But  we  also  reverse  the 
z  to  w  motion,  for  the  point  at  the  extremity  of  the 
positive  0  axis  is  now  at  the  extremity  of  the  negative  z 
axis,  and  since  we  have  not  interfered  with  its  motion  it 
goes  in  the  direction  of  position  w.  Hence  we  have  y  to 
x  and  w  to  0,  which  is  the  same  as  x  to  y  and  z  to  w. 
Thus  both  components  are  reversed,  and  there  is  the  A 
motion  over  again.  The  B  kind  is  the  semi-negative, 
with  only  one  component  reversed. 

Hence  a  system  of  molecules  with  the  A  motion  would 
not  destroy  it  in  one  another,  and  would  impart  it  to  a 
body  in  contact  with  them.  Thus  A  and  B  motions 
possess  the  first  requisite  which  must  be  demanded  in 
any  mode  of  motion  representative  of  electricity. 

Let  us  trace  out  the  consequences  of  defining  positive 
electricity  as  an  A  motion  and  negative  electricity  as  a  B 
motion.  The  combination  of  positive  and  negative 


228  THE   FOURTH   DIMENSION 

electricity  produces  a  current.  Imagine  a  vortex  in  the 
ether  of  the  A  kind  and  unite  with  this  one  of  the  B  kind. 
An  A  motion  and  B  motion  produce  rotation  round  a  plane, 
which  is  in  the  ether  a  vortex  round  an  axial  surface. 
It  is  a  vortex  of  the  kind  we  represent  as  a  part  of  a 
sphere  turning  inside  out.  Now  such  a  vortex  must  have 
its  rim  on  a  boundary  of  the  ether — on  a  body  in  the 
ether. 

Let  us  suppose  that  a  conductor  is  a  body  which  has 
the  property  of  serving  as  the  terminal  abutment  of  such 
a  vortex.  Then  the  conception  we  must  form  of  a  closed 
current  is  of  a  vortex  sheet  having  its  edge  along  the 
circuit  of  the  conducting  wire.  The  whole  wire  will  then 
be  like  the  centres  on  which  a  spindle  turns  in  three- 
dimensional  space,  and  any  interruption  of  the  continuity 
of  the  wire  will  produce  a  tension  in  place  of  a  continuous 
revolution. 

As  the  direction  of  the  rotation  of  the  vortex  is  from  a 
three-space  direction  into  the  fourth  dimension  and  back 
again,  there  will  be  no  direction  of  flow  to  the  current ; 
but  it  will  have  two  sides,  according  to  whether  z  goes 
to  w  or  z  goes  to  negative  w. 

We  can  draw  any  line  from  one  part  of  the  circuit  to 
another ;  then  the  ether  along  that  line  is  rotating  round 
its  points. 

This  geometric  image  corresponds  to  the  definition  of 
an  electric  circuit.  It  is  known  that  the  action  does  not 
lie  in  the  wire,  but  in  the  medium,  and  it  is  known  that 
there  is  no  direction  of  flow  in  the  wire. 

No  explanation  has  been  offered  in  three-dimensional 
mechanics  of  how  an  action  can  be  impressed  throughout 
a  region  and  yet  necessarily  run  itself  out  along  a  closed 
boundary,  as  is  the  case  in  an  electric  current.  But  this 
phenomenon  corresponds  exactly  to  the  definition  of  a 
four-dimensional  vortex. 


RECAPITULATION    AND   EXTENSION  229 

If  we  take  a  very  long  magnet,  so  long  that  one  of  its 
poles  is  practically  isolated,  and  pat  this  pole  in  the 
vicinity  of  an  electric  circuit,  we  find  that  it  moves. 

Now,  assuming  for  the  sake  of  simplicity  that  the  wire 
which  determines  the  current  is  in  the  form  of  a  circle, 
if  we  take  a  number  of  small  magnets  and  place  them  all 
pointing  in  the  same  direction  normal  to  the  plane  of  the 
circle,  so  that  they  fill  it  and  the  wire  binds  them  round, 
we  find  that  this  sheet  of  magnets  has  the  same  effect  on 
the  magnetic  pole  that  the  current  has.  The  sheet  of 
magnets  may  be  curved,  but  the  edge  of  it  must  coincide 
with  the  wire.  The  collection  of  magnets  is  then 
equivalent  to  the  vortex  sheet,  and  an  elementary  magnet 
to  a  part  of  it.  Thus,  we  must  think  of  a  .magnet  as 
conditioning  a  rotation  in  the  ether  round  the  plane 
which  bisects  at  right  angles  the  line  joining  its  poles. 

If  a  current  is  started  in  a  circuit,  we  must  imagine 
vortices  like  bowls  turning  themselves  inside  out,  starting 
from  the  contour.  In  reaching  a  parallel  circuit,  if  the 
vortex  sheet  were  interrupted  and  joined  momentarily  to 
the  second  circuit  by  a  free  rim,  the  axis  plane  would  lie 
between  the  two  circuits,  and  a  point  on  the  second  circuit 
opposite  a  point  on  the  first  would  correspond  to  a  point 
opposite  to  it  on  the  first;  hence  we  should  expect  a 
current  in  the  opposite  direction  in  the  second  circuit. 
Thus  the  phenomena  of  induction  are  not  inconsistent 
with  the  hypothesis  of  a  vortex  about  an  axial  plane. 

In  four-dimensional  space,  in  which  all  four  dimensions 
were  commensurable,  the  intensity  of  the  action  transmitted 
by  the  medium  would  vary  inversely  as  the  cube  of  the 
distance.  Now,  the  action  of  a  current  on  a  magnetic 
pole  varies  inversely  as  the  square  of  the  distance ;  hence, 
over  measurable  distances  the  extension  of  the  ether  in 
the  fourth  dimension  cannot  be  assumed  as  other  than 
small  in  comparison  with  those  distances. 


230  THE   FOUBTH    DIMENSION 

If  we  suppose  the  ether  to  be  filled  with  vortices  in  the 
shape  of  four-dimensional  spheres  rotating  with  the  A 
motion,  the  B  motion  would  correspond  to  electricity  in 
the  one-fluid  theory.  There  would  thus  be  a  possibility 
of  electricity  existing  in  two  forms,  statically,  by  itself, 
and,  combined  with  the  universal  motion,  in  the  form  of 
a  current. 

To  arrive  at  a  definite  conclusion  it  will  be  necessary  to 
investigate  the  resultant  pressures  which  accompany  the 
collocation  of  solid  vortices  with  surface  ones. 

To  recapitulate : 

The  movements  and  mechanics  of  four-dimensional 
space  are  definite  and  intelligible.  A  vortex  with  a 
surface  as  its  axis  affords  a  geometric  image  of  a  closed 
circuit,  and  there  are  rotations  which  by  their  polarity 
afford  a  possible  definition  of  statical  electricity.  * 

*  These  double  rotations  of  the  A  and  B  kinds  I  should  like  to  call 
Hamiltons  and  co-Hamiltons,  for  it  is  a  singular  fact  that  in  his 
"Quaternions"  Sir  Wm.  Eowan  Hamilton  has  given  the  theory  of 
either  the  A  or  the  B  kind.  They  follow  the  laws  of  his  symbols, 
I,  J,  K. 

Hamiltons  and  co-Hamiltons  seem  to  be  natural  units  of  geometrical 
expression.  In  the  paper  in  the  "  Proceedings  of  the  Royal  Irish 
Academy,"  Nov.  1903,  already  alluded  to,  I  have  shown  something  of 
the  remarkable  facility  which  is  gained  in  dealing  with  the  composition 
of  three-  and  four-dimensional  rotations  by  an  alteration  in  Hamilton's 
notation,  which  enables  his  system  to  be  applied  to  both  the  A  and  B 
kinds  of  rotations. 

The  objection  which  has  been  often  made  to  Hamilton's  system) 
namely,  that  it  is  only  under  special  conditions  of  application  that  his 
processes  give  geometrically  interpretable  results,  can  be  removed,  if 
we  assume  that  he  was  really  dealing  with  a  four-dimensional  motion, 
and  alter  his  notation  to  bring  this  circumstance  into  explicit 
recognition, 


APPENDIX    I 
THE   MODELS 

IN  Chapter  XI.  a  description  has  been  given  which  will 
enable  any  one  to  make  a  set  of  models  illustrative  of  the 
tesseract  and  its  properties.  The  set  here  supposed  to  be 
employed  consists  of : — 

1.  Three  sets  of  twenty-seven  cubes  each. 

2.  Twenty- seven  slabs. 

3.  Twelve  cubes  with  points,  lines,  faces,  distinguished 

by  colours,  which  will  be  called  the  catalogue  cubes. 

The  preparation  of  the  twelve  catalogue  cubes  involves 
the  expenditure  of  a  considerable  amount  of  time.  It  is 
advantageous  to  use  them,  but  they  can  be  replaced  by 
the  drawing  of  the  views  of  the  tesseract  or  by  a  reference 
to  figs.  103,  104,  105,  106  of  the  text. 

The  slabs  are  coloured  like  the  twenty-seven  cubes  of 
the  first  cubic  block  in  fig.  101,  the  one  with  red, 
white,  yellow  axes. 

The  colours  of  the  three  sets  of  twenty-seven  cubes  are 
those  of  the  cubes  shown  in  fig.  101. 

The  slabs  are  used  to  form  the  representation  of  a  cube 
in  a  plane,  and  can  well  be  dispensed  with  by  any  one 
who  is  accustomed  to  deal  with  solid  figures.  But  the 
whole  theory  depends  on  a  careful  observation  of  how  the 
cube  would  be  represented  by  these  slabs. 

Jn  the  first  step,  that  of  forming  a  clear  idea  how  a. 

28} 


232  THE   FOURTH    DIMENSION 

plane  being  would  represent  three-dimensional  space,  only 
one  of  the  catalogue  cubes  and  one  of  the  three  blocks  is 
needed. 


APPLICATION  TO  THE  STEP  FROM  PLANE  TO  SOLID. 

Look  at  fig.  1  of  the  views  of  the  tesseract,  or,  what 
comes  to  the  same  thing,  take  catalogue  cube  No.  1  and 
place  it  before  you  with  the  red  line  running  up,  the 
white  line  running  to  the  right,  the  yellow  line  running 
away.  The  three  dimensions  of  space  are  then  marked 
out^  by  these  lines  or  axes.  Now  take  a  piece  of  card- 
board, or  a  book,  and  place  it  so  that  it  forms  a  wall 
extending  up  and  down  not  opposite  to  you,  but  run- 
ning away  parallel  to  the  wall  of  the  room  on  your 
left  hand. 

Placing  the  catalogue  cube  against  this  wall  we  see 
that  it  comes  into  contact  with  it  by  the  red  and  yellow 
lines,  and  by  the  included  orange  face. 

In  the  plane  being's  world  the  aspect  he  has  of  the 
cube  would  be  a  square  surrounded  by  red  and  yellow 
lines  with  grey  points. 

Now,  keeping  the  red  line  fixed,  turn  the  cube  about  it 
so  that  the  yellow  line  goes  out  to  the  right,  and  the 
white  line  comes  into  contact  with  the  plane. 

In  this  case  a  different  aspect  is  presented  to  the  plane 
being,  a  square,  namely,  surrounded  by  red  and  white 
lines  and  grey  points.  You  should  particularly  notice 
that  when  the  yellow  line  goes  out,  at  right  angles  to  the 
plane,  and  the  white  comes  in,  the  latter  does  not  run  in 
the  same  sense  that  the  yellow  did. 

From  the  fixed  grey  point  at  the  base  of  the  red  line 
the  yellow  line  ran  away  from  you.  The  white  line  now 
runs  towards  you.  This  turning  at  right  angles  makes 
the  line  which  was  out  of  the  plane  before,  come  into  it 


APPENDIX  I  233 

in  an  opposite  sense  to  that  in  which  the  line  ran  which 
has  just  left  the  plane.  If  the  cube  does  not  break 
through  the  plane  this  is  always  the  rule. 

Again  turn  the  cube  back  to  the  normal  position  with 
red  running  up,  white  to  the  right,  and  yellow  away,  and 
try  another  turning. 

You  can  keep  the  yellow  line  fixed,  and  turn  the  cube 
about  it.  In  this  case  the  red  line  going  out  to  the 
right  the  white  line  will  come  in  pointing  downwards. 

You  will  be  obliged  to  elevate  the  cube  from  the  table 
in  order  to  carry  out  this  turning.  It  is  always  necessary 
when  a  vertical  axis  goes  out  of  a  space  to  imagine  a 
movable  support  which  will  allow  the  line  which  ran  out 
before  to  come  in  below. 

Having  looked  at  the  three  ways  of  turning  the  cube 
so  as  to  present  different  faces  to  the  plane,  examine  what 
would  be  the  appearance  if  a  square  hole  were  cut  in  the 
piece  of  cardboard,  and  the  cube  were  to  pass  through  it. 
A  hole  can  be  actually  cut,  and  it  will  be  seen  that  in  the 
normal  position,  with  red  axis  running  up,  yellow  away, 
and  white  to  the  right,  the  square  first  perceived  by  the 
plane  being — the  one  contained  by  red  and  yellow  lines — 
would  be  replaced  by  another  square  of  which  the  line 
towards  you  is  pink — the  section  line  of  the  pink  face. 
The  line  above  is  light  yellow,  below  is  light  yellow  and 
on  the  opposite  side  away  from  you  is  pink. 

In  the  same  way  the  cube  can  be  pushed  through  a 
square  opening  in  the  plane  from  any  of  the  positions 
which  you  have  already  turned  it  into.  In  each  case 
the  plane  being  will  perceive  a  different  set  of  contour 
lines. 

Having  observed  these  facts  about  the  catalogue  cube, 
turn  now  to  the  first  block  of  twenty-seven  cubes. 

You  notice  that  the  colour  scheme  on  the  catalogue  cube 
and  that  of  this  set  of  blocks  is  the  same, 


234  THE   FOURTH   DIMENSION 

Place  them  before  you,  a  grey  or  null  cube  on  the 
table,  above  it  a  red  cube,  and  on  the  top  a  null  cube 
again.  Then  away  from  you  place  a  yellow  cube,  and 
beyond  it  a  null  cube.  Then  to  the  right  place  a  white 
cube  and  beyond  it  another  null.  Then  complete  the 
block,  according  to  the  scheme  of  the  catalogue  cube, 
putting  in  the  centre  of  all  an  ochre  cube. 

You  have  now  a  cube  like  that  which  is  described  in 
the  text.  For  the  sake  of  simplicity,  in  some  cases,  this 
cubic  block  can  be  reduced  to  one  of  eight  cubes,  by 
leaving  out  the  terminations  in  each  direction.  Thus, 
instead  of  null,  red,  null,  three  cubes,  you  can  take  null, 
red,  two  cubes,  and  so  on. 

It  is  useful,  however,  to  practise  the  representation  in 
a  plane  of  a  block  of  twenty-seven  cubes.  For  this 
purpose  take  the  slabs,  and  build  them  up  against  the 
piece  of  cardboard,  or  the  book  in  such  a  way  as  to 
represent  the  different  aspects  of  the  cube. 

Proceed  as  follows : — 

First,  cube  in  normal  position. 

Place  nine  slabs  against  the  cardboard  to  represent  the 
nine  cubes  in  the  wall  of  the  red  and  yellow  axes,  facing 
the  cardboard  ;  these  represent  the  aspect  of  the  cube  as  it 
touches  the  plane. 

Now  push  these  along  the  cardboard  and  make  a 
different  set  of  nine  slabs  to  represent  the  appearance 
which  the  cube  would  present  to  a  plane  being,  if  it  were 
to  pass  half  way  through  the  plane. 

There  would  be  a  white  slab,  above  it  a  pink  one,  above 
that  another  white  one,  and  six  others,  representing  what 
would  be  the  nature  of  a  section  across  the  middle  of  the 
block  of  cubes.  The  section  can  be  thought  of  as  a  thin 
slice  cut  out  by  two  parallel  cuts  across  the  cube. 
Having  arranged  these  nine  slabs,  push  them  along  the 
plane,  and  make  another  set  of  nine  to  represent  what 


APPENDIX  I  235 

would  be  the  appearance  of  the  cube  when  it  had  almost 
completely  gone  through.  This  set  of  nine  will  be  the 
same  as  the  first  set  of  nine. 

Now  we  have  in  the  plane  three  sets  of  nine  slabs 
each,  which  represent  three  sections  of  the  twenty-seven 
block. 

They  are  put  alongside  one  another.  We  see  that  it 
does  not  matter  in  what  order  the  sets  of  nine  are  put. 
As  the  cube  passes  through  the  plane  they  represent  ap- 
pearances which  follow  the  one  after  the  other.  If  they 
were  what  they  represented,  they  could  not  exist  in  the 
same  plane  together. 

This  is  a  rather  important  point,  namely,  to  notice  that 
they  should  not  co-exist  on  the  plane,  and  that  the  order 
in  which  they  are  placed  is  indifferent.  When  we 
represent  a  four-dimensional  body  our  solid  cubes  are  to 
us  in  the  same  position  that  the  slabs  are  to  the  plane 
being.  You  should  also  notice  that  each  of  these  slabs 
represents  only  the  very  thinnest  slice  of  a  cube.  The 
set  of  nine  slabs  first  set  up  represents  the  side  surface  of 
the  block.  It  is,  as  it  were,  a  kind  of  tray — a  beginning 
from  which  the  solid  cube  goes  off.  The  slabs  as  we  use 
them  have  thickness,  but  this  thickness  is  a  necessity  of 
construction.  They  are  to  be  thought  of  as  merely  of  the 
thickness  of  a  line. 

If  now  the  block  of  cubes  passed  through  the  plane  at 
the  rate  of  an  inch  a  minute  the  appearance  to  a  plane 
being  would  be  represented  by  : — 

1.  The  first  set  of  nine  slabs  lasting  for  one  minute. 

2.  The  second  set  of  nine  slabs  lasting  for  one  minute. 

3.  The  third  set  of  nine  slabs  lasting  for  one  minute. 
Now  the   appearances  which  the   cube  would   present 

to  the  plane  being  in  other  positions  can  be  shown  by 
means  of  these  slabs.  The  use  of  such  slabs  would  be 
the  means  by  which  a  plane  being  could  acquire  a 


236  THE    FOURTH    DIMENSION 

familiarity  with  our  cube.  Turn  the  catalogue  cube  (or 
imagine  the  coloured  figure  turned)  so  that  the  red  line 
runs  up,  the  yellow  line  out  to  the  right,  and  the  white 
line  towards  you.  Then  turn  the  block  of  cubes  to 
occupy  a  similar  position. 

The  block  has  now  a  different  wall  in  contact  with 
the  plane.  Its  appearance  to  a  plane  being  will  not  be 
the  same  as  before.  He  has,  however,  enough  slabs  to 
represent  this  new  set  of  appearances.  Bat  he  must 
remodel  his  former  arrangement  of  them. 

He  must  take  a  null,  a  red,  and  a  null  slab  from  the  first 
of  his  sets  of  slabs,  then  a  white,  a  pink,  and  a  white  from 
the  second,  and  then  a  null,  a  red,  and  a  null  from  the 
thiid  set  of  slabs. 

He  takes  the  first  column  from  the  first  set,  the  first 
column  from  the  second  set,  and  the  first  column  from 
the  third  set. 

To  represent  the  half- way-through  appearance,  which 
is  as  if  a  very  thin  slice  were  cut  out  half  way  through  the 
block,  he  must  take  the  second  column  of  each  of  his 
sets  of  slabs,  and  to  represent  the  final  appearance,  the 
third  column  of  each  set. 

Now  turn  the  catalogue  cube  back  to  the  normal 
position,  and  also  the  block  of  cubes. 

There  is  another  turning — a  turning  about  the  yellow 
line,  in  which  the  white  axis  comes  below  the  support, 

You  cannot  break  through  the  surface  of  the  table,  so 
you  must  imagine  the  old  support  to  be  raised.  Then 
the  top  of  the  block  of  cubes  in  its  new  position  is  at  the 
level  at  which  the  base  of  it  was  before. 

Now  representing  the  appearance  on  the  plane,  we  must 
draw  a  horizontal  line  to  represent  the  old  base.  The 
line  should  be  drawn  three  inches  high  on  the  cardboard. 

Below  this  the  representative  slabs  can  be  arranged. 

Jt  is  easy  to  see  what  they  are.     The  old  arrangements 


APPENDIX  I  237 

have  to  be  broken  up,  and  the  layers  taken  in  order,  the 
fir^t  layer  of  each  for  the  representation  of  the  aspect  ol 
the  block  as  it  touches  the  plane. 

Then  the  second  layers  will  represent  the  appearance 
half  way  through,  and  the  third  layers  will  represent  the 
final  appearance. 

It  is  evident  that  the  slabs  individually  do  not  represent 
the  same  portion  of  the  cube  in  these  different  presenta- 
tions. 

In  the  first  case  each  slab  represents  a  section  or  a  face 
perpendicular  to  the  white  axis,  in  the  second  case  a 
face  or  a  section  which  runs  perpendicularly  to  the  yellow 
axis,  and  in  the  third  case  a  section  or  a  face  perpendicular 
to  the  red  axis. 

But  by  means  of  these  nine  slabs  the  plane  being  can 
represent  the  whole  of  the  cubic  block.  He  can  touch 
and  handle  each  portion  of  the  cubic  block,  there  is  no 
part,  of  it  which  he  cannot  observe.  Taking  it  bit  by  bit, 
two  axes  at  a  time,  he  can  examine  the  whole  of  it. 

OUR  REPRESENTATION  OF  A  BLOCK  OF  TESSERACTS. 

Look  at  the  views  of  the  tesseract  1,  2,  3,  or  take  the 
catalogue  cubes  1,  2,  3,  and  place  them  in  front  of  you, 
in  any  order,  say  rum  ing  from  left  to  right,  placing  1  in 
the  normal  position,  the  red  axis  running  up,  the  white 
to  the  right,  and  yellow  away. 

Now  notice  that  in  catalogue  cube  2  the  colours  of  each 
region  are  derived  from  those  of  the  corresponding  region 
of  cube  1  by  the  addition  of  blue.  Thus  null  +  blue  = 
blue,  and  the  corners  of  number  2  are  blue.  Again, 
red -f-blue  =  purple,  and  the  vertical  lines  of  2  are  purple. 
Blue  +  yellow  =  green,  and  the  line  which  runs  away  is 
coloured  green. 

By  means  of  these  observations  you  may  be  sure  that 


238  THE   FOUETH  DIMENSION 

catalogue  cube  2  is  rightly  placed.  Catalogue  cube  3  is 
just  like  number  1. 

Having  these  cubes  in  what  we  may  call  their  normal 
position,  proceed  to  build  up  the  three  sets  of  blocks. 

This  is  easily  done  in  accordance  with  the  colour  scheme 
on  the  catalogue  cubes. 

The  first  block  we  already  know.  Build  up  the  second 
block,  beginning  with  a  blue  corner  cube,  placing  a  purple 
on  it,  and  so  on. 

Having  these  three  blocks  we  have  the  means  of 
representing  the  appearances  of  a  group  of  eighty-one 
tesseracts. 

Let  us  consider  a  moment  what  the  analogy  in  the  case 
of  the  plane  being  is. 

He  has  his  three  sets  of  nine  slabs  each.  We  have  our 
three  sets  of  twenty-seven  cubes  each. 

Our  cubes  are  like  his  slabs.  As  his  slabs  are  not  the 
things  which  they  represent  to  him,  so  our  cubes  are  not 
the  things  they  represent  to  us. 

The  plane  being's  slabs  are  to  him  the  faces  of  cubes. 

Our  cubes  then  are  the  faces  of  tesseracts,  the  cubes  by 
which  they  are  in  contact  with  our  space. 

As  each  set  of  slabs  in  the  case  of  the  plane  being 
might  be  considered  as  a  sort  of  tray  from  which  the  solid 
contents  of  the  cubes  came  out,  so  our  three  blocks  of 
cubes  may  be  considered  as  three-space  trays,  each  of 
which  is  the  beginning  of  an  inch  of  the  solid  contents 
of  the  four-dimensional  solids  starting  from  them. 

We  want  now  to  use  the  names  null,  red,  white,  etc., 
for  tesseracts.  The  cubes  we  use  are  only  tesseract  faces. 
Let  us  denote  that  fact  by  calling  the  cube  of  null  colour, 
null  face ;  or,  shortly,  null  f.,  meaning  that  it  is  the  face 
of  a  tesseract. 

To  determine  which  face  it  is  let  us  look  at  the  catalogue 
cube  1  or  the  first  of  the  views  of  the  tesseract,  which 


APPENDIX   I  239 

can  be  used  instead  of  the  models.  It  has  three  axes, 
red,  white,  yellow,  in  our  space.  Hence  the  cube  deter- 
mined by  these  axes  is  the  face  of  the  tesseract  which  we 
now  have  before  us.  It  is  the  ochre  face.  It  is  enough, 
however,  simply  to  say  null  f.,  red  f.  for  the  cubes  which 
we  use. 

To  impress  this  in  your  mind,  imagine  that  tesseracts 
do  actually  run  from  each  cube.  Then,  when  you  move  the 
cubes  about,  you  move  the  tesseracts  about  with  them. 
You  move  the  face  but  the  tesseract  follows  with  it,  as  the 
cube  follows  when  its  face  is  shifted  in  a  plane. 

The  cube  null  in  the  normal  position  is  the  cube  which 
has  in  it  the  red,  yellow,  white  axes.  It  is  the  face 
having  these,  but  wanting  the  blue.  In  this  way  you  can 
define  which  face  it  is  you  are  handling.  I  will  write  an 
"  f."  after  the  name  of  each  tesseract  just  as  the  plane 
being  might  call  each  of  his  slabs  null  slab,  yellow  slab, 
etc.,  to  denote  that  they  were  representations. 

We  have  then  in  the  first  block  of  twenty-seven  cubes, 
the  following — null  f.,  red  f.,  null  f.,  going  up ;  white  f.,  null 
f.,  lying  to  the  right,  and  so  on.  Starting  from  the  null 
point  and  travelling  up  one  inch  we  are  in  the  null  region, 
the  same  for  the  away  and  the  right-hand  directions. 
And  if  we  were  to  travel  in  the  fourth  dimension  for  an 
inch  we  should  still  be  in  a  null  region.  The  tesseract 
stretches  equally  all  four  ways.  Hence  the  appearance  we 
have  in  this  first  block  would  do  equally  well  if  the 
tesseract  block  were  to  move  across  our  space  for  a  certain 
distance.  For  anything  less  than  an  inch  of  their  trans- 
verse motion  we  should  still  have  the  same  appearance. 
You  must  notice,  however,  that  we  should  not  have  null 
face  after  the  motion  had  begun. 

When  the  tesseract,  null  for  instance,  had  moved  ever 
so  little  we  should  not  have  a  face  of  null  but  a  section  of 
null  in  our  space.  Hence,  when  we  think  of  the  motion 


240  THE   FOURTH   DIMENSION 

a  cross  our  space  we  must  call  our  cubes  tesseraci  sections. 
Thus  on  null  pass-ing  across  we  should  see  first  null  f.,  then 
null  s.,  and  then,  finally,  null  f.  again. 

Imagine  now  the  whole  first  block  of  twenty-seven 
tesseracts  to  have  moved  tranverse  to  our  space  a  distance 
of  one  inch.  Then  the  second  set  of  tesseracts,  which 
originally  were  an  inch  distant  from  our  space,  would  be 
ready  to  come  in. 

Their  colours  are  shown  in  the  second  block  of  twenty- 
seven  cubes  which  you  have  before  you.  These  represent 
the  tesseract  faces  of  the  set  of  tesseracts  that  lay  before 
an  inch  away  from  our  space.  They  are  ready  now  to 
come  in,  and  we  can  observe  their  colours.  In  the  place 
which  null  f.  occupied  before  we  have  blue  f.,  in  place  of 
red  f.  we  have  purple  f.,  and  so  on.  Each  tesseract  is 
coloured  like  the  one  whose  place  it  takes  in  this  motion 
with  the  addition  of  blue. 

Now  if  the  tesseract  block  goes  on  moving  at  the  rate 
of  an  inch  a  minute,  this  next  set  of  tesseracts  will  occupy 
a  minute  in  passing  across.  We  shall  see,  to  take  the  null 
one  for  instance,  first  of  all  null  face,  then  null  section, 
then  null  face  again. 

At  the  end  of  the'  second  minute  the  second  set  of 
tesseracts  has  gone  through,  and  the  third  set  comes  in, 
This,  as  you  see,  is  coloured  just  like  the  first.  Altogether, 
these  three  sets  extend  three  inches  in  the  fourth  dimension, 
making  the  tesseract  block  of  equal  magnitude  in  all 
dimensions. 

We  have  now  before  us  a  complete  catalogue  of  all  the 
tesseracts  in  our  group.  We  have  seen  them  all,  and  we 
shall  refer  to  this  arrangement  of  the  blocks  as  the 
"  normal  position."  We  have  seen  as  much  of  each 
tesseract  at  a  time  as  could  be  done  in  a  three-dimen- 
sional space.  Each  part  of  each  tesseract  has  been  in 
our  space,  and  we  could  have  touched  it. 


APPENDIX  I  241 

The  fourth  dimension  appeared  to  us  as  the  duration 
of  the  block. 

If  a  bit  of  our  matter  were  to  be  subjected  to  the  same 
motion  it  would  be  instantly  removed  out  of  our  space. 
Being  thin  in  the  fourth  dimension  it  is  at  once  taken 
out  of  our  space  by  a  motion  in  the  fourth  dimension. 

But  the  tesseract  block  we  represent  having  length  in 
the  fourth  dimension  remains  steadily  before  our  eyes  for 
three  minutes,  when  it  is  subjected  to  this  transverse 
motion. 

We  have  now  to  form  representations  of  the  other 
views  of  the  same  tesseract  group  which  are  possible  in 
our  space. 

Let  us  then  turn  the  block  of  tesseracts  so  that  another 
face  of  it  comes  into  contact  with  our  space,  and  then 
by  observing  what  we  have,  and  what  changes  come  when 
the  block  traverses  our  space,  we  shall  have  another  view 
of  it.  The  dimension  which  appeared  as  duration  before 
will  become  extension  in  one  of  our  known  dimensions, 
and  a  dimension  which  coincided  with  one  of  our  space 
dimensions  will  appear  as  duration. 

Leaving  catalogue  cube  1  in  the  normal  position, 
remove  the  other  two,  or  suppose  them  removed.  We 
have  in  space  the  red,  the  yellow,  and  the  white  axes. 
Let  the  white  axis  go  out  into  the  unknown,  and  occupy 
the  position  the  blue  axis  holds.  Then  the  blue  axis, 
which  runs  in  that  direction  now  will  come  into  space. 
But  it  will  not  come  in  pointing  in  the  same  way  that 
the  white  axis  does  now.  It  will  point  in  the  opposite 
sense.  It  will  come  in  running  to  the  left  instead  of 
running  to  the  right  as  the  white  axis  does  now. 

When  this  turning  takes  place  every  part  of  the  cube  1 
will  disappear  except  the  left-hand  face — the  orange  face. 

And  the  new  cube  that  appears  in  our  space  will  run  to 
the  left  from  this  orange  face,  having  axes,  red,  yellow,  blue. 

16 


242  THE   FOURTH   DIMENSION 

Take  models  4,  5,  6.  Place  4,  or  suppose  No.  4  of  the 
tesseract  views  placed,  with  its  orange  face  coincident  with 
the  orange  face  of  1 ,  red  line  to  red  line,  and  yellow  line 
to  yellow  line,  with  the  blue  line  pointing  to  the  left. 
Then  remove  cube  1  and  we  have  the  tesseract  face 
which  comes  in  when  the  white  axis  runs  in  the  positive 
unknown,  and  the  blue  axis  comes  into  our  space. 

Now  place  catalogue  cube  5  in  some  position,  it  does 
not  matter  which,  say  to  the  left ;  and  place  it  so  that 
there  is  a  correspondence  of  colour  corresponding  to  the 
colour  of  the  line  that  runs  out  of  space.  The  line  that 
runs  out  of  space  is  white,  hence,  every  part  of  this 
cube  5  should  differ  from  the  corresponding  part  of  4  by 
an  alteration  in  the  direction  of  white. 

Thus  we  have  white  points  in  5  corresponding  to  the 
null  points  in  4.  We  have  a  pink  line  corresponding  to 
a  red  line,  a  light  yellow  line  corresponding  to  a  yellow 
line,  an  ochre  face  corresponding  to  an  orange  face.  This 
cube  section  is  completely  named  in  Chapter  XI.  Finally 
cube  6  is  a  replica  of  1. 

These  catalogue  cubes  will  enable  us  to  set  up  our 
models  of  the  block  of  tesseracts. 

First  of  all  for  the  set  of  tesseracts,  which  beginning 
in  our  space  reach  out  one  inch  in  the  unknown,  we  have 
the  pattern  of  catalogue  cube  4. 

We  see  that  we  can  build  up  a  block  of  twenty-seven 
tesseract  faces  after  the  colour  scheme  of  cube  4,  by 
taking  the  left-hand  wall  of  block  1,  then  the  left-hand 
wall  of  block  2,  and  finally  that  of  block  3.  We  take, 
that  is,  the  three  first  walls  of  our  previous  arrangement 
to  form  the  first  cubic  block  of  this  new  one. 

This  will  represent  the  cubic  faces  by  which  the  group 
of  tesseracts  in  its  new  position  touches  our  space. 
We  have  running  up,  null  f.,  red  f.,  null  f.  In  the  next 
vertical  line,  on  the  side  remote  from  us,  we  have  yellow  f., 


APPENDIX  I  243 

orangfr  f.,  yellow  f.,  and  then  the  first  colours  over  again. 
Then  the  three  following  columns  are,  blue  f.,  purple  f., 
blue  f. ;  green  f.,  brown  f.,  green  f. ;  blue  f.,  purple  f.,  blue  f. 
The  last  three  columns  are  like  the  first. 

These  tesseracts  touch  our  space,  and  none  of  them  are 
by  any  part  of  them  distant  more  than  an  inch  from  it. 
What  lies  beyond  them  in  the  unknown  ? 

This  can  be  told  by  looking  at  catalogue  cube  5. 
According  to  its  scheme  of  colour  we  see  that  the  second 
wall  of  each  of  our  old  arrangements  must  be  taken. 
Putting  them  together  we  have,  as  the  corner,  white  f. 
above  it,  pink  f.  above  it,  white  f.  The  column  next  to 
this  remote  from  us  is  as  follows  : — light  yellow  f.,  ochre  f., 
light  yellow  f.,  and,  beyond  this  a  column  like  the  first. 
Then  for  the  middle  of  the  block,  light  blue  f.,  above 
it  light  purple,  then  light  blue.  The  centre  column  has, 
at  the  bottom,  light  green  f.,  light  brown  f.  in  the  centre 
and  at  the  top  light  green  f.  The  last  wall  is  like  the 
first. 

The  third  block  is  made  by  taking  the  third  walls  of 
our  previous  arrangement,  which  we  called  the  normal 
one. 

You  may  ask  what  faces  and  what  sections  our  cubes 
represent.  To  answer  this  question  look  at  what  axes 
you  have  in  our  space.  You  have  red,  yellow,  blue. 
Now  these  determine  brown.  The  colours  red, 
yellow,  blue  are  supposed  by  us  when  mixed  to  produce 
a  brown  colour.  And  that  cube  which  is  determined 
by  the  red,  yellow,  blue  axes  we  call  the  brown  cube. 

When  the  tesseract  block  in  its  new  position  begins  to 
move  across  our  space  each  tesseract  in  it  gives  a  section 
in  our  space.  This  section  is  transverse  to  the  white 
axis,  which  now  runs  in  the  unknown. 

As  the  tesseract  in  its  present  position  passes  across 
our  space,  we  should  see  first  of  all  the  first  of  the  blocks 


244  THE  FOURTH   DIMENSION 

of  cubic  faces  we  have  put  up — these  would  last  for  a 
minute,  then  would  come  the  second  block  and  then  the 
third.  At  first  we  should  have  a  cube  of  tesseract  faces, 
each  of  which  would  be  brown.  Directly  the  movement 
began,  we  should  have  tesseract  sections  transverse  to  the 
white  line. 

There  are  two  more  analogous  positions  in  which  the 
block  of  tesseracts  can  be  placed.  To  find  the  third 
position,  restore  the  blocks  to  the  normal  arrangement. 

Let  us  make  the  yellow  axis  go  out  into  the  positive 
unknown,  and  let  the  blue  axis,  consequently,  come  in 
running  towards  us.  The  yellow  ran  away,  so  the  blue 
will  come  in  running  towards  us. 

Put  catalogue  cube  1  in  its  normal  position.  Take 
catalogue  cube  7  and  place  it  so  that  its  pink  face 
coincides  with  the  pink  face  of  cube  1,  making  also  its 
red  axis  coincide  with  the  red  axis  of  1  and  its  white 
with  the  white.  Moreover,  make  cube  7  come 
towards  us  from  cube  1.  Looking  at  it  we  see  in  our 
space,  red,  white,  and  blue  axes.  The  yellow  runs  out. 
Place  catalogue  cube  8  in  the  neighbourhood  of 
7 — observe  that  every  region  in  8  has  a  change  in 
the  direction  of  yellow  from  the  corresponding  region 
in  7.  This  is  because  it  represents  what  you  come 
to  now  in  going  in  the  unknown,  when  the  yellow  axis 
runs  out  of  our  space.  Finally  catalogue  cube  9, 
which  is  like  number  7,  shows  the  colours  of  the  third 
set  of  tesseracts.  Now  evidently,  starting  from  the 
normal  position,  to  make  up  our  three  blocks  of  tesseract 
faces  we  have  to  take  the  near  wall  from  the  first  block, 
the  near  wall  from  the  second,  and  then  the  near  wall 
from  the  third  block.  This  gives  us  the  cubic  block 
formed  by  the  faces  of  the  twenty-seven  tesseracts  which 
are  now  immediately  touching  our  space. 

Following   the    colour   scheme    of   catalogue  cube    8, 


APPENDIX  I  245 

we  make  the  next  set  of  twenty-seven  tesseract  faces, 
representing  the  tesseracts,  each  of  which  begins  one  inch 
off  from  our  space,  by  putting  the  second  walls  of  our 
previous  arrangement  together,  and  the  representation 
of  the  third  set  of  tessaracts  is  the  cubic  block  formed  of 
the  remaining  three  walls. 

Since  we  have  red,  white,  blue  axes  in  our  space  to 
begin  with,  the  cubes  we  see  at  first  are  light  purple 
tesseract  faces,  and  after  the  transverse  motion  begins 
we  have  cubic  sections  transverse  to  the  yellow  line. 

Restore  the  blocks  to  the  normal  position,  there 
remains  the  case  in  which  the  red  axis  turns  out  of 
space.  In  this  case  the  blue  axis  will  come  in  down- 
wards, opposite  to  the  sense  in  which  the  red  axis  ran. 

In  this  case  take  catalogue  cubes  10,  11,  12.  Lift  up 
catalogue  cube  1  and  put  10  underneath  it,  imagining 
that  it  goes  down  from  the  previous  position  of  1. 

We  have  to  keep  in  space  the  white  and  the  yellow 
axes,  and  let  the  red  go  out,  the  blue  come  in. 

Now,  you  will  find  on  cube  10  a  light  yellow  face ;  this 
should  coincide  with  the  base  of  1,  and  the  white  and 
yellow  lines  on  the  two  cubes  should  coincide.  Then  the 
blue  axis  running  down  you  have  the  catalogue  cube 
correctly  placed,  and  it  forms  a  guide  for  putting  up  the 
first  representative  block. 

Catalogue  cube  11  will  represent  what  lies  in  the  fourth 
dimension — now  the  red  line  runs  in  the  fourth  dimen- 
sion. Thus  the  change  from  10  to  11  should  be  towards 
red,  corresponding  to  a  null  point  is  a  red  point,  to  a 
white  line  is  a  pink  line,  to  a  yellow  line  an  orange 
line,  and  so  on. 

Catalogue  cube  12  is  like  10.  Hence  we  see  that  to 
build  up  our  blocks  of  tesseract  faces  we  must  take  the 
bottom  layer  of  the  first  block,  hold  that  up  in  the  air, 
underneath  it  place  the  bottom  layer  of  the  second  block) 


246  THE  FOURTH  DIMENSION 

and  finally  underneath  this  last  the  bottom  layer  of  the 
last  of  our  normal  blocks. 

Similarly  we  make  the  second  representative  group  by 
taking  the  middle  courses  of  our  three  blocks.  The  last 
is  made  by  taking  the  three  topmost  layers.  The  three 
axes  in  our  space  before  the  transverse  motion  begins  are 
blue,  white,  yellow,  so  we  have  light  green  tesseract 
faces,  and  after  the  motion  begins  sections  transverse  to 
the  red  light. 

These  three  blocks  represent  the  appearances  as  the 
tesseract  group  in  its  new  position  passes  across  our  space. 
The  cubes  of  contact  in  this  case  are  those  determinal  by 
the  three  axes^in  our  space,  namely,  the  white,  the 
yellow,  the  blue.  Hence  they  are  light  green. 

It  follows  from  this  that  light  green  is  the  interior 
cube  of  the  first  block  of  representative  cubic  faces. 

Practice  in  the  manipulations  described,  with  a 
realization  in  each  case  of  the  face  or  section  which 
is  in  our  space,  is  one  of  the  best  means  of  a  thorough 
comprehension  of  the  subject. 

We  have  to  learn  how  to  get  any  part  of  these  four- 
dimensional  figures  into  space,  so  that  we  can  look  at 
them.  We  must  first  learn  to  swing  a  tesseract,  and  a 
group  of  tesseracts  about  in  any  way. 

When  these  operations  have  been  repeated  and  the 
method  of  arrangement  of  the  set  of  blocks  has  become 
familiar,  it  is  a  good  plan  to  rotate  the  axes  of  the  normal 
cube  1  about  a  diagonal,  and  then  repeat  the  whole  series 
of  turnings. 

Thus,  in  the  normal  position,  red  goes  up,  white  to  the 
right,  yellow  away.  Make  white  go  up,  yellow  to  the  right, 
and  red  away.  Learn  the  cube  in  this  position  by  putting 
up  the  set  of  blocks  of  the  normal  cube,  over  and  over 
again  till  it  becomes  as  familiar  to  you  as  in  the  normal 
position.  Then  when  this  is  learned,  and  the  corre- 


APPENDIX  I  247 

spending  changes  in  the  arrangements  of  the  tesseract 
groups  are  made,  another  change  should  be  made :  let, 
in  the  normal  cube,  yellow  go  up,  red  to  the  right,  and 
white  away. 

Learn  the  normal  block  of  cubes  in  this  new  position 
by  arranging  them  and  re-arranging  them  till  you  know 
without  thought  where  each,  one  goes.  Then  carry  out 
all  the  tesseract  arrangements  and  turnings. 

If  you  want  to  understand  the  subject,  but  do  not  see 
your  way  clearly,  if  it  does  not  seem  natural  and  easy  to 
you,  practise  these  turnings.  Practise,  first  of  all,  the 
turning  of  a  block  of  cubes  round,  so  that  you  know  it 
in  every  position  as  well  as  in  the  normal  one.  Practise 
by  gradually  putting  up  the  set  of  cubes  in  their  new 
arrangements.  Then  put  up  the  tesseract  blocks  in  their 
arrangements.  This  will  give  you  a  working  conception 
of  higher  space,  you  will  gain  the  feeling  of  it,  whether 
you  take  up  the  mathematical  treatment  of  it  or  not. 


APPENDIX    II 
A  LANGUAGE   OP   SPACE 

THE  mere  naming  the  parts  of  the  figures  we  con  Rider 
involves  a  certain  amount  of  time  and  attention.  This 
time  and  attention  leads  to  no  result,  for  with  each 
new  figure  the  nomenclature  applied  is  completely 
changed,  every  letter  or  symbol  is  used  in  a  different 
significance. 

Surely  it  must  be  possible  in  some  way  to  utilise  the 
labour  thus  at  present  wasted  ! 

Why  should  we  not  make  a  language  for  space  itself,  so 
that  every  position  we  want  to  refer  to  would  have  its  own 
name  ?  Then  every  time  we  named  a  figure  in  order  to 
demonstrate  its  properties  we  should  be  exercising 
ourselves  in  the  vocabulary  of  place. 

If  we  use  a  definite  system  of  names,  and  always  refer 
to  the  same  space  position  by  the  same  name,  we  create 
as  it  were  a  multitude  of  little  hands,  each  prepared  to 
grasp  a  special  point,  position,  or  element,  and  hold  it 
for  us  in  its  proper  relations. 

We  make,  to  use  another  analogy,  a  kind  of  mental 
paper,  which  has  somewhat  of  the  properties  of  a  sensitive 
plate,  in  that  it  will  register,  without  effort,  complex, 
visual,  or  tactual  impressions. 

But  of  far  more  importance  than  the  applications  of  a 
space  language  to  the  plane  and  to  solid  space  is  the 

248 


APPENDIX   II  249 

facilitation  it  brings  with  it  to  the  study  of  four-dimen- 
sional shapes. 

I  have  delayed  introducing  a  space  language  because 
all  the  systems  I  made  turned  out,  after  giving  them  a 
fair  trial,  to  be  intolerable.  I  have  now  come  upon  one 
which  seems  to  present  features  of  permanence,  and  I  will 
here  give  an  outline  of  it,  so  that  it  can  be  applied  to 
the  subject  of  the  text,  and  in  order  that  it  may  be 
subjected  to  criticism. 

The  principle  on  which  the  language  is  constructed  is 
to  sacrifice  every  other  consideration  for  brevity. 

It  is  indeed  curious  that  we  are  able  to  talk  and 
converse  on  every  subject  of  thought  except  the  funda- 
mental one  of  space.  The  only  way  of  speaking  about 
the  spatial  configurations  that  underlie  every  subject 
of  discursive  thought  is  a  co-ordinate  system  of  numbers. 
This  is  so  awkward  and  incommodious  that  it  is  never 
used.  In  thinking  also,  in  realising  shapes,  we  do  not 
use  it ;  we  confine  ourselves  to  a  direct  visualisation. 

Now,  the  use  of  words  corresponds  to  the  storing  up 
of  our  experience  in  a  definite  brain  structure.  A  child, 
in  the  endless  tactual,  visual,  mental  manipulations  it 
makes  for  itself,  is  best  left  to  itself,  but  in  the  course 
of  instruction  the  introduction  of  space  names  would 
make  the  teachers  work  more  cumulative,  and  the  child's 
knowledge  more  social. 

Their  full  use  can  only  be  appreciated,  if  they  are 
introduced  early  in  the  course  of  education ;  but  in  a 
minor  degree  any  one  can  convince  himself  of  their 
utility,  especially  in  our  immediate  subject  of  handling 
four-dimensional  shapes.  The  sum  total  of  the  results 
obtained  in  the  preceding  pages  can  be  compendiously  and 
accurately  expressed  in  nine  words  of  the  Space  Language. 

In  one  of  Plato's  dialogues  Socrates  makes  an  experi- 
ment on  a  slave  boy  standing  by.  He  makes  certain 


250  THE   FOURTH   DIMENSION 

perceptions  of  space  awake  in  the  mind  of  Meno's  slave 
by  directing  his  close  attention  on  some  simple  facts  of 
geometry. 

By  means  of  a  few  words  and  some  simple  forms  we  can 
repeat  Plato's  experiment  on  new  ground. 

Do  we  by  directing  our  close  attention  on  the  facts  of 
four  dimensions  awaken  a  latent  faculty  in  ourselves  ? 
The  old  experiment  of  Plato's,  it  seems  to  me,  has  come 
down  to  us  as  novel  as  on  the  day  he  incepted  it,  and  its 
significance  not  better  understood  through  all  the  dis- 
cussion of  which  it  has  been  the  subject. 

Imagine  a  voiceless  people  living  in  a  region  where 
everything  had  a  velvety  surface,  and  who  were  thus 
deprived  of  all  opportunity  of  experiencing  what  sound  is. 
They  could  observe  the  slow  pulsations  of  the  air  caused 
by  their  movements,  and  arguing  from  analogy,  they 
would  no  doubt  infer  that  more  rapid  vibrations  were 
possible.  From  the  theoretical  side  they  could  determine 
all  about  these  more  rapid  vibrations.  They  merely  differ, 
they  would  say,  from  slower  ones,  by  the  number  that 
occur  in  a  given  time;  there  is  a  merely  formal  difference. 

But  suppose  they  were  to  take  the  trouble,  go  to  the 
pains  of  producing  these  more  rapid  vibrations,  then  a 
totally  new  sensation  would  fall  on  their  rudimentary  ears. 
Probably  at  first  they  would  only  be  dimly  conscious  of 
Sound,  but  even  from  the  first  they  would  become  aware 
that  a  merely  formal  difference,  a  mere  difference  in  point 
of  number  in  this  particular  respect,  made  a  great  difference 
practically,  as  related  to  them.  And  to  us  the  difference 
between  three  and  four  dimensions  is  merely  formal, 
numerical.  We  can  tell  formally  all  about  four  dimensions, 
calculate  the  relations  that  would  exist.  But  that  the 
difference  is  merely  formal  does  not  prove  that  it  is  a 
futile  and  empty  task,  to  present  to  ourselves  as  closely  as 
we  can  the  phenomena  of  four  dimensions.  In  our  formal 


APPENDIX  H  251 

knowledge  of  it,  the  whole  question  of  its  actual  relation 
to  us,  as  we  are,  is  left  in  abeyance. 

Possibly  a  new  apprehension  of  nature  may  come  to  us 
through  the  practical,  as  distinguished  from  the  mathe- 
matical and  formal,  study  of  four  dimensions.  As  a  child 
handles  and  examines  the  objects  with  which  he  comes  in 
contact,  so  we  can  mentally  handle  and  examine  four- 
dimensional  objects.  The  point  to  be  determined  is  this. 
Do  we  find  something  cognate  and  natural  to  our  faculties, 
or  are  we  merely  building  up  an  artificial  presentation  of 
a  scheme  only  formally  possible,  conceivable,  but  which 
has  no  real  connection  with  any  existing  or  possible 
experience  ? 

This,  it  seems  to  me,  is  a  question  which  can  only  be 
settled  by  actually  trying.  This  practical  attempt  is  the 
logical  and  direct  continuation  of  the  experiment  Plato 
devised  in  the  "Meno." 

Why  do  we  think  true?  Why,  by  our  processes  oi 
thought,  can  we  predict  what  will  happen,  and  correctly 
conjecture  the  constitution  of  the  things  around  us  ? 
This  is  a  problem  which  every  modern  philosopher  has 
considered,  and  of  which  Descartes,  Leibnitz,  Kant,  to 
name  a  few,  have  given  memorable  solutions.  Plato  was 
the  first  to  suggest  it.  And  as  he  had  the  unique  position 
of  being  the  first  devisor  of  the  problem,  so  his  solution 
is  the  most  unique.  Later  philosophers  have  talked  about 
consciousness  and  its  laws,  sensations,  categories.  But 
Plato  never  used  such  words.  Consciousness  apart  from  a 
conscious  being  meant  nothing  to  him.  His  was  always 
an  objective  search.  He  made  man's  intuitions  the  basis 
of  a  new  kind  of  natural  history. 

In  a  few  simple  words  Plato  puts  us  in  an  attitude 
with  regard  to  psychic  phenomena — the  mind — the  ego — 
"what  we  are,"  which  is  analogous  to  the  attitude  scientific 
men  of  the  present  day  have  with  regard  to  the  phenomena 


252  THE   FOURTH   DIMENSION 

of  outward  nature.  Behind  this  first  apprehension  of  ours 
of  nature,  there  is  an  infinite  depth  to  be  learned  and 
known.  Plato  said  that  behind  the  phenomena  of  mind 
that  Meno's  slave  boy  exhibited,  there  was  a  vast,  an 
infinite  perspective.  And  his  singularity,  his  originality, 
comes  out  most  strongly  marked  in  this,  that  the  per- 
spective, the  complex  phenomena  beyond  were,  according 
to  him,  phenomena  of  personal  experience.  A  footprint 
in  the  sand  means  a  man  to  a  being  that  has  the  con- 
ception of  a  man.  But  to  a  creature  that  has  no  such 
conception,  it  means  a  curious  mark,  somehow  resulting 
from  the  concatenation  of  ordinary  occurrences.  Such  a 
being  would  attempt  merely  to  explain  how  causes  known 
to  him  could  so  coincide  as  to  produce  such  a  result ; 
he  would  not  recognise  its  significance. 

Plato  introduced  the  conception  which  made  a  new 
kind  of  natural  history  possible.  He  said  that  Meno's 
slave  boy  thought  true  about  things  he  had  never 
learned,  because  his  "  soul  "  had  experience.  I  know  this 
will  sound  absurd  to  some  people,  and  it  flies  straight 
in  the  face  of  the  maxim,  that  explanation  consists  in 
showing  how  an  effect  depends  on  simple  causes.  But 
what  a  mistaken  maxim  that  is!  Can  any  single  instance 
be  shown  of  a  simple  cause  ?  Take  the  behaviour  of 
spheres  for  instance ;  say  those  ivory  spheres,  billiard  balls, 
for  example.  We  can  explain  their  behaviour  by  supposing 
they  are  homogeneous  elastic  solids.  We  can  give  formulae 
which  will  account  for  their  movements  in  every  variety. 
But  are  they  homogeneous  elastic  solids  ?  No,  certainly 
not.  They  are  complex  in  physical  and  molecular  structure, 
and  atoms  and  ions  beyond  open  an  endless  vista.  Our 
simple  explanation  is  false,  false  as  it  can  be.  The  balls 
act  as  if  they  were  homogeneous  elastic  spheres.  There  is 
a  statistical  simplicity  in  the  resultant  of  very  complex 
conditions,  which  makes  that  artificial  conception  useful. 


APPENDIX  n  253 

But  its  usefulness  must  not  blind  us  to  the  fact  that  it  is 
artificial.  If  we  really  look  deep  into  nature,  we  find  a 
much  greater  complexity  than  we  at  first  suspect.  And 
so  behind  this  simple  "  I,"  this  myself,  is  there  not  a 
parallel  complexity  ?  Plato's  "  soul "  would  be  quite 
acceptable  to  a  large  class  of  thinkers,  if  by  "  soul "  and 
the  complexity  he  attributes  to  it,  he  meant  the  product 
of  a  long  course  of  evolutionary  changes,  whereby  simple 
forms  of  living  matter  endowed  with  rudimentary  sensation 
had  gradually  developed  into  fully  conscious  beings. 

But  Plato  does  not  mean  by  "  soul "  a  being  of  such  a 
kind.  His  soul  is  a  being  whose  faculties  are  plogged  by 
its  bodily  environment,  or  at  least  hampered  by  the 
difficulty  of  directing  its  bodily  frame — a  being  which 
is  essentially  higher  than  the  account  it  gives  of  itself 
through  its  organs.  At  the  same  time  Plato's  soul  is 
not  incorporeal.  It  is  a  real  being  with  a  real  experience. 
The  question  of  whether  Plato  had  the  conception  of  non- 
spatial  existence  has  been  much  discussed.  The  verdict 
is,  I  believe,  that  even  his  "  ideas  "  were  conceived  by  him 
as  beings  in  space,  or,  as  we  should  say,  real.  Plato's 
attitude  is  that  of  Science,  inasmuch  as  he  thinks  of  a 
world  in  Space.  But,  granting  this,  it  cannot  be  denied 
that  there  is  a  fundamental  divergence  between  Plato's 
conception  and  the  evolutionary  theory,  and  also  an 
absolute  divergence  between  his  conception  and  the 
genetic  account  of  the  origin  of  the  human  faculties. 
The  functions  and  capacities  of  Plato's  "  soul "  are  not 
derived  by  the  interaction  of  the  body  and  its  environment. 

Plato  was  engaged  on  a  variety  of  problems,  and  his 
religious  and  ethical  thoughts  were  so  keen  and  fertile 
that  the  experimental  investigation  of  his  soul  appears 
involved  with  many  other  motives.  In  one  passage  Plato 
will  combine  matter  of  thought  of  all  kinds  and  from  all 
sources,  overlapping,  interrunning.  And  in  no  case  is  he 


254  THE  FOURTH   DIMENSION 

more  involved  and  rich  than  in  this  question  of  the  soul. 
In  fact,  I  wish  there  were  two  words,  one  denoting  that 
being,  corporeal  and  real,  but  with  higher  faculties  than 
we  manifest  in  our  bodily  actions,  which  is  to  be  taken  as 
the  subject  of  experimental  investigation ;  and  the  other 
word  denoting  "  soul "  in  the  sense  in  which  it  is  made 
the  recipient  and  the  promise  of  so  much  that  men  desire. 
It  is  the  soul  in  the  former  sense  that  I  wish  to  investigate, 
and  in  a  limited  sphere  only.  I  wish  to  find  out,  in  con- 
tinuation of  the  experiment  in  the  Meno,  what  the  "  soul " 
in  us  thinks  about  extension,  experimenting  on  the 
grounds  laid  down  by  Plato.  He  made,  to  state  the 
matter  briefly,  the  hypothesis  with  regard  to  the  thinking 
power  of  a  being  in  us,  a  "  soul."  This  soul  is  not  acces- 
sible to  observation  by  sight  or  touch,  but  it  can  be 
observed  by  its  functions ;  it  is  the  object  of  a  new  kind 
of  natural  history,  the  materials  for  constructing  which 
lie  in  what  it  is  natural  to  us  to  think.  With  Plato 
"  thought "  was  a  very  wide-reaching  term,  but  still  I 
would  claim  in  his  general  plan  of  procedure  a  place  for 
the  particular  question  of  extension. 

The  problem  comes  to  be,  "  What  is  it  natural  to  us  to 
think  about  matter  qua  extended  ?  " 

Fir.  t  of  all,  I  find  that  the  ordinary  intuition  of  any 
simple  object  is  extremely  imperfect.  Take  a  block  of 
differently  marked  cubes,  for  instance,  and  become  ac- 
quainted with  them  in  their  positions.  You  may  think 
you  know  them  quite  well,  but  when  you  turn  them  round 
— rotate  the  block  round  a  diagonal,  for  instance — you 
will  find  that  you  have  lost  track  of  the  individuals  in 
their  new  positions.  You  can  mentally  construct  the 
block  in  its  new  position,  by  a  rule,  by  taking  the  remem- 
bered sequences,  but  you  don't  know  it  intuitively.  By 
observation  of  a  block  of  cubes  in  various  positions,  and 
very  expeditiously  by  a  use  of  Space  names  applied  to  the 


n  265 

cubes  in  their  different  presentations,  it  is  possible  to  get 
an  intuitive  knowledge  of  the  block  of  cubes,  which  is  not 
disturbed  by  any  displacement.  Now,  with  regard  to  this 
intuition,  we  moderns  would  say  that  I  had  formed  it  by 
my  tactual  visual  experiences  (aided  by  hereditary  pre- 
disposition). Plato  would  say  that  the  soul  had  been 
stimulated  to  recognise  an  instance  of  shape  which  it 
knew.  Plato  would  consider  the  operation  of  learning 
merely  as  a  stimulus;  we  as  completely  accounting  for 
the  result.  The  latter  is  the  more  common-sense  view. 
But,  on  the  other  hand,  it  presupposes  the  generation  of 
experience  from  physical  changes.  The  world  of  sentient 
experience,  according  to  the  modern  view,  is  closed  and 
limited ;  only  the  physical  world  is  ample  and  large  and 
of  ever-to-be-discovered  complexity.  Plato's  world  of  soul, 
on  the  other  hand,  is  at  least  as  large  and  ample  as  the 
world  of  things. 

Let  us  now  try  a  crucial  experiment.  Can  I  form  an 
intuition  of  a  four-dimensional  object  ?  Such  an  object 
is  not  given  in  the  physical  range  of  my  sense  contacts. 
All  I  can  do  is  to  present  to  myself  the  sequences  of  solids, 
which  would  mean  the  presentation  to  me  under  my  con- 
ditions of  a  four-dimensional  object.  All  I  can  do  is  to 
visualise  and  tactualise  different  series  of  solids  which  are 
alternative  sets  of  sectional  views  of  a  four-dimensional 
shape. 

If  now,  on  presenting  these  sequences,  I  find  a  power 
in  me  of  intuitively  passing  from  one  of  these  sets  of 
sequences  to  another,  of,  being  given  one,  intuitively 
constructing  another,  not  using  a  rule,  but  directly  appre- 
hending it,  then  I  have  found  a  new  fact  about  my  soul, 
that  it  has  a  four-dimensional  experience ;  I  have  observed 
it  by  a  function  it  has. 

I  do  not  like  to  speak  positively,  for  I  might  occasion 
a  loss  of  time  on  the  part  of  others,  if,  as  may  very  well 


256  THE   FOURTH   DIMENSION 

be,  I  am  mistaken.  But  for  my  own  part,  I  think  there 
are  indications  of  such  an  intuition  ;  from  the  results  of 
my  experiments,  I  adopt  the  hypothesis  that  that  which 
thinks  in  us  has  an  ample  experience,  of  which  the  intui- 
tions we  use  in  dealing  with  the  world  of  real  objects 
are  a  part;  of  which  experience,  the  intuition  of  four- 
dimensional  forms  and  motions  is  also  a  part.  The  process 
we  are  engaged  in  intellectually  is  the  reading  the  obscure 
signals  of  our  nerves  into  a  world  of  reality,  by  means  of 
intuitions  derived  from  the  inner  experience. 

The  image  I  form  is  as  follows.  Imagine  the  captain 
of  a  modern  battle-ship  directing  its  course.  He  has 
his  charts  before  him ;  he  is  in  communication  with  his 
associates  and  subordinates ;  can  convey  his  messages  and 
commands  to  every  part  of  the  ship,  and  receive  informa- 
tion from  the  conn  ing-tower  and  the  engine-room.  Now 
suppose  the  captain  immersed  in  the  problem  of  the 
navigation  of  his  ship  over  the  ocean,  to  have  so  absorbed 
himself  in  the  problem  of  the  direction  of  his  craft  over 
the  plane  surface  of  the  sea  that  he  forgets  himself.  All 
that  occupies  his  attention  is  the  kind  of  movement  that 
his  ship  makes.  The  operations  by  which  that  movement 
is  produced  have  sunk  below  the  threshold  of  his  con- 
sciousness, his  own  actions,  by  which  he  pushes  the  buttons, 
gives  the  orders,  are  so  familiar  as  to  be  automatic,  his 
mind  is  on  the  motion  of  the  ship  as  a  whole.  In  such 
a  case  we  can  imagine  that  he  identifies  himself  with  his 
ship ;  all  that  enters  his  conscious  thought  is  the  direction 
of  its  movement  over  the  plane  surface  of  the  ocean. 

Such  is  the  relation,  as  I  imagine  it,  of  the  soul  to  the 
body.  A  relation  which  we  can  imagine  as  existing 
momentarily  in  the  case  of  the  captain  is  the  normal 
one  in  the  case  of  the  soul  with  its  craft.  As  the  captain 
is  capable  of  a  kind  of  movement,  an  amplitude  of  motion, 
which  does  not  enter  into  his  thoughts  with  regard  to  the 


APPENDIX   II  257 

directing  the  ship  over  the  plane  surface  of  the  ocean,  so 
the  soul  is  capable  of  a  kind  of  movement,  has  an  ampli- 
tude of  motion,  which  is  not  used  in  its  task  of  directing 
the  body  in  the  three-dimensional  region  in  which  the 
body's  activity  lies.  If  for  any  reason  it  became  necessary 
for  the  captain  to  consider  three-dimensional  motions  with 
regard  to  his  ship,  it  would  not  be  difficult  for  him  to 
gain  the  materials  for  thinking  about  such  motions ;  all 
he  has  to  do  is  to  call  his  own  intimate  experience  into 
play.  As  far  as  the  navigation  of  the  ship,  however,  is 
concerned,  he  is  not  obliged  to  call  on  such  experience. 
The  ship  as  a  whole  simply  moves  on  a  surface.  The 
problem  of  three-dimensional  movement  does  not  ordinarily 
concern  its  steering.  And  thus  with  regard  to  ourselves 
all  those  movements  and  activities  which  characterise  our 
bodily  organs  are  three-dimensional ;  we  never  need  to 
consider  the  ampler  movements.  But  we  do  more  than 
use  the  movements  of  our  body  to  effect  our  aims  by 
direct  means ;  we  have  now  come  to  the  pass  when  we  act 
indirectly  on  nature,  when  we  call  processes  into  play 
which  lie  beyond  the  reach  of  any  explanation  we  can 
give  by  the  kind  of  thought  which  has  been  sufficient  for 
the  steering  of  our  craft  as  a  whole.  When  we  come  to 
the  problem  of  what  goes  on  in  the  minute,  and  apply 
ourselves  to  the  mechanism  of  the  minute,  we  find  our 
habitual  conceptions  inadequate. 

The  captain  in  us  must  wake  up  to  his  own  intimate 
nature,  realise  those  functions  of  movement  which  are  his 
own,  and  in  virtue  of  his  knowledge  of  them  apprehend 
how  to  deal  with  the  problems  he  has  come  to. 

Think  of  the  history  of  man.  When  has  there  been  a 
time,  in  which  his  thoughts  of  form  and  movement  were 
not  exclusively  of  such  varieties  as  were  adapted  for  his 
bodily  performance  ?  We  have  never  had  a  demand  to 
conceive  what  our  own  most  intimate  powers  are.  But, 

17 


258  THE  FOUfcTH  DIMENSION 

just  as  little  as  by  immersing  himself  in  the  steering  of 
his  ship  over  the  plane  surface  of  the  ocean,  a  captain 
can  loose  the  faculty  of  thinking  about  what  he  actually 
does,  so  little  can  the  soul  loose  its  own  nature.  It 
can  be  roused  to  an  intuition  that  is  not  derived  from 
the  experience  which  the  senses  give.  All  that  is 
necessary  is  to  present  some  few  of  those  appearances 
which,  while  inconsistent  with  three-dimensional  matter, 
are  yet  consistent  with  our  formal  knowledge  of  four- 
dimensional  matter,  in  order  for  the  soul  to  wake  up  and 
not  begin  to  learn,  but  of  its  own  intimate  feeling  fill  up 
the  gaps  in  the  presentiment,  grasp  the  full  orb  of  possi- 
bilities from  the  isolated  points  presented  to  it.  In  relation 
to  this  question  of  our  perceptions,  let  me  suggest  another 
illustration,  not  taking  it  too  seriously,  only  propounding 
it  to  exhibit  the  possibilities  in  a  broad  and  general  way. 

In  the  heavens,  amongst  the  multitude  of  stars,  there 
are  some  which,  when  the  telescope  is  directed  on  them, 
seem  not  to  be  single  stars,  but  to  be  split  up  into  two. 
Regarding  these  twin  stars  through  a  spectroscope,  an 
astronomer  sees  in  each  a  spectrum  of  bands  of  colour  and 
black  lines.  Comparing  these  spectrums  with  one  another, 
he  finds  that  there  is  a  slight  relative  shifting  of  the  dark 
lines,  and  from  that  shifting  he  knows  that  the  stars  are 
rotating  round  one  another,  and  can  tell  their  relative 
velocity  with  regard  to  the  earth.  By  means  of  his 
terrestrial  physics  he  reads  this  signal  of  the  skies.  This 
shifting  of  lines,  the  mere  slight  variation  of  a  black  line 
in  a  spectrum,  is  very  unlike  that  which  the  astronomer 
knows  it  means.  But  it  is  probably  much  more  like  what 
it  means  than  the  signals  which  the  nerves  deliver  are 
like  the  phenomena  of  the  outer  world. 

No  picture  of  an  object  is  conveyed  through  the  nerves. 
No  picture  of  motion,  in  the  sense  in  which  we  postulate 
its  existence,  is  conveyed  through  the  nerves.  The  actual 


APPENDIX    II  259 

deliverances  of  which  our  consciousness  takes  account  are 
probably  identical  for  eye  and  ear,  sight  and  touch. 

If  for  a  moment  I  take  the  whole  earth  together  and 
regard  it  as  a  sentient  being,  I  find  that  the  problem  of 
its  apprehension  is  a  very  complex  one,  and  involves  a 
long  series  of  personal  and  physical  events.  Similarly  the 
problem  of  our  apprehension  is  a  very  complex  one.  I 
only  use  this  illustration  to  exhibit  my  meaning.  It  has 
this  especial  merit,  that,  as  the  process  of  conscious 
apprehension  takes  place  in  our  case  in  the  minute,  so, 
with  regard  to  this  earth  being,  the  corresponding  process 
takes  place  in  what  is  relatively  to  it  very  minute. 

Now,  Plato's  view  of  a  soul  leads  us  to  the  hypothesis 
that  that  which  we  designate  as  an  act  of  apprehension 
may  be  a  very  complex  event,  both  physically  and  per- 
sonally. He  does  not  seek  to  explain  what  an  intuition 
is;  he  makes  it  a  basis  from  whence  he  sets  out  on  a 
voyage  of  discovery.  Knowledge  means  knowledge  ;  he 
puts  conscious  being  to  account  for  conscious  being.  He 
makes  an  hypothesis  of  the  kind  that  is  so  fertile  in 
physical  science — an  hypothesis  making  no  claim  to 
finality,  which  marks  out  a  vista  of  possible  determination 
behind  determination,  like  the  hypothesis  of  space  itself, 
the  type  of  serviceable  hypotheses. 

And,  above  all,  Plato's  hypothesis  is  conducive  to  ex- 
periment. He  gives  the  perspective  in  which  real  objects 
can  be  determined ;  and,  in  our  present  enquiry,  we  are 
making  the  simplest  of  all  possible  experiments — we  are 
enquiring  what  it  is  natural  to  the  soul  to  think  of  matter 
as  extended. 

Aristotle  says  we  always  use  a  "  phantasm  "  in  thinking, 
a  phantasm  of  our  corporeal  senses  a  visualisation  or  a 
tactualisation.  But  we  can  so  modify  that  visualisation 
or  tactualisation  that  it  represents  something  not  known 
by  the  senses.  Do  we  by  that  representation  wake  up  an 


260  THE   FOURTH   DIMENSION 

intuition  of  the  soul?  Can  we  by  the  presentation  of 
these  hypothetical  forms,  that  are  the  subject  of  our 
present  discussion,  wake  ourselves  up  to  higher  intuitions  ? 
And  can  we  explain  the  world  around  by  a  motion  that  we 
only  know  by  our  souls  ? 

Apart  from  all  speculation,  however,  it  seems  to  me 
that  the  interest  of  these  four-dimensional  shapes  and 
motions  is  sufficient  reason  for  studying  them,  and  that 
they  are  the  way  by  which  we  can  grow  into  a  fuller 
apprehension  of  the  world  as  a  concrete  whole. 

SPACE  NAMES. 

If  the  words  written  in  the  squares  drawn  in  fig.  1  are 
used  as  the  names  of  the   squares  in   the  positions  in 
which  they  are  placed,  it  is  evident  that 
a  combination  of  these  names  will  denote 
a    figure    composed    of    the    designated 
squares.      It  is  found  to  be  most  con- 
venient to  take  as  the  initial  square  that 
marked   with   an   asterisk,   so    that    the 
Fig.  i.  directions  of  progression  are  towards  the 

observer  and  to  his  right.  The  directions 
of  progression,  however,  are  arbitrary,  and  can  be  chosen 
at  will. 

Thus  et,  at,  it,  an,  al  will  denote  a  figure  in  the  form 
of  a  cross  composed  of  five  squares. 

Here,  by  means  of  the  double  sequence,  e,a,i  and  n,t,l,  it 
is  possible  to  name  a  limited  collection  of  space  elements. 
The  system  can  obviously  be  extended  by  using  letter 
sequences  of  more  members. 

But,  without  introducing  such  a  complexity,  the 
principles  of  a  space  language  can  be  exhibited,  and  a 
nomenclature  obtained  adequate  to  all  the  considerations 
of  the  preceding  pages. 


APPENDIX   II 


261 


I.  Extension. 

Call  the  large  squares  in  fig.  2  by  the  name  written 
in  them.  It  is  evident  that  each 
can  be  divided  as  shown  in  fig.  1. 
Thens  the  small  square  marked  1 
will  be  "en"  in  "En,"  or  "  Enen." 
The  square  marked  2  will  be  "  et " 
in  "  En  "  or  «  Enet,"  while  the 
square  marked  4  will  be  "  en "  in 
"  Et "  or  "  Eten."  Thus  the  square 
5  will  be  called"  Ilil." 

This  principle  of  extension   can 
be  applied  in  any  number  of  dimensions. 


Fig.  2. 


2.  Application  to  Three-Dimensional  Space. 
To  name  a  three-dimensional  collocation  of  cubes  take 
the   upward   direction   first,  secondly   the 
direction  towards  the  observer,  thirdly  the 
direction  to  his  right  hand. 

These  form  a  word  in  which  the  first 
letter  gives  the  place  of  the  cube  upwards, 
the  second  letter  its  place  towards  the 
observer,  the  third  letter  its  place  to  the 
right. 

We  have  thus  the  following  scheme, 
which  represents  the  set  of  cubes  of 
column  1,  fig.  101,  page  165. 

We  begin  with  the  remote  lowest  cube 
at  the  left  hand,  where  the  asterisk  is 
placed  (this  proves  to  be  by  far  the  most 
convenient  origin  to  take  for  the  normal 
system). 

Thus  "nen"  is  a  "null"  cube,  "ten" 
a  red  cube   on   it,  and  "  len  "  a   "  null " 
cube  above  "  ten." 


262 


THE   FOURTH    DIMENSION 


By  using  a  more  extended  sequence  of  consonants  and 
vowels  a  larger  set  of  cubes  can  be  named. 

To  name  a  four-dimensional  block  of  tesseracts  it  is 
simply  necessary  to  prefix  an  "  e,"  an  "  a,"  or  an  "  i "  to 
the  cube  names. 

Thus  the  tesseract  blocks  schematically  represented  on 
page  165,  fig.  101  are  named  as  follows  : — 


2.  DERIVATION  OF  POINT,  LINE,  FACE,  ETC.,  NAMES. 

The  principle  of  derivation  can  be  shown  as  follows 
Taking  the  square  of  squares 


APPENDIX 


263 


the   number  of  squares   in  it   can  be  enlarged  and  the 
whole  kept  the  same  size. 


Compare  fig.  79,  p.  138,  for  instance,  or  the  bottom  layer 
of  fig.  84. 

Now  use  an  initial  "  s  "  to  denote  the  result  of  carrying 
this  process  on  to  a  great  extent,  and  we  obtain  the  limit 
names,  that  is  the  point,  line,  area  names  for  a  square. 
"  Sat "  is   the  whole   interior.     The   corners   are    "  sen," 
"sel,"  "sin,"   "  sil,"  while   the   lines 
.*..       *.        sei        are  "  san,"  "  sal,"  "  set,"  "  sit." 

I    find    that    by   the   use    of   the 
initial  "  s "  these  names  come  to  be 
practically  entirely  disconnected  with 
the  systematic  names  for  the  square 
from  which  they  are  derived.     They 
are  easy  to  learn,  and  when  learned 
can    be   used    readily   with    the    axes    running    in    any 
direction. 

To  derive  the  limit  names  for  a  four-dimensional  rect- 
angular figure,  like  the  tesseract,  is  a  simple  extension  of 
this  process.  These  point,  line,  etc.,  names  include  those 
which  apply  to  a  cube,  as  will  be  evident  on  inspection 
of  the  first  cube  of  the  diagrams  which  follow. 

All  that  is  necessary  is  to  place  an  "  s  "  before  each  of  the 
names  given  for  a  tesseract  block.  We  then  obtain 
apellatires  which,  like  the  colour  names  on  page  174, 
fig.  103,  apply  to  all  the  points,  lines,  faces,  solids,  and  to 


264 


THE   FOURTH   DIMENSION 


the  hypersolid  of  the  tesseract.  These  names  have  the 
advantage  over  the  colour  marks  that  each  point,  line,  etc., 
has  its  own  individual  name. 

In  the  diagrams  I  give  the  names  corresponding  to 
the  positions  shown  in  the  coloured  plate  or  described  on 
p.  174.  By  comparing  cubes  1,  2,  3  with  the  first  row  of 
cubes  in  the  coloured  plate,  the  systematic  names  of  each 
of  the  points,  lines,  faces,  etc.,  can  be  determined.  The 
asterisk  shows  the  origin  from  which  the  names  run. 

These  point,  line,  face,  etc.,  names  should  be  used  in 
connection  with  the  corresponding  colours.  The  names 
should  call  up  coloured  images  of  the  parts  named  in  their 
right  connection. 

It  is  found  that  a  certain  abbreviation  adds  vividness  of 
distinction  to  these  names.  If  the  final  "  en  "  be  dropped 
wherever  it  occurs  the  system  is  improved.  Thus  instead 
of  "  senen,"  "seten,"  "  selen,"  it  is  preferable  to  abbreviate 
to  "sen,"  "set,"  "sel,"  and  also  use  "  san,"  "sin"  for 
"  sanen,"  "  sinen." 


5tt(n     S*Uf    5el«l         5aW     Siltt     5<jl«l        5ilen.    Oilet      Stlel 


or&l 

\5ela^      N^/>  °Cx> 

\^  5ala>  x<>ors'<L 

\^>  5ilaf     N^y 

Srv 

Win       5«liT\  ^ 

N,^ 

t  !•           s  l-K     CoU 
Sal  in        .'XiMX  gjj» 

X-5'lin       5iW\ 

_ 

Sel  it 

? 

.t 

Salit 

3 

-                Silt^ 

^, 

C/"»                           C/> 

\ 

2^                  &" 

^        0>                        <s> 

Snra 
or,W« 

^ 

?  ^Seh't      ^ 

trjn 

\, 

5-5.Hl      ^ 

Sinen 
or5in 

\>,       ?                           ^ 

\ 

Stni'n 

Seiil         \ 

\imrt 

oauit         \  Ociun 

5*nil"                                     Soiil                                    Snn.it 

Cr. 

\ 

M    ? 

\ 

«»  gN 

tf 

f*«s 

Sanel   JJiitl 

"^ 

.r 

C-ffiJ    '^ 

*\         ,            ^ 

V""  \ 

X                   ^ 

V     Santf     ^V 

\       Smit'V 

Stntl 


fjvtsri.r    5«t*t 


iTiferior     Jj^f  luTfrior     Oil 


APPENDIX   II 


265 


Sil — Mia       Siltt    Saia  M     Slid  5al,l  M 


OantL 


f            o 
Sit        2, 

X 

Sifcl     ^ 

\ 

Cx* 

\ 

./ 

t/: 

(*r 

1* 

<5dii-      .Sen. 

H 

3anef<S*«et 

^ 

S      I.S    1 

e<v 

\Jana.  ^ 

•Sanat      Xtfx 

^mr^ 

\        OiniL     ^y 

Inferior    5arai 


inltrior   Jat^ 


iianjcur  5;m      Sill*  Si  m   S:?ii 


Sfniil 


5itet     S 
5niJ     ii'tif^ 

\ 

^ 

«  g 

\ 

X 

5m  if   Jmil 

X 

\ 

\J^ 

Ny   5a  n.dt 

X4'"1' 

266 


THE   FOURTH   DIMENSION 


Se.t      Qtle.t     OcTet  Sel 

Cx-        N&x  c    !-•          f  f^*       f  f 


Sel.  I 


Inferior    Oan*t 


Interior  iat.t 


We  can  now  name  any  section.  Take  £.(7.  the  line  in 
the  first  cube  from  senin  to  senel,  we  should  call  the  line 
running  from  senin  to  senel,  senin  senat  senel,  a  line 
light  yellow  in  colour  with  null  points. 

Here  senat  is  the  name  for  all  of  the  line  except  its  ends. 
Using  "  senat "  in  this  way  does  not  mean  that  the  line  is 
the  whole  of  senat,  but  what  there  is  of  it  is  senat.  It  is 
a  part  of  the  senat  region.  Thus  also  the  triangle,  which 
has  its  three  vertices  in  senin,  senel,  selen,  is  named  thus  : 

Area :  setat. 

Sides  :  setan,  senat,  setet. 

Vertices :  senin,  senel,  sel. 

The  tetrahedron  section  of  the  tesseract  can  be  thought 
of  as  a  series  of  plane  sections  in  the  successive  sections  of 
the  tesseract  shown  in  fig.  114,  p.  191.  In  b0  the  section 
is  the  one  written  above.  In  bj  the  section  is  made  by  a 


APPENDIX   II  267 

plane  which  cuts  the  three  edges  from  sanen  intermediate 
of  their  lengths  and  thus  will  be : 

Area :  satat. 

Sides  :  satan,  sanat,  satet. 

Vertices  :  sanan,  sanet,  sat. 

The  sections  in  ba,  b3  will  be  like  the  section  in  b,  but 
smaller. 

Finally  in  b4  the  section  plane  simply  passes  through  the 
corner  named  sin. 

Hence,  putting  these  sections  together  in  their  right 
relation,  from  the  face  setat,  surrounded  by  the  lines  and 
points  mentioned  above,  there  run  : 

3  faces :  satan,  sanat,  satet 
3  lines  :  sanan,  sanet,  sat 

and  these  faces  and  lines  run  to  the  point  sin.  Thus 
the  tetrahedron  is  completely  named. 

The  octahedron  section  of  the  tesseract,  which  can  be 
traced  from  fig.  72,  p.  129  by  extending  the  lines  there 
drawn,  is  named : 

Front  triangle  selin,  selat,  selel,  setal,  senil,  setit,  selin 
with  area  setat. 

The  sections  between  the  front  and  rear  triangle,  of 
which  one  is  shown  in  Ib  another  in  2b,  are  thus  named, 
points  and  lines,  salan,  salat,  salet,  satet,  satel,  satal,  sanal, 
sanat,  sanit,  satit,  satin,  satan,  salan. 

The  rear  triangle  found  in  3b  by  producing  lines  is  sil, 
sitet,  sinel,  sinat,  sinin,  sitan,  sil. 

The  assemblage  of  sections  constitute  the  solid  body  of 
the  octahedron  satat  with  triangular  faces.  The  one  from 
the  line  selat  to  the  point  sil,  for  instance,  is  named 


268 


THE    FOURTH   DIMENSION 


selin,    selat,    selel,    salet,   salat,    salan,    sil.      The  whole 
interior  is  salat. 

Shapes  can  easily  be  cut  out  of  cardboard  which,  when 
folded  together,  form  not  only  the  tetrahedron  and  the 
octohedron,  but  also  samples  of  all  the  sections  of  the 
tesseract  taken  as  it  passes  cornerwise  through  our  space. 
To  name  and  visualise  with  appropriate  colours  a  series  of 
these  sections  is  an  admirable  exercise  for  obtaining 
familiarity  with  the  subject. 


EXTENSION  AND  CONNECTION  WITH  NUMBERS. 

By  extending  the  letter  sequence  it  is  of  course  possible 
to  name  a  larger  field.  By  using  the  limit  names  the 
corners  of  each  square  can  be  named. 

Thus  "  en  sen,"  "  an  sen,"  etc.,  will  be  the  names  of  the 
points  nearest  the  origin  in  "  en  "  and  in  "  an." 

A  field  of  points  of  which  each  one  is  indefinitely  small 
is  given  by  the  names  written  below. 


ensen 

eisen 

! 

clsen 



•  4 

k  

arisen 

atsen 

alben 

insen 

itscn 

ilsen  ! 

1 

The  squares  are  shown  in  dotted  lines,  the  names 
denote  the  points.  These  points  are  not  mathematical 
points,  but  really  minute  areas. 

Instead  of  starting  with  a  set  of  squares  and  naming 
them,  we  can  start  with  a  set  of  points. 

By  an  easily  remembered  convention  we  can  give 
names  to  such  a  region  of  points. 


269 

Let  the  space  names  with  a  final  "  e  "  added  denote  the 
mathematical  points  at  the  corner  of  each  square  nearest 
the  origin.  We  have  then 


ene 

etc 

ele     ! 

i 

ane 

ate 

ale 

ine 

L... 

K-  4 

ite 

iTe" 

for  the  set  of  mathematical  points  indicated.  This 
system  is  really  completely  independent  of  the  area 
system  and  is  connected  with  it  merely  for  the  purpose 
of  facilitating  the  memory  processes.  The  word  "  ene  "  is 
pronounced  like  "  eny,"  with  just  sufficient  attention  to 
the  final  vowel  to  distinguish  it  from  the  word  "  en." 

Now,  connecting  the  numbers  0,  1,  2  with  the  sequence 
e,  a,  i,  and  also  with  the  sequence  n,  t,  1,  we  have  a  set  of 
points  named  as  with  numbers  in  a  co-ordinate  system. 
Thus  "ene"  is  (0,  0)  "ate"  is  (1,  1)  "ite"  is  (2,  1). 
To  pass  to  the  area  system  the  rule  is  that  the  name  of 
the  square  is  formed  from  the  name  of  its  point  nearest 
to  the  origin  by  dropping  the  final  e. 

By  using  a  notation  analogous  to  the  decimal  system 
a  larger  field  of  points  can  be  named.  It  remains  to 
assign  a  letter  sequence  to  the  numbers  from  positive  0 
to  positive  9,  and  from  negative  0  to  negative  9,  to  obtain 
a  system  which  can  be  used  to  denote  both  the  usual 
co-ordinate  system  of  mapping  and  a  system  of  named 
squares.  The  names  denoting  the  points  all  end  with  e. 
Those  that  denote  squares  end  with  a  consonant. 

There  are  many  considerations  which  must  be  attended 
to  in  extending  the  sequences  to  be  used,  such  as 
uniqueness  in  the  meaning  of  the  words  formed,  ease 
of  pronunciation,  avoidance  of  awkward  combinations. 


270  THE   FOURTH    DIMENSION 

I  drop  "s"  altogether  from  the  consonant  series  and 
short  "  u "  from  the  vowel  series.  It  is  convenient  to 
have  unsignificant  letters  at  disposal.  '  A  double  consonant 
like  "  st "  for  instance  can  be  referred  to  without  giving  it 
a  local  significance  by  calling  it  "ust."  I  increase  the 
number  of  vowels  by  considering  a  sound  like  "  ra  "  to 
be  a  vowel,  using,  that  is,  the  letter  "r"  as  forming  a 
compound  vowel. 

The  series  is  as  follows  : — 


CONSONANTS. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

positive 

n 

t 

1 

P 

f 

sh 

k 

ch 

nt 

st 

negative 

z 

d 

th 

b 

V 

m 

g 

j 

nd 

sp 

VOWELS. 

012345        678         9 
positive         e       a       i       ee      ae      ai       ar       ra       ri       ree 
negative       er      o      oo     io      oe      iu       or       ro       roo    rio 

Pronunciation. — e  as  in  men  ;  a  as  in  man ;  i  as  in  in ; 
ee  as  in  between  ;  ae  as  ay  in  may ;  ai  as  i  in  mine ;  ar  as 
in  art ;  er  as  ear  in  earth  ;  o  as  in  on ;  oo  as  oo  in  soon ; 
io  as  in  clarion  ;  oe  as  oa  in  oat ;  iu  pronounced  like  yew. 

To  name  a  point  such  as  (23,  41)  it  is  considered  as 
(3,  1)  on  from  (20,  40)  and  is  called  "  ifeete."  It  is  the 
initial  point  of  the  square  ifeet  of  the  area  system. 

The  preceding  amplification  of  a  space  language  has 
been  introduced  merely  for  the  sake  of  completeness.  As 
has  already  been  said  nine  words  and  their  combinations, 
applied  to  a  few  simple  models  suffice  for  the  purposes  of 
our  present  enquiry. 


Printed  by  Hazell,  Watson  <k  Viney,  Ld,,  London  and  Aylttbury. 


By  THE  SAME  AUTHOR 

SCIENTIFIC    ROMANCES 

TWO  SERIES 

SERIES  I:  (i)  What  is  the  Fourth  Dimension?  (2)  The 
Persian  King ;  or,  The  Law  of  the  Valley ;  (3)  A  Picture 
of  our  Universe;  (4)  Casting  out  the  Self;  (5)  A  Plane 
World. 

SERIES  II :  (6)  Education  of  the  Imagination ;  (7)  Many 
Dimensions;  (8)  Stella;  (9)  An  Unfinished  Communi- 
cation. 

Crown  8vo,  Cloth,  6s.  each 

PALL  MALL  GAZETTE  :  "  It  is  a  treatise  of  admirable  clearness. 
Mr.  Hinton  brings  us  panting,  but  delighted,  to  at  least  a  momentary  faith 
in  the  Fourth  Dimension,  and  tipon  the  eye  there  opens  a  vista  of  interesting 
problems.  It  exhibits  a  boldness  of  speculation  and  a  power  of  conceiving 
and  expressing  even  the  inconceivable,  which  rouses  one* s  faculties  like  a 
tonic."  

AN    EPISODE    OF    FLAT-LAND; 

or,  How  a  Plane  Folk  Discovered  the  Third  Dimension 
Crown  8vo,  Cloth,  3s.  6d. 

SCOTSMAN  :  "  The  Higher  Criticism  has  come  very  largely  into  con- 
temporary fiction,  but  the  Higher  Mathematics  rarely  inspires  a  story.  This 
tale  of  Mr.  Hinton's  will  be  all  the  more  welcome  to  persons  skilled  in 
geometry,  because  the  ignorant  will  scarcely  be  able  to  understand.  It  will 
be  best  appreciated  by  mathematicians  who  like  to  speculate  about  a  fourth 
dimension  of  space,  and  who  will  be  entertained  by  a  well-sustained  account 
of  people  who  know  only  two." 


LONDON 

GEORGE    ALLEN    &    CO.,    LTD. 

RUSKIN     HOUSE 
44,   45   RATHBONE  PLACE 


Views  of  the  Tessaract. 


No.  1. 


No.  2. 


No.  3. 


No.  4. 


No.  5. 


No.  6. 


No.  7. 


No.  8. 


No.  9. 


No.  10. 


No.  11. 


No.  12