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THE  LIBRARY 
OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

LOS  ANGELES 


LECTURES   AND   ESSAYS 


< 


London,  PuttosTuJ,  fyJfaaff 


LECTURES  AND  ESSAYS 


BY   THE   LATE 


WILLIAM   KINGDON   CLIFFORD,   F.R.S. 

LATE   PROFESSOR   OF    APPLIED    MATHEMATICS  AND    MECHANICS 

IN    UNIVERSITY  COLLEGE,    LONDON  J    AND    SOMETIME 

FELLOW   OF   TRINITY   COLLEGE,    CAMBRIDGE 


EDITED    BY 

LESLIE   STEPHEN 

AND 

SIR   FREDERICK   POLLOCK 


"  La  verite  est  toute  pour  tous." — PAUL-LOUIS  COURIER 


IN  TWO  VOLUMES 
VOL.  I 


iLontJon 
MACMILLAN   AND    CO.,  LIMITED 

NEW  YORK  :   THE  MACMILLAN   COMPANY 
1901 


First  Edition,  a  Vols.     &vo.     ifyg. 
Second  Edition,  i.  Vol.     Crown  Zvo.     1886. 
Third  Edition,  Eversley  Series,  2  Vols.    Globe  Zvo. 


Eh 

Sciu 


% 


CONTENTS 


INTRODUCTION 

PART  PACK 

I.    BlOGRAPHICAI I 

II.  SELECTIONS  FROM  LETTERS,  ETC.         .        .        .  56 


LECTURES  AND   ESSAYS 

ON  SOME  OF  THE  CONDITIONS  OF  MENTAL  DEVELOP- 
MENT.       ,        .        ,_v  .  ^   .  •••,,    .»        •        •  79 
ON  THEORIES  OF  THE  PHYSICAL  FORCES  .        .        .120 
ON  THE   AIMS    AND    INSTRUMENTS   OF    SCIENTIFIC 

THOUGHT  .        ...=.'.        .        .        .  139 

ATOMS.        .        .        .    ,...,   ,    ..  ,<:.        .        .        .  181 

THE  FIRST  AND  THE  LAST  CATASTROPHE  .        .        .  222 

THE  UNSEEN  UNIVERSE 268 

THE  PHILOSOPHY  OF  THE  PURE  SCIENCES         .        .  301 


if 


INTRODUCTION 

PART  I 
BIOGRAPHICAL1 

IT  is  an  open  secret  to  the  few  who  know  it, 
but  a  mystery  and  a  stumbling-block  to  the 
many,  that  Science  and  Poetry  are  own  sisters  ; 
insomuch  that  in  those  branches  of  scientific  in- 
quiry which  are  most  abstract,  most  formal,  and 
most  remote  from  the  grasp  of  the  ordinary 
sensible  imagination,  a  higher  power  of 
imagination  akin  to  the  creative  insight  of 
the  poet  is  most  needed  and  most  fruitful  of 
lasting  work.  This  living  and  constructive 
energy  projects  itself  out  into  the  world  at  the 
same  time  that  it  assimilates  the  surrounding 
world  to  itself.  When  it  is  joined  with  quick 
perception  and  delicate  sympathies,  it  can  work 
the  miracle  of  piercing  the  barrier  that  separates 

1  Written  in  1879.  A  few  sentences  have  now  (1886)  been 
added.  Some  verbal  alterations,  mostly  rendered  necessary  by  the 
lapse  of  time,  will  explain  themselves.  — F.  P. 

VOL.  I  B 


2  INTRODUCTION 

one  mind  from  another,  and  becomes  a  personal 
charm.  It  can  be  known  only  in  its  operation, 
and  is  by  its  very  nature  incommunicable  and 
indescribable.  Yet  this  faculty,  when  a  man  is 
gifted  with  it,  seems  to  gather  up  the  best  of 
his  life,  so  that  the  man  always  transcends 
every  work  shapen  and  sent  forth  by  him  ;  his 
presence  is  full  of  it,  and  it  lightens  the  air  his 
friends  breathe  ;  it  commands  not  verbal  assent 
to  propositions  or  intellectual  acquiescence  in 
arguments,  but  the  conviction  of  being  in  the 
sphere  of  a  vital  force  for  which  nature  must 
make  room.  Therefore  when,  being  happy  in 
that  we  knew  and  saw  these  things,  and  have 
received  the  imperishable  gifts,  we  must  un- 
happily speak  of  the  friend  who  gave  them  as 
having  passed  from  us,  it  becomes  nothing  less 
than  a  duty  to  attempt  the  impossible  task,  to 
describe  that  which  admits  of  no  description, 
and  communicate  that  for  which  words  are  but 
blundering  messengers.  And  perhaps  it  may 
not  be  in  vain  ;  for  a  voice  which  is  in  itself 
weak  may  strengthen  the  kindred  notes  that 
vibrate  in  other  memories  touched  by  the  same 
power,  and  those  we  know  to  be  very  many. 
For  this  power,  when  it  works  for  fellowship 
and  not  ambition,  wins  for  its  wearer  the  love 
of  all  sorts  and  conditions  of  men,  and  this  was 
marked  in  Clifford  by  all  who  had  to  do  with 
him  even  a  little.  More  than  this,  our  words 


BIOGRAPHICAL  3 

may  peradventure  strike  further,  though  by  no 
force  or  skill  of  their  own,  and  stir  some  new 
accord  in  imaginations  favourably  attuned  for 
the  impulse.  The  discourses  and  writings 
collected  in  this  book  will  indeed  testify  to  the 
intellectual  grasp  and  acuteness  that  went  to 
the  making  of  them.  Clifford's  earnestness  and 
simplicity, 'too,  are  fairly  enough  presented  to 
the  reader,  and  the  clearness  of  his  expression 
is  such  that  any  comment  by  way  of  mere  ex- 
planation would  be  impertinent.  But  of  the 
winning  felicity  of  his  manner,  the  varied  and 
flexible  play  of  his  thought,  the  almost  bound- 
less range  of  his  human  interests  and  sym- 
pathies, his  writing  tells — at  least,  so  it  seems 
to  those  who  really  knew  him — nothing  or  very 
little.  To  say  a  word  or  two  in  remembrance 
of  one's  friend  is  but  natural ;  and  in  these 
days  excuse  is  hardly  needed  for  saying  it  in 
public.  But  here  this  is  the  least  part  of  the 
matter  in  hand.  Personal  desires  and  aims  are 
merged  in  the  higher  responsibility  of  telling 
the  world  that  it  has  lost  a  man  of  genius  ;  a 
responsibility  which  must  be  accepted  even  with 
the  knowledge  that  it  cannot  be  adequately 
discharged. 

Not  many  weeks  had  passed  of  my  first 
year  at  Trinity  when  it  began  to  be  noised 
about  that  among  the  new  minor  scholars  there 
was  a  young  man  of  extraordinary  mathematical 


4  INTRODUCTION 

powers,  and  eccentric  in  appearance,  habits,  and 
opinions.  He  was  reputed,  and  at  the  time 
with  truth,  an  ardent  High  Churchman.  I 
think  it  was  then  a  more  remarkable  thing  at 
Cambridge  than  it  would  be  now,  the  evangelical 
tradition  of  Simeon  and  his  school  being  still 
prevalent  This  was  the  first  I  heard  of  Clifford ; 
and  for  some  two  years  he  continued  to  be 
nothing  more  to  me  than  a  name  and  a  some- 
what enigmatic  person.  In  the  course  of  our 
third  year  circumstances  brought  us  together  : 
it  is  difficult  to  remember  the  beginnings  of  a 
friendship  that  seems  as  if  it  must  always  have 
been,  but  to  the  best  of  my  recollection  there 
was  nothing  very  sudden  or  rapid  in  our  closer 
approach.  I  should  assign  about  six  months 
as  the  interval  filled  by  the  transition  from 
acquaintance  to  intimacy.  At  an  early  stage 
in  my  knowledge  of  him  I  remember  being 
struck  by  the  daring  versatility  of  his  talk. 
Even  then  there  was  no  subject  on  which  he 
was  not  ready  with  something  in  point,  generally 
of  an  unexpected  kind  ;  and  his  unsurpassed 
power  of  mathematical  exposition  was  already 
longing  to  find  exercise.  I  shall  be  pardoned 
for  giving  a  concrete  instance  which  may  be  in 
itself  trivial.  In  the  analytical  treatment  of 
statics  there  occurs  a  proposition  called  Ivory's 
Theorem  concerning  the  attractions  of  an  ellip- 
soid. The  text -books  demonstrate  it  by  a 


BIOGRAPHICAL  5 

formidable  apparatus  of  co-ordinates  and  in- 
tegrals, such  as  we  were  wont  to  call  a  grind. 
On  a  certain  day  in  the  Long  Vacation  of  1 866, 
which  Clifford  and  I  spent  at  Cambridge,  I  was 
not  a  little  exercised  by  the  theorem  in  question, 
as  I  suppose  many  students  have  been  before  and 
since.  The  chain  of  symbolic  proof  seemed 
artificial  and  dead ;  it  compelled  the  under- 
standing but  failed  to  satisfy  the  reason.  After 
reading  and  learning  the  proposition  one  still 
failed  to  see  what  it  was  all  about.  Being  out 
for  a  walk  with  Clifford,  I  opened  my  per- 
plexities to  him  ;  I  think  I  can  recall  the  very 
spot.  What  he  said  I  do  not  remember  in 
detail,  which  is  not  surprising,  as  I  have  had  no 
occasion  to  remember  anything  about  Ivory's 
Theorem  these  twelve  years.  But  I  know  that 
as  he  spoke  he  appeared  not  to  be  working  out 
a  question,  but  simply  telling  what  he  saw. 
Without  any  diagram  or  symbolic  aid  he 
described  the  geometrical  conditions  on  which 
the  solution  depended,  and  they  seemed  to 
stand  out  visibly  in  space.  There  was  no 
longer  consequences  to  be  deduced,  but  real 
and  evident  facts  which  only  required  to  be 
seen.  And  this  one  instance,  fixed  in  my 
memory  as  the  first  that  came  to  my  know- 
ledge, represents  both  Clifford's  theory  of  what 
teaching  ought  to  be,  and  his  constant  way  of 
carrying  it  out  in  his  discourses  and  conversa- 


6  INTRODUCTION 

tion  on  mathematical  and  scientific  subjects. 
So  whole  and  complete  was  the  vision  that  for 
the  time  the  only  strange  thing  was  that  any- 
body should  fail  to  see  it  in  the  same  way. 
When  one  endeavoured  to  call  it  up  again,  and 
not  till  then,  it  became  clear  that  the  magic  of 
genius  had  been  at  work,  and  that  the  common 
sight  had  been  raised  to  that  higher  perception 
by  the  power  which  makes  and  transforms 
ideas,  th^  conquering  and  masterful  quality  of 
the  human  mind  which  Goethe  called  in  one 
Word  das  Damonische. 

A  soul  eager  for  new  mastery  and  ever 
looking  forward  cares  little  to  dwell  upon  the 
past ;  and  Clifford  was  not  much  apt  to  speak 
of  his  own  earlier  life,  or  indeed  of  himself  at 
all.  Hence  I  am  indebted  to  his  wife  and  to 
other  friends  for  what  little  I  am  able  to  say  of 
the  time  before  I  knew  him.  William  Kingdon 
Clifford  was  born  at  Exeter  on  May  4,  1845  ; 
his  father  was  a  well-known  and  active  citizen, 
and  filled  the  office  of  justice  of  the  peace. 
His  mother  he  lost  early  in  life  ;  he  inherited 
from  her  probably  some  of  his  genius,  and 
almost  certainly  the  deep-seated  constitutional 
weakness,  ill  paired  with  restless  activity  of 
nerve  and  brain,  which  was  the  cause  of  his 
premature  loss.  He  was  educated  at  Exeter 
till  1 860,  when  he  was  sent  to  King's  College, 
London,  not  without  distinction  already  won  in 


BIOGRAPHICAL  7 

the  University  Local  Examinations.  At  school 
he  showed  little  taste  for  the  ordinary  games, 
but  made  himself  proficient  in  gymnastics  ;  a 
pursuit  which  at  Cambridge  he  carried  out,  in 
fellowship  with  a  few  like-minded  companions, 
not  only  into  the  performance  of  the  most 
difficult  feats  habitual  to  the  gymnasium,  but 
into  the  invention  of  other  new  and  adventurous 
ones.  But  (as  he  once  said  himself  of  Dr. 
Whewell)  his  nature  was  to  touch  nothing 
without  leaving  some  stamp  of  invention  upon 
it.  His  accomplishments  of  this  kind  were  the 
only  ones  in  which  he  ever  manifested  pride. 
When  he  took  his  degree  there  was  a  paragraph 
in  Bell's  Life  pointing  out,  for  the  rebuke  of  those 
who  might  suppose  manly  exercises  incompatible 
with  intellectual  distinction,  that  the  Second 
Wrangler,  Mr.  Clifford,  was  also  one  of  the  most 
daring  athletes  of  the  University.  This  paragraph 
gave  him  far  more  lively  pleasure  than  any  of 
the  more  serious  and  academical  marks  of 
approval  which  he  had  earned.  In  1 869  he 
wrote  from  Cambridge  : — "  I  am  at  present  in 
a  very  heaven  of  joy  because  my  corkscrew 
was  encored  last  night  at  the  assault  of  arms  : 
it  consists  in  running  at  a  fixed  'upright  pole 
which  you  seize  with  both  hands  and  spin 
round  and  round  descending  in  a  corkscrew 
fashion."  In  after  years  he  did  not  keep  up 
his  gymnastic  practice  with  anything  like 


regularity;  but  he  was  with  great  difficulty 
induced  to  accept  the  necessity  of  completely 
abandoning  it  when  it  was  known  to  be  posi- 
tively injurious  to  his  health.  A  friend  who 
was  his  companion  in  gymnastics  writes  to 
me : — "  His  neatness  and  dexterity  were  un- 
usually great,  but  the  most  remarkable  thing 
was  his  great  strength  as  compared  with  his 
weight,  as  shown  in  some  exercises.  At  one 
time  he  could  pull  up  on  the  bar  with  either 
hand,  which  is  well  known  to  be  one  of  the 
greatest  feats  of  strength.  His  nerve  at 
dangerous  heights  was  extraordinary.  I  am 
appalled  now  to  think 'that  he  climbed  up  and 
sat  on  the  cross  bars  of  the  weathercock  on  a 
church  tower,  and  when  by  way  of  doing  some- 
thing worse  I  went  up  and  hung  by  my  toes  to 
the  bars  he  did  the  same." 

At  King's  College  Clifford's  peculiar  mathe- 
matical abilities  came  to  the  front,  but  not  so 
as  to  exclude  attention  to  other  subjects.  He 
was  at  various  times  and  in  various  ways  marked 
out  for  honourable  mention  in  classics,  modern 
history,  and  English  literature.  His  knowledge 
of  the  classics,  though  he  did  not  cultivate  the 
niceties  of  scholarship,  was  certainly  as  sound 
and  extensive  as  that  of  many  professedly 
classical  students  ;  and,  like  all  his  knowledge, 
it  was  vital.  If  he  made  use  of  it  for  quota- 
tion or  otherwise,  it  was  not  because  the 


BIOGRAPHICAL  9 

passage  or  circumstance  was  classical,  but 
because  it  was  the  thing  he  wanted  to  illustrate 
his  own  thought.  Of  history  he  knew  a  good 
deal  ;  he  was  fond  of  historical  reading 
throughout  his  life,  and  had  a  ready  corr.mand 
of  parallels  and  analogies  between  widely 
remote  times  and  countries,  sometimes  too 
ingenious  to  bear  criticism.  I  doubt  if  he 
studied  historical  works  critically  ;  it  seems  to 
me  that  he  regarded  history  in  a  poetical  rather 
than  a  scientific  spirit,  seeing  events  in  a  series 
of  vivid  pictures  which  had  the  force  of  present 
realities  as  each  came  in  turn  before  the  mind's 
eye.  Thus  he  threw  himself  into  the  past  with 
a  dramatic  interest  and  looked  on  the  civilised 
world  as  a  field  where  the  destinies  of  man  are 
fought  out  in  a  secular  contest  between  the 
powers  of  good  and  evil,  rather  than  as  a  scene 
of  the  development  and  interaction  of  infinite 
and  infinitely  complex  motives.  This  indeed, 
in  a  meagre  and  far  cruder  form,  is  essentially 
the  popular  view ;  the  sort  of  history  upon 
which  most  people  are  still  brought  up  divides 
men,  actions,  and  institutions  into  good  and 
bad  according  to  the  writer's  present  notions 
of  what  might  and  ought  to  be,  and  distributes 
blessing  and  cursing  without  more  ado.1  Only 
Clifford,  accepting  to  some  extent  the  popular 
or  pictorial  way  of  looking  at  history,  took  on 

1  As  children  learning  history  say — "  But  was  he  a  good  man  ?  " 


io  INTRODUCTION 

most  questions  the  unpopular  side,  and  so 
found  himself  in  collision  with  current  opinions. 
He  had  a  fair  general  knowledge  of  English 
literature  (by  which  I  mean  considerably  more 
than  is  yet  supposed  necessary  for  an  English- 
man's education),  with  a  preference  for  modern 
poetry,  and  especially  for  such  as  gave  expres- 
sion to  his  own  ideas.  Milton's  prose  had  also 
a  special  attraction  for  him.  I  do  not  think 
he  cared  much  for  the  use  of  language  as  a 
fine  art,  though  he  had  a  great  appreciation  of 
arrangement  and  composition.  His  own  style, 
always  admirably  clear  and  often  eloquent,  was 
never  elaborate  ;  for  we  cannot  fairly  count  the 
studied  ornament  of  his  College  declamations, 
which  were  not  only  produced  while  he  was 
an  undergraduate,  but  for  an  occasion  which 
justified  some  special  aiming  at  rhetorical  effect. 
Much  of  his  best  work  was  actually  spoken 
before  it  was  written.  He  gave  most  of  his 
public  lectures  with  no  visible  preparation 
beyond  very  short  notes,  and  the  outline 
seemed  to  be  filled  in  without  effort  or  hesita- 
tion. Afterwards  he  would  revise  the  lecture 
from  a  shorthand-writer's  report,  or  sometimes 
write  down  from  memory  almost  exactly  what 
he  had  said.  It  fell  out  now  and  then,  how- 
ever, that  neither  of  these  things  was  done  ; 
and  in  such  cases  there  is  now  no  record  of 
the  lecture  at  all.  Once  or  twice  he  tried 


BIOGRAPHICAL  11 

writing  part  of  the  lecture  beforehand,  but 
found  it  only  an  embarrassment  in  the  delivery. 
I  believe  the  only  one  wholly  put  in  writing  in 
the  first  instance  was  "  Ethics  of  Religion," 
which  he  was  unable  to  deliver  himself.  I 
cannot  find  anything  showing  early  aptitude 
for  acquiring  languages  ;  but  that  he  had  it 
and  was  fond  of  exercising  it  in  later  life  is 
certain.  One  practical  reason  for  it  was  the 
desire  of  being  able  to  read  mathematical 
papers  in  foreign  journals  ;  but  this  would  not 
account  for  his  taking  up  Spanish,  of  which  he 
acquired  a  competent  knowledge  in  the  course 
of  a  tour  to  the  Pyrenees.  When  he  was  at 
Algiers  in  1876  he  began  Arabic,  and  made 
progress  enough  to  follow  in  a  general  way 
a  course  of  lessons  given  in  that  language. 
He  read  modern  Greek  fluently,  and  at  one 
time  he  was  curious  about  Sanskrit.  He  even 
spent  some  time  on  hieroglyphics.  A  new 
language  is  a  riddle  before  it  is  conquered,  a 
power  in  the  hand  afterwards  :  to  Clifford 
every  riddle  was  a  challenge,  and  every  chance 
of  new  power  a  divine  opportunity  to  bd  seized. 
Hence  he  was  likewise  interested  in  the  various 
modes  of  conveying  and  expressing  language 
invented  for  special  purposes,  such  as  the 
Morse  alphabet  and  shorthand.  One  of  his 
ideas  about  education  was  that  children  might 
learn  these  things  at  an  early  age,  perhaps  in 


12  INTRODUCTION 

play,  so  as  to  grow  up  no  less  familiar  with 
them  than  with  common  printing  and  writing. 
I  have  forgotten  to  mention  his  command  of 
French  and  German,  the  former  of  which  he 
knew  very  well,  and  the  latter  quite  sufficiently  ; 
I  think  his  German  reading  was  mostly  in  the 
direction  of  philosophy  and  mathematics. 

In  1863  Clifford  came  up  with  a  minor 
scholarship  to  Trinity  College,  Cambridge  ;  in 
his  third  year  (to  continue  for  the  present  on 
the  line  of  his  literary  accomplishments)  he 
won  the  College  declamation  prize1  with  a 
very  brilliant  discourse  on  Sir  W.  Raleigh, 
partly  cast  in  the  form  of  quasi  -  dramatic 
dialogues,  and  accordingly  had  to  deliver  the 
annual  oration  at  the  Commemoration  of 
Benefactors  in  December.  His  subject  was  a 
panegyric  of  the  late  Master  of  the  College, 
Dr.  Whewell,  whose  death  was  then  recent  It 
was  treated  in  an  original  and  unexpected 
manner,  Dr.  Whewell's  claim  to  admiration 
and  emulation  being  put  on  the  ground  of  his 
intellectual  life  exemplifying  in  an  eminent 
degree  the  active  and  creating  faculty. 
"  Thought  is  powerless  except  it  make  some- 
thing outside  of  itself:  the  thought  which 
conquers  the  world  is  not  contemplative  but 

1  He  was  bracketed  with  Mr.  C.  A.  Elliott  for  the  first  prize  ; 
but  (I  now  forget  for  what  reason)  the  office  of  delivering  the 
Oration  fell  to  Clifford  alone. 


BIOGRAPHICAL  13 

active.  And  it  is  this  that  I  am  asking  you 
to  worship  to-day."  Taking  this  oration  as  a 
whole,  it  must  be  considered  as  a  tour  de  force, 
giving  glimpses  and  undetermined  promises  of 
speculative  power.  But  there  occurred  in  it  an 
apologue  which  caught  the  attention  of  some 
good  judges  at  the  time,  and  so  well  illustrates 
the  fanciful  and  sportive  side  of  Clifford's  mind 
that  I  shall  here  transcribe  it. 

"  Once  upon  a  time — much  longer  than  six 
thousand  years  ago — the  Trilobites  were  the 
only  people  that  had  eyes  ;  and  they  were  only 
just  beginning  to  have  them,  and  some  even 
of  the  Trilobites  had  as  yet  no  signs  of  coming 
sight.  So  that  the  utmost  they  could  know 
was  that  they  were  living  in  darkness,  and  that 
perhaps  there  was  such  a  thing  as  light.  But  at 
last  one  of  them  got  so  far  advanced  that  when 
he  happened  to  come  to  the  top  of  the  water 
in  the  daytime  he  saw  the  sun.  So  he  went 
down  and  told  the  others  that  in  general  the 
world  was  light,  but  there  was  one  great  light 
which  caused  it  all.  Then  they  killed  him  for 
disturbing  the  commonwealth  ;  but  they  con- 
sidered it  impious  to  doubt  that  in  general  the 
world  was  light,  and  that  there  was  one  great 
light  which  caused  it  all.  And  they  had  great 
disputes  about  the  manner  in  which  they  had 
come  to  know  this.  Afterwards  another  of 
them  got  so  far  advanced  that  when  he 


,4  INTRODUCTION 

happened  to  come  to  the  top  of  the  water  in 
the  night-time  he  saw  the  stars.  So  he  went 
down  and  told  the  others  that  in  general  the 
world  was  dark,  but  that  nevertheless  there 
was  a  great  number  of  little  lights  in  it.  Then 
they  killed  him  for  maintaining  false  doctrines  : 
but  from  that  time  there  was  a  division  amongst 
them,  and  all  the  Trilobites  were  split  into  two 
parties,  some  maintaining  one  thing  and  some 
the  other,  until  such  time  as  so  many  of  them 
had  learned  to  see  that  there  could  be  no  doubt 
about  the  matter." 

The  interpretation  was  barely  indicated  on 
this  occasion  ;  but  it  is  worked  out  in  another 
Cambridge  MS.  of  somewhat  later  date,1  in 
which  the  apologue  stands  first  as  a  kind  of 
text  It  was  nothing  less  than  a  theory  of  the 
intellectual  growth  of  mankind  ;  and  the  posi- 
tion was  that,  as  the  physical  senses  have  been 
gradually  developed  out  of  confused  and  un- 
certain impressions,  so  a  set  of  intellectual 
senses  or  insights  are  still  in  course  of  develop- 
ment, the  operation  of  which  may  ultimately  be 
expected  to  be  as  certain  and  immediate  as  our 
ordinary  sense-perceptions. 

This  theory  may  be  traced  in  the  discourse 

1  It  has  now  (1886)  been  ascertained  that  this  MS.,  which  was 
found  among  Clifford's  papers  fairly  written  out,  but  without  title 
or  indication  of  date,  was  used  for  a  lecture  delivered  to  a  military 
audience  at  Woolwich  in  1869.  Still  the  ideas  distinctly  belong 
to  an  early  and  tentative  stage. 


BIOGRAPHICAL  15 

"  On  some  of  the  Conditions  of  Mental  Develop- 
ment," delivered  in  March  1868,  which  stands 
first  in  the  present  collection  ;  and  for  that 
reason  I  make  special  mention  of  it.  Other- 
wise it  was  only  one  inventive  experiment 
among  many.  I  should  far  exceed  my  limits 
if  I  were  to  attempt  any  account  of  the  various 
forms  of  speculation,  physical,  metaphysical, 
social,  and  ethical,  through  which  Clifford 
ranged  in  the  first  few  years  after  his  degree. 
Not  that  he  was  constantly  changing  his 
opinions,  as  a  superficial  observer  might  have 
thought ;  he  was  seeking  for  definite  principles, 
and  of  set  purpose  made  his  search  various  and 
widespread.  He  had  a  singular  power  of  taking 
up  any  theory  that  seemed  to  him  at  all  worth 
investigating,  realising  it,  working  it  out,  and 
making  it  completely  his  own  for  the  time  being, 
and  yet  all  the  while  consciously  holding  it  as 
an  experiment,  and  being  perfectly  ready  to 
give  it  up  when  found  wanting. 

Clifford's  mathematical  course  at  Cambridge 
was  a  struggle  between  the  exigencies  of  the 
Tripos  and  his  native  bent  for  independent 
reading  and  research  going  far  beyond  the  sub- 
jects of  the  examination  ;  and  the  Tripos  had 
very  much  the  worst  of  it.  If  there  was  any 
faculty  in  which  he  was  entirely  wanting,  it  was 
the  examination-faculty.  On  this  subject  I  am 
not  competent  to  speak  with  certainty,  but  it  is 


,6  INTRODUCTION 

my  belief  that,  from  the  point  of  view  to  which 
the  class-list  is  an  end  in  itself,  Clifford  omitted 
most  of  the  things  he  ought  to  have  read,  and 
read  everything  he  ought  not  to  have  read. 
Nevertheless  his  powers  of  original  work  carried 
him  so  far  that  he  came  out  Second  Wrangler 
in  the  Tripos  of  1867,  and  was  also  Second 
Smith's  Prizeman.  I  am  fortunately  able  to 
quote  on  this  head  the  statement  of  one  of  our 
first  living  analysts,  Professor  Sylvester  : — 

"  Like  the  late  Dr.  Whewell,  Professor 
Clerk  Maxwell,  and  Sir  William  Thomson, 
Mr.  Clifford  was  Second  Wrangler  at  the 
University  of  Cambridge.  I  believe  there  is 
little  doubt  that  he  might  easily  have  been  first 
of  his  year  had  he  chosen  to  devote  himself  ex- 
clusively to  the  University  curriculum  instead 
of  pursuing  his  studies,  while  still  an  under- 
graduate, in  a  more  extended  field,  and  with 
a  view  rather  to  self-culture  than  to  the  acquisi- 
tion of  immediate  honour  or  emolument." 

This  pursuit  of  knowledge  for  its  own  sake, 
and  without  even  such  regard  to  collateral  in- 
terests as  most  people  would  think  a  matter  of 
common  prudence,  was  the  leading  character  of 
Clifford's  work  throughout  his  life.  The  dis- 
covery of  truth  was  for  him  an  end  in  itself, 
and  the  proclamation  of  it,  or  of  whatever 
seemed  to  lead  to  it,  a  duty  of  primary  and 
paramount  obligation.  This  had  something  to 


BIOGRAPHICAL  17 

do  with  the  fascination  of  his  teaching ;  he 
never  seemed  to  be  imposing  dogmas  on  his 
hearers,  but  to  be  leading  them  into  the  enjoy- 
ment of  a  common  possession.  He  did  not 
tell  them  that  knowledge  was  priceless  and 
truth  beautiful ;  he  made  them  feel  it.  He  gave 
them  not  formulas,  but  ideas.  Again  I  can 
appeal  to  a  witness  of  undoubted  authority. 
The  following  words  were  written  in  1871  by 
a  man  who  was  in  no  way  given  to  unmeasured 
expression  of  his  mind,  and  who  was  as  eminent 
in  mathematical  physics  as  the  author  of  the 
statement  I  have  already  cited  is  in  pure  mathe- 
matics— I  mean  Clerk  Maxwell : — 

"  The  peculiarity  of  Mr.  Clifford's  researches, 
which  in  my  opinion  points  him  out  as  the 
right  man  for  a  chair  of  mathematical  science, 
is  that  they  tend  not  to  the  elaboration  of 
abstruse  theorems  by  ingenious  calculations, 
but  to  the  elucidation  of  scientific  ideas  by  the 
concentration  upon  them  of  clear  and  steady 
thought.  The  pupils  of  such  a  teacher  not 
only  obtain  clearer  views  of  the  subjects  taught, 
but  are  encouraged  to  cultivate  in  themselves 
that  power  of  thought  which  is  so  liable  to  be 
neglected  amidst  the  appliances  of  education." 

I  shall  not  attempt  to  enter  in  more  detail 
on  the  amount  and  character  of  Clifford's  sub- 
sequent contributions  to  mathematical  science. 
But  in  an  introduction  to  his  philosophical 

VOL.  i  c 


,8  INTRODUCTION 

writings  it  is  fitting  to  call  attention  to  the 
manner  in  which  he  brought  mathematical  con- 
ceptions to  bear  upon  philosophy.  He  took 
much  pleasure  in  the  speculative  constructions 
of  imaginary  or  non-Euclidean  systems  of  space- 
relations  which  have  been  achieved  by  Conti- 
nental geometers,  partly  because  they  afforded 
a  congenial  field  for  the  combined  exercise  of 
scientific  intuition  and  unbridled  fancy.  He 
liked  talking  about  imaginary  geometry,  as  a 
matter  of  pure  amusement,  to  any  one  interested 
in  it.  But  at  the  same  time  he  attached  a 
serious  import  to  it  He  was  the  first  in 
this  country,  as  Helmholtz  in  Germany,  to  call 
attention  to  the  philosophical  importance  of 
these  new  ideas  with  regard  to  the  question  of 
the  nature  and  origin  of  geometrical  knowledge. 
His  opinion  on  this  point  is  briefly  expressed 
in  the  lectures  "  On  the  Philosophy  of  the  Pure 
Sciences."  He  intended  to  recast  and  expand 
these,  and  doubtless  would  have  amplified  this 
particular  discussion.  It  will  be  seen  that  he 
considered  Kant's  position  in  the  matter  of 
"transcendental  aesthetic"  to  be  wholly  un- 
assailable if  it  was  once  admitted  that  geo- 
metrical knowledge  is  really  exact  and  universal. 
The  ordinary  arguments  for  the  derivative  nature 
of  axioms  appeared  to  him  ingenious  but  hope- 
less attempts  to  escape  from  this  fatal  admission. 
And  it  may  be  said  in  general  terms  that  he 


BIOGRAPHICAL  19 

had  a  much  fuller  appreciation  of  the  merit 
and  the  necessity  of  Kant's  work  than  most 
adherents  of  the  English  school  of  psychology. 
Of  course  I  do  not  include  Professor  Huxley, 
whose  testimony  to  Kant  in  his  little  book  on 
Hume  is  as  unmistakable  as  it  is  weighty. 

Few  words  will  suffice  to  set  down  the  re- 
maining facts  of  Clifford's  life,  or  what  we  are 
accustomed  to  call  facts  because  they  can  be 
dated  and  made  equally  known  to  everybody, 
as  if  that  made  them  somehow  more  real  than 
the  passages  and  events  which  in  truth  decide 
the  issues  of  life  and  fix  the  courses  of  a  man's 
work.  In  1868  he  was  elected  a  Fellow  of 
Trinity  College,  and  after  spending  rather  more 
than  two  years  at  Cambridge,  he  was  in  1871 
appointed  to  the  Professorship  of  Applied 
Mathematics  at  University  College,  London. 
Meanwhile  he  had  taken  part  in  the  English 
Eclipse  expedition  of  1870  :  his  letters  of  that 
time  show  keen  enjoyment  of  the  new  ex- 
perience of  men  and  cities,  and  of  the  natural 
beauty  of  the  Mediterranean  coasts,  which  he 
was  to  visit  again,  as  fate  would  have  it,  only 
on  the  sad  and  fruitless  errand  of  attempting 
to  recover  strength  when  it  was  too  late.  In 
June  1874  he  was  elected  a  Fellow  of  the 
Royal  Society ;  he  might  have  been  proposed 
at  a  much  earlier  time,  but  had  then  declined, 
turning  it  off  with  the  remark  that  he  did  not 


20  INTRODUCTION 

want  to  be  respectable  yet  And  such  was  the 
absence  in  him  of  anything  like  vanity  or  self- 
assertion,  that  when  his  scruples  were  overcome, 
and  his  election  took  place,  he  was  the  last 
person  from  whom  his  friends  heard  of  it.  I 
did  not  know  it  myself  till  several  months  later. 
On  April  7,  1875,  he  married  Lucy,  daughter 
of  Mr.  John  Lane,  formerly  of  Barbados.  This 
was  the  occasion  of  the  only  voluntary  leave  of 
absence  he  ever  took  from  his  lectures  at 
University  College,  when  he  characteristically 
informed  his  class  that  he  was  obliged  to  be 
absent  on  important  business  which  would  prob- 
ably not  occur  again.  Clifford's  house  was 
thenceforward  (as,  indeed,  his  rooms,  both  at 
Cambridge  and  in  London,  had  already  been) 
the  meeting -point  of  a  numerous  body  of 
friends,  in  which  almost  every  possible  variety 
of  taste  and  opinion  was  represented,  and  many 
of  whom  had  nothing  else  in  common.  The 
scientific  element  had  naturally  a  certain  pre- 
dominance ;  and  with  Clifford,  as  with  other 
men,  a  close  friendship  implied,  as  a  rule,  some 
sort  of  general  coincidence  in  sentiments  and 
aims,  personal  and  intellectual  concord  being 
apt  to  go  together.  But  he  cared  for  sympathy, 
not  for  agreement ;  coincidence  in  actual  results 
was  indifferent  to  him.  He  wrote  of  a  very 
near  and  dear  friend  (G.  Crotch,  of  St.  John's 
College,  Cambridge),  whose  death  preceded  his 


BIOGRAPHICAL  21 

own  by  some  years  :  "  We  never  agreed  upon 
results,  but  we  always  used  the  same  method 
with  the  same  object."  Much  more  would  it 
be  a  mistake  to  suppose  that  Clifford  was  a 
scientific  fanatic  who  reserved  his  social 
qualities  for  such  persons  as  happened  to 
accept  his  theories,  or  that  he  could  not  be  at 
his  ease  and  make  the  charm  of  his  presence 
felt  among  those  who  did  not  care  for  theories 
at  all.  It  was  possible  to  take  offence  at  certain 
passages  in  his  writings,  but  impossible  not  to 
like  the  man ;  and  some  of  those  to  whom 
Clifford's  published  opinions  were  naturally 
most  repugnant,  but  who  had  the  opportunity 
of  personal  intercourse  with  him,  were  by  no 
means  the  last  to  express  their  sympathy  and 
anxiety  when  the  threatenings  of  the  disease 
which  carried  him  off  became  apparent.  This 
charm  remained  with  him  to  his  very  last  days ; 
even  when  he  was  in  an  enfeebled  and  almost 
prostrate  condition  there  were  those  who  con- 
ceived for  him  and  his,  upon  sudden  and  casual 
acquaintance,  an  affection  and  good-will  which 
bore  such  fruit  of  kindly  deeds  as  men  usually 
look  for  only  from  the  devotion  ripened  by 
long  familiarity.  Something  of  this  was  due 
to  the  extreme  openness  and  candour  of  his 
conversation  ;  something  to  the  quickness  with 
which  he  read  the  feelings  of  others,  and  the 
delicacy  and  gentleness  with  which  he  adapted 


22  INTRODUCTION 

himself  to  them  ;  something,  perhaps  most,  to 
a  certain  undefinable  simplicity  in  which  the 
whole  man  seemed  to  be  revealed,  and  the 
whole  moral  beauty  of  his  character  to  be 
grounded.  It  was  by  this  simplicity,  one  may 
suppose,  that  he  was  endeared  from  his  early 
days  to  children.  He  always  took  delight  in 
being  with  them,  and  appeared  to  have  a  special 
gift  of  holding  their  attention.  That  he  did 
not  live  to  teach  his  own  children  is  deeply  to 
be  regretted  not  only  for  their  sake,  but  in 
the  interest  of  education  as  a  science  and  an 
art.  What  he  could  do  for  the  amusement  of 
children  (and  of  all  persons  healthy  enough  not 
to  be  ashamed  of  childishness)  was  shown  to 
the  world  in  his  contributions  to  a  collection  of 
fairy  tales  called  The  Little  People.  One  of 
these  ("The  Giant's  Shoes")  is  one  of  the 
choicest  pieces  of  pure  nonsense  ever  put 
together  ;  and  he  doubtless  enjoyed  writing  it 
as  much  as  any  child  could  enjoy  hearing  it. 
A  children's  party  was  one  of  Clifford's  greatest 
pleasures.  At  one  such  party  he  kept  a  wax- 
work show,  children  doing  duty  for  the  figures  ; 
but  he  reproached  himself  for  several  days  after- 
wards because  he  had  forgotten  to  wind  up  the 
Siamese  twins.  He  seemed  to  have  an  in- 
exhaustible store  of  merriment  at  all  times  : 
not  merely  a  keen  perception  of  the  ludicrous, 
but  an  ever  fresh  gaiety  and  gladness  in  the 


BIOGRAPHICAL  23 

common  pleasures  of  life.  His  laughter  was 
free  and  clear  like  a  child's,  and  as  little  re- 
strained by  any  consideration  of  conventional 
gravity.  And  he  carried  his  mirth  and  humour 
into  all  departments  of  life,  by  no  means  ex- 
cepting philosophy.  When  he  came  home 
from  the  meetings  of  the  Metaphysical  Society 
(attending  which  was  one  of  his  greatest 
pleasures,  and  most  reluctantly  given  up  when 
going  abroad  after  sunset  was  forbidden  him), 
he  would  repeat  the  discussion  almost  at  length, 
giving  not  only  the  matter  but  the  manner  of 
what  had  been  said  by  every  speaker,  and  now 
and  then  making  his  report  extremely  comic 
by  a  touch  of  plausible  fiction.  There  was  an 
irresistible  affectation  of  innocence  in  his  manner 
of  telling  an  absurd  story,  as  if  the  drollery  of 
it  were  an  accident  with  which  he  had  nothing 
to  do.  It  was  hardly  possible  to  be  depressed 
in  his  company  :  and  this  was  so  not  only  in 
his  best  days,  but  as  long  as  he  had  strength 
to  sustain  conversation  at  all.  The  charm  of 
his  countenance  and  talk  banished  for  the  time 
the  anxiety  we  felt  for  him  (only  too  justly) 
whenever  we  were  not  with  him. 

On  the  intellectual  side  this  character  of 
simplicity  manifested  itself  in  the  absolute 
straightforwardness  of  everything  he  said  and 
did  ;  and  this,  being  joined  to  subtlety  and  a 
wide  range  of  vision,  became  in  speculation 


24  INTRODUCTION 

and  discussion  a  very  formidable  power.  If 
there  was  anything  for  which  he  had  no  tolera- 
tion, and  with  which  he  would  enter  into  no 
compromise,  it  was  insincerity  in  thought,  word, 
or  deed.  He  expressed  his  own  opinions 
plainly  and  strongly  because  he  held  it  the 
duty  of  every  man  so  to  do ;  he  could  not 
discuss  great  subjects  in  a  half-hearted  fashion 
under  a  system  of  mutual  conventions.  As  for 
considerations  of  policy  or  expediency  that 
seemed  to  interfere  in  any  way  with  the 
downright  speaking  of  truth  for  the  truth's 
sake,  he  was  simply  incapable  of  entertaining 
them.  "  A  question  of  right  and  wrong,"  he 
once  wrote  to  me,  "  knows  neither  time,  place, 
nor  expediency."  Being  always  frank,  he  was 
at  times  indiscreet ;  but  consummate  discretion 
has  never  yet  been  recognised  as  a  necessary 
or  even  a  very  appropriate  element  of  moral 
heroism.  This  must  be  borne  in  mind  in  esti- 
mating such  passages  of  his  writings  as,  judged 
by  the  ordinary  rules]  of  literary  etiquette,  may 
seem  harsh  and  violent 

Personal  enmity  was  a  thing  impossible  to 
Clifford.     Once  he  wrote  :  "  A  great  misfortune 

has  fallen  upon  me  ;   I  shook  hands  with . 

I  believe  if  all  the  murderers  and  all  the  priests 
and  all  the  liars  in  the  world  were  united  into 
one  man,  and  he  came  suddenly  upon  me  round 
a  corner  and  said,  '  How  do  you  do  ?'  in  a 


BIOGRAPHICAL  25 

smiling  way,  I  could  hot  be  rude  to  him  upon 
the  instant."  And  it  was  the  bare  truth. 
Neither  did  he  ever  make  an  enemy  that  I 
know  of ;  I  do  not  count  one  or  two  blundering 
attacks  which,  however  far  they  might  go  beyond 
the  fair  bounds  of  controversy  or  satire,  were 
made  by  people  who  only  guessed  at  the  man 
from  a  superficial  inspection  of  his  writings, 
and  were  incapable  of  understanding  either. 
Yet  he  carried  about  with  him  as  deadly  a  foe 
as  could  have  been  wished  him  by  any  of  those 
who  fear  and  hate  the  light  he  strove  so  man- 
fully to  spread  abroad.  This  was  the  perilous 
excess  in  his  own  frame  of  nervous  energy  over 
constitutional  strength  and  endurance.  He  was 
able  to  call  upon  himself,  with  a  facility  which 
in  the  result  was  fatal,  for  the  expenditure  of 
power  in  ways  and  to  an  extent  which  only  a 
strong  constitution  could  have  permanently 
supported  ;  and  here  the  constitution  was  feeble. 
He  tried  experiments  on  himself  when  he  ought 
to  have  been  taking  precautions.  He  thought, 
I  believe,  that  he  was  really  training  his  body 
to  versatility  and  disregard  of  circumstances, 
and  fancied  himself  to  be  making  investments 
when  he  was  in  fact  living  on  his  capital.  At 
Cambridge  he  would  constantly  sit  up  most  of 
the  night  working  or  talking.  In  London  it 
was  not  very  different,  and  once  or  twice  he 
wrote  the  whole  night  through  ;  and  this  without 


36  INTRODUCTION 

any  proportionate  reduction  of  his  occupations 
in  more  usual  hours.  The  paper  on  "The 
Unseen  Universe "  was  composed  in  this  way, 
except  a  page  or  two  at  the  beginning,  at  a 
single  sitting  which  lasted  from  a  quarter  to 
ten  in  the  evening  till  nine  o'clock  the  following 
morning.  So,  too,  was  the  article  on  Virchow's 
address.  But  Clifford's  rashness  extended 
much  further  than  this  one  particular.  He 
could  not  be  induced,  or  only  with  the  utmost 
difficulty,  to  pay  even  moderate  attention  to 
the  cautions  and  observances  which  are 
commonly  and  aptly  described  as  taking  care 
of  one's  self.  Had  he  been  asked  if  it  was 
wrong  to  neglect  the  conditions  of  health  in 
one's  own  person,  as  well  as  to  approve  or 
tolerate  their  neglect  on  a  larger  scale,  he 
would  certainly  have  answered  yes.  But  to  be 
careful  about  himself  was  a  thing  that  never 
occurred  to  him.  Even  when,  in  the  spring 
of  1 876,  distinct  and  grave  indications  of 
pulmonary  disease  were  noted,  his  advisers  and 
friends  could  hardly  persuade  him  that  there 
was  anything  more  serious  than  could  be  set 
right  by  two  or  three  weeks'  rest  in  the  country. 
Here,  however,  there  came  into  play  something 
more  than  incredulity  or  indifference  ;  the  spirit 
of  the  worker  and  inventor  rebelled  against 
thus  being  baffled.  His  repugnance  was  like 
that  of  a  wounded  soldier  who  thinks  himself 


BIOGRAPHICAL  27 

dishonoured  if  he  quits  the  field  while  his  limbs 
can  bear  him.  Reluctantly  and  almost  in- 
dignantly he  accepted  six  months'  leave  of 
absence,  and  spent  the  summer  of  that  year  in 
a  journey  to  Algiers  and  the  south  of  Spain. 
He  came  back  recruited  for  the  time,  and  was 
allowed  to  winter  in  England  on  pledges  of 
special  care  and  avoidance  of  exposure.  These 
were  in  the  main  observed,  and  so  matters  went 
on  for  a  year  and  a  half  more,  as  it  seemed 
with  fair  prospects  of  ultimate  recovery  and 
tolerably  secure  enjoyment  of  life.  What 
mischief  was  already  done  could  not  be  undone  ; 
but  the  spread  of  it  seemed  in  a  way  to  be 
permanently  arrested.  But  in  the  early  months 
of  1878  there  came  a  sudden  change  for  the 
worse.  His  father's  death,  which  happened  at 
this  time,  was  a  grievous  blow,  and  the  conjunc- 
tion of  this  with  exciting  literary  work,  done 
under  pressure  of  time,  threw  upon  him  a  strain 
which  he  was  wholly  unable  to  resist.  The 
essay  on  Virchow's  address,  which  closes  the 
present  collection,  is  both  in  my  opinion  and 
in  that  of  other  and  more  competent  judges 
one  of  Clifford's  best  and  most  mature  perform- 
ances. But  it  was  produced  at  a  fearful  cost, 
we  have  already  seen  in  what  manner.  A  few 
days  after  the  MS.  had  left  his  hands  he 
received  a  peremptory  warning  that  he  was  in 
a  state  of  such  imminent  danger  that  he  must 


28  INTRODUCTION 

give  up  all  work  and  leave  England  forthwith. 
This  time  the  warning  was  too  stern  to  admit 
of  doubt  or  even  delay.  Yet,  while  the  neces- 
sary preparations  were  in  hand,  he  would  not 
leave  his  official  duties  until  he  actually  broke 
down  in  the  attempt  to  complete  a  lecture. 
He  was  now  suffering,  not  from  any  inroad  of 
specific  local  disease,  but  from  a  rapid  and 
alarming  collapse  of  general  strength  which 
made  it  seem  doubtful  if  he  could  live  many 
weeks.  But  his  constitutional  frailty  was 
accompanied  withal  by  a  wonderful  power  of 
rallying  from  prostration  ;  and  one  could  not 
help  entertaining  a  dim  hope,  even  to  the  last, 
that  this  vitality  was  somehow  the  deepest 
thing  in  his  nature,  and  would  in  the  long  run 
win  the  day.  In  April  that  year,  Clifford  and 
his  wife  left  England  for  the  Mediterranean  ; 
the  accounts  they  sent  home  were  various  and 
often  anxious  ;  but  after  voyages  and  short 
halts  which  embraced  Gibraltar,  Venice,  and 
Malta,  they  rested  for  some  weeks  at  Monte 
Generoso,  and  there  for  the  first  time  there  was 
the  appearance  of  steady  improvement  setting 
in.  From  this  place  Clifford  wrote  long  letters 
with  his  own  hand,  full  of  his  usual  spirit  and 
manifold  interest  in  everything  about  him.  I 
may  mention  here  that  his  letters  were  the 
more  valuable  because  they  were  always  spon- 
taneous and  could  seldom  be  counted  on  before- 


BIOGRAPHICAL  29 

hand.  He  wrote  quickly  and  easily  ;  and  yet 
for  some  obscure  reason  letter-writing,  especially 
as  a  matter  of  business,  was  beyond  measure 
irksome  and  difficult  to  him.  He  would  rather 
take  almost  any  trouble  than  answer  a  letter, 
and  the  painfulness  of  answering  was  at  its 
height  when  (as  pretty  often  happened)  old 
acquaintances  applied  to  him  for  testimonials. 
For  in  this  case  it  was  aggravated  by  the  utter 
impossibility  of  lending  himself  to  the  petty 
exaggerations  and  dissimulations  which  custom 
allows  to  pass  current  for  such  purposes,  and 
which  are  almost  thought  to  be  required  by 
civility.  One  such  application,  from  a  man  he 
had  known  before  but  had  lost  sight  of,  vexed 
him  extremely  ;  he  did  not  know  what  to  do 
with  it,  for  he  could  honestly  have  certified  only 
as  to  the  past,  and  he  carried  the  letter  about 
with  him  till  it  was  ragged,  being  newly  vexed 
every  time  he  saw  it.  There  were  many 
letters  of  friends  which  he  regretted  to  the  last 
not  having  answered.  Several  received  in  the 
last  months  or  weeks  of  his  life  he  intended  to 
answer  if  he  had  ever  become  strong  enough. 
Yet  now  and  then  he  would  write  unsought  to 
some  one  he  was  intimate  with,  and  throw  him- 
self completely  into  his  letter ;  and  then  his 
descriptions  were  so  full  of  life  and  colour  that 
they  might  well  be  taken  as  models  by  any  one 
minded  to  study  the  art  of  correspondence,  not 


30  INTRODUCTION 

uncommonly  alleged  to  be  lost  since  the  intro- 
duction of  cheap  and  rapid  communications. 
Such  letters  he  sent  to  England  from  Spain  and 
Sicily  in  1870,  and  from  Algiers  in  1876. 
Some  of  them  are  printed  farther  on. 

In  August  1878,  there  being  signs  of  im- 
provement, and  a  warm  climate  not  being 
judged  necessary  or  very  desirable  at  that 
season,  leave  was  given  for  a  short  return  to 
England.  Clifford  came  home  looking  very  ill 
and  feeble  to  ordinary  observation,  but  much 
better  to  those  who  had  seen  him  before  he 
started.  He  was  incapable  of  continuous  exer- 
tion of  any  kind,  but  much  of  the  old  animation 
had  come  back,  and  his  conversation  had  lost 
nothing  of  its  vigour  and  brilliancy.  The 
object  of  the  summer  journey  had  been  rest 
and  freedom  from  care  above  all  things :  now 
it  was  planned  that  with  the  first  days  of 
autumn  he  should  again  go  in  search  of  condi- 
tions which  might  be  not  only  rest-giving  but 
curative.  But  all  plans  were  cut  short  by  a 
relapse  which  took  place  late  in  September, 
induced  by  fatigue.  From  that  day  the  fight 
was  a  losing  one,  though  fought  with  such 
tenacity  of  life  that  sometimes  the  inevitable 
end  seemed  as  if  it  might  yet  be  put  far  off. 
Clifford's  patience,  cheerfulness,  unselfishness, 
and  continued  interest  in  his  friends  and  in 
what  was  going  on  in  the  world,  were  unbroken 


BIOGRAPHICAL  31 

and  unabated  through  all  that  heavy  time. 
Far  be  it  from  me,  as  it  was  far  from  him,  to 
grudge  to  any  man  or  woman  the  hope  or 
comfort  that  may  be  found  in  sincere  expecta- 
tion of  a  better  life  to  come.  But  (let  this  be 
set  down  and  remembered,  plainly  and  openly, 
for  the  instruction  and  rebuke  of  those  who 
fancy  that  their  dogmas  have  a  monopoly  of 
happiness,  and  will  not  face  the  fact  that  there 
are  true  men,  ay  and  women,  to  whom  the 
dignity  of  manhood  and  the  fellowship  of  this 
life,  undazzled  by  the  magic  of  any  revelation, 
unholpen  of  any  promises  holding  out  aught  as 
higher  or  more  enduring  than  the  fruition  of 
human  love  and  the  fulfilment  of  human  duties, 
are  sufficient  to  bear  the  weight  of  both  life 
and  death.  Here  was  a  man  who  utterly 
dismissed  from  his  thoughts,  as  being  unprofit- 
able or  worse,  all  speculations  on  a  future  or 
unseen  world  ;  a  man  to  whom  life  was  holy 
and  precious,  a  thing  not  to  be  despised,  but  to 
be  used  with  joyfulness  ;  a  soul  full  of  life  and 
light,  ever  longing  for  activity,  ever  counting 
what  was  achieved  as  not  worthy  to  be  reckoned 
in  comparison  of  what  was  left  to  do.  And  this 
is  the  witness  of  his  ending,  that  as  never  man 
loved  life  more,  so  never  man  feared  death  less. 
He  fulfilled  well  and  truly  that  great  saying  of 
Spinoza,  often  in  his  mind  and  on  his  lips :  Homo 
liber  de  nulla  re  minus  quam  de  morte  cogitat. 


32  INTRODUCTION 

One  last  stand  was  made,  too  late  to  be 
permanently  successful  (if  ever  it  could  have  so 
far  availed),  but  yet  not  wholly  in  vain.  At 
the  opening  of  the  year  1 879  Clifford's  remnant 
of  strength  was  visibly  diminishing.  The  peril 
of  attempting  a  journey  was  great,  but  no  peril 
could  be  greater  than  that  which  he  already  lay 
in.  Medicine  had  no  new  thing  to  recommend, 
and  almost  nothing  to  forbid  :  a  last  experiment 
could  only  be  tried.  Clifford  sailed  for  Madeira, 
his  friends  hardly  expecting  him  to  live  out  the 
voyage.  Of  the  friendship  and  devotion  that 
accompanied  and  tended  him  there  it  is  not 
fitting  that  I  should  speak.  So  it  was,  how- 
ever, that  he  arrived  safely  in  the  island,  and 
some  weeks  were  added  to  his  life.  The 
change  from  the  bitterest  of  recent  English 
winters  to  the  fair  and  temperate  air  of 
Madeira  had  no  power  to  restore  the  waning 
forces  ;  but  it  enabled  him  to  spend  his  last 
days  in  ease  and  comparative  enjoyment.  He 
could  once  more  look  on  the  glories  of  a  bounti- 
ful world,  and  breathe  under  a  free  sky.  Some- 
thing of  spirit  and  even  of  strength  revived  ; 
his  powers  of  conversation,  which  had  been 
restrained  by  mere  physical  weakness  in  his 
last  days  in  England,  returned  to  some  extent, 
and  in  that  short  time,  with  all  the  disadvantages 
of  a  stranger  and  an  invalid,  he  made  new 
friends :  one  such  (though  in  spirit  not  a 


BIOGRAPHICAL  33 

stranger  before)  of  whose  friendship  even  he 
might  have  been  proud.  There  was  a  glimmer 
of  hope,  faint,  uncertain,  but  perceptible  ;  there 
was  a  possibility  that  if  amendment  once  began, 
it  might  go  further  than  we  had  dared  to 
speculate  upon.  But  it  was  not  to  be.  In 
the  last  days  of  February  we  learnt  that  his 
condition  was  hopeless;  on  the  3rd  of  March  the 
end  came.  For  a  week  he  had  known  that  it 
might  come  at  any  moment,  and  looked  to  it 
steadfastly.  So  calmly  had  he  received  the 
warning  which  conveyed  this  knowledge  that  it 
seemed  at  the  instant  as  if  he  did  not  understand 
it.  He  gave  careful  and  exact  directions  as  to 
the  disposal  of  his  works,  which  are  partly 
carried  out  in  this  volume,  and  have  been  sub- 
stantially fulfilled  as  to  his  mathematical 
remains  also.  His  work  was,  indeed,  the  only 
thing  personal  to  himself  that  he  took  much 
thought  for  ;  and  that  not  because  it  was  his 
own  possession,  but  because  he  felt  that  it  was 
his  own  to  do  and  to  make  a  possession  for 
others.  He  loved  it  for  the  work's  and  the 
truth's  sake,  not  for  his  own.  More  than 
this,  his  interest  in  the  outer  world,  his  affec- 
tion for  his  friends  and  his  pleasure  in  their 
pleasures,  did  not  desert  him  to  the  very 
last.  He  still  followed  the  course  of  events, 
and  asked  for  public  news  on  the  morning 
of  his  death  :  so  strongly  did  he  hold  fast  his 
VOL.  I  D 


34  INTRODUCTION 

part  in  the  common  weal  and  in  active  social 
life. 

It  has  been  mentioned  how  unwilling  Clifford 
was  to  throw  up,  even  under  necessity,  his  work 
at  University  College.  His  friends  and  col- 
leagues there  were  equally  unwilling  to  lose 
him  ;  and  when  it  became  evident  that  he 
could  never  permanently  resume  his  lectures, 
they  still  cast  about  for  means  to  retain  him  as 
one  of  their  number.  In  1879  the  Senate,  in 
reviewing  the  whole  question  of  the  teaching  of 
mathematics  and  physics,  recommended  that 
Clifford  should  "remain  in  possession  of  his 
chair,  and  that  if,  against  the  expectation,  but 
in  accordance  with  the  most  earnest  desire  of 
his  colleagues,  he  should  so  far  recover  health 
as  to  be  able  to  lecture,  he  should  be  invited  to 
lecture  upon  special  subjects  in  mathematics,  to 
which  he  could  bring  his  own  rare  qualities  of 
mind  without  being  subjected  to  any  strain  of 
constant  necessary  work."  This  recommenda- 
tion only  awaited  the  assent  of  the  Council  to 
take  effect,  and  that  assent  would  almost 
certainly  have  been  given ;  but  before  the 
matter  could  be  submitted  to  the  Council  it 
was  known  that  the  time  of  expectation  was 
over,  and  desire  quelled  by  the  final  certainty 
of  loss. 

The  essays  here  brought  together  represent, 
with  few  if  any  exceptions,  the  general  view  of 


BIOGRAPHICAL  35 

the  world  and  human  knowledge  which  Clifford 
had  definitely  arrived  at  in  his  later  years.  I  do 
not  mean  that  he  had  got  a  fixed  set  of  results 
and  meant  to  rest  in  them  ;  he  admitted  no 
finality  of  that  sort.  But  he  did  believe  very 
decidedly  that  the  difference  between  right  and 
wrong  method  is  everywhere  important,  and 
that  there  is  only  one  right  method  for  all  de- 
partments of  knowledge.  He  held  that  meta- 
physical and  theological  problems  ought  to  be 
discussed  with  exactly  the  same  freedom  from 
preconceived  conclusions  and  fearlessness  of 
consequences  as  any  other  problems.  And  he 
further  held  that,  as  the  frank  application  of  the 
right  method  of  search  to  the  physical  sciences 
has  put  them  on  a  footing  of  steady  progress, 
though  they  differ  in  the  amount  and  certainty 
of  the  knowledge  already  won  in  their  respective 
fields,  so  the  like  effects  might  be  expected 
when  philosophical  speculation  was  taken  in 
hand  by  the  light  of  science  and  with  scientific 
impartiality  and  earnestness.  For  the  popular 
or  unscientific  rhetoric  which  frequently  assumes 
the  name  of  philosophy  Clifford  had  as  much 
contempt  as  he  permitted  himself  to  feel  for 
anything.  Once  he  said  of  an  acquaintance 
who  was  believed  to  be  undertaking  something 
in  this  kind  :  "  He  is  writing  a  book  on  meta- 
physics, and  is  really  cut  out  for  it ;  the  clear- 
ness with  which  he  thinks  he  understands 


36  INTRODUCTION 

things  and  his  total  inability  to  express  what 
little  he  knows  will  make  his  fortune  as  a 
philosopher."  But  he  never  accepted,  and  I  do 
not  think  he  was  ever  tempted  to  accept,  the 
doctrine  that  all  metaphysical  inquiries  ought 
to  be  put  aside  as  unprofitable.  Indeed  he 
went  beyond  most  English  psychologists, 
though  in  a  general  way  he  must  be  classed 
with  the  English  school,  in  his  estimate  of  the 
possibility  of  constructing  a  definite  meta- 
physical system  on  scientific  principles.  With 
regard  to  the  application  of  his  philosophical 
ideas  to  theological  conceptions,  it  may  perhaps 
be  said  that  he  aimed  at  doing  for  dogmatic 
and  natural  theology  something  like  what  the 
Tubingen  school  in  Germany  have  done  for 
historical  theology,  namely,  bringing  them  to  the 
light  of  unbiassed  common  sense,  including 
therein  as  an  important  element  the  healthy 
moral  sense  of  civilised  men.  Whether  Clifford 
had  any  feeling  that  his  line  of  work  was  com- 
plementary to  the  historical  criticism  of  dogmas 
I  cannot  say  :  but  so  it  was  that  he  paid  no 
special  attention  to  the  historical  side  of  these 
questions,  either  because  it  did  not  particularly 
interest  him,  or  because  he  thought  it  outside 
his  competence.  In  ethics,  on  the  other  hand, 
he  attached  the  utmost  importance  to  the 
historical  facts  of  moral  culture  as  affording  the 
key  of  the  speculative  position  and  indicating 


BIOGRAPHICAL  37 

the  profitable  directions  of  inquiry.  And  it 
may  be  noted  as  an  instance  of  the  freshness 
and  openness  of  his  mind  that  the  importance 
of  this  point  of  view,  set  forth  in  "  The  Scientific 
Basis  of  Morals"  and  the  papers  following  it, 
was  perceived  by  him  only  after  he  left  Cam- 
bridge. The  main  points  of  the  last-named 
essay  were  stated  by  Clifford  himself  in  a  letter 
written  when  he  had  nearly  finished  it.  He 
described  it  as  "  showing  that  moral  maxims 
are  ultimately  of  the  same  nature  as  the  maxims 
of  any  other  craft :  if  you  want  to  live  together 
successfully,  you  must  do  so-and-so.  .  .  .  That 
conscience  is  developed  out  of  experience  by 
healthy  natural  processes.  .  .  .  That  responsi- 
bility is  founded  on  such  order  as  we  can 
observe,  and  not  upon  such  disorder  as  we  can 
conjecture."  This  is  quite  a  different  line  from 
that  which  his  speculations  on  the  nature  of 
duty  were  wont  to  take  at  Cambridge,  both  in 
the  conversations  I  remember,  and  in  various 
MS.  fragments  of  that  period  which  are  now 
before  me. 

A  letter  of  the  autumn  of  1874,  written  by 
Clifford  to  his  wife  during  their  engagement, 
bears  upon  his  practical  conception  of  ethics 
and  is  otherwise  interesting.  "  At  the  Savile 
I  found  C,  who  had  just  done  dinner,  but  sat 
down  while  I  ate  mine,  and  we  solved  the 
universe  with  great  delight  until  A.  came  in 


38  INTRODUCTION 

and  wanted  to  take  him  off  to  explain  coins  to 
somebody.  Of  course  I  would  not  let  him  go. 
.  .  .  We  walked  about  in  the  New  Road 
solving  more  universe.  He  says  the  people  in 
the  middle  ages  had  a  closer  connection  between 
theory  and  practice ;  a  fellow  would  get  a 
practical  idea  into  his  head,  be  cock-sure  it 
was  right,  and  then  get  up  and  snort  and  just 
have  it  carried  through.  Nowadays  we  don't 
have  prophets  with  the  same  fire  and  fervour 
and  insight.  To  which  it  may  be  said  that 
our  problems  are  infinitely  more  complex,  and 
that  we  can't  be  so  cock-sure  of  the  right  thing 
to  do.  He  quoted  the  statesmanship  of  the 
great  emperors,  e.g.  Frederic  II.;  and  some  of 
the  saints,  as  St.  Francis  and  St.  Catherine  of 
Siena.  Still  there  is  room  for  some  earnest 
person  to  go  and  preach  around  in  a  simple 
way  the  main  straightforward  rules  that  society 
has  unconsciously  worked  out  and  that  are 
floating  in  the  air  ;  to  do  as  well  as  possible 
what  one  can  do  best ;  to  work  for  the  im- 
provement of  the  social  organisation  ;  to  seek 
earnestly  after  truth  and  only  to  accept  pro- 
visionally opinions  one  has  not  inquired  into  ; 
to  regard  men  as  comrades  in  work  and  their 
freedom  as  a  sacred  thing  ;  in  fact,  to  recognise 
the  enormous  and  fearful  difference  between 
truth  and  falsehood,  right  and  wrong,  and  how 
truth  and  right  are  to  be  got  at  by  free  inquiry 


BIOGRAPHICAL  39 

and  the  love  of  our  comrades  for  their  own 
sakes  and  nobody  else's.  Mazzini  has  done  a 
great  deal  in  this  direction,  and  formed  the 
conception  of  the  world  as  a  great  workshop 
where  we  all  have  to  do  our  best  to  make 
something  good  and  beautiful  with  the  help  of 
the  others.  Such  a  preaching  to  the  people  of 
the  ideas  taught  by  the  great  Rabbis  was  (as 
near  as  we  can  make  out)  the  sort  of  work  that 
Christ  did  ;  but  he  differed  from  the  Rabbis 
and  resembled  all  other  Jew  prophets  in  not 
being  able  to  stand  priests." 

It  will  not  be  amiss  to  go  back  to  the  time 
when  we  left  Clifford  celebrating  the  late  Master 
of  Trinity  in  parables,  and  to  take  up  more 
continuously  than  we  have  yet  done  the  growth 
of  his  philosophic  ideas.  Before  he  took  his 
degree,  and  I  think  for  some  little  time  after, 
he  was  (as  before  mentioned)  a  High  Churchman ; 
but  there  was  an  intellectual  and  speculative 
activity  about  his  belief  which  made  it  impossible 
that  it  should  remain  permanently  at  that  stage. 
On  the  one  hand  he  acquired  a  far  more  ac- 
curate knowledge  of  Catholic  theology  than  is 
often  met  with  in  England  even  among  those 
who  discuss  theological  questions  ;  he  was  pretty 
well  read  in  St.  Thomas  Aquinas,  and  would 
maintain  the  Catholic  position  on  most  points 
with  extreme  ingenuity,  not  unfrequently  adding 
scientific  arguments  and  analogies  of  his  own. 


4o  INTRODUCTION 

On  the  other  hand,  believing  from  the  first 
in  the  unity  or  at  least  the  harmony  of  all 
truth,  he  never  slackened  in  the  pursuit  of 
scientific  knowledge  and  ideas.  For  a  while 
he  experimented  in  schemes  for  the  juxta- 
position of  science  and  dogma.  Religious 
beliefs  he  regarded  as  outside  the  region  of 
scientific  proof,  even  when  they  can  be  made 
highly  probable  by  reasoning;  for,  as  he 
observes  in  a  MS.  fragment  of  this  time,  they 
are  received  and  held  not  as  probable  but  as 
certain.  And  he  actually  defined  superstition 
as  "a  belief  held  on  religious  or  theological 
grounds,  but  capable  of  scientific  proof  or 
disproof."  He  also  held  that  there  was  a 
special  theological  faculty  or  insight,  analogous 
to  the  scientific,  poetic,  and  artistic  faculty ; 
and  that  the  persons  in  whom  this  genius  is 
exceptionally  developed  are  the  founders  of 
new  religions  and  religious  orders.  He  seems 
to  have  been  always  and  equally  dissatisfied 
with  attempts  at  proving  theological  pro- 
positions, especially  in  the  usual  manner  of 
Protestant  divinity,  and  with  the  theological 
version  of  natural  history  commonly  called 
Natural  Theology.  There  are  indications  in 
his  note -books  of  that  which  might  have 
become,  under  other  conditions,  a  spiritual 
vocabulary  no  whit  less  original  than  William 
Blake's.  Underlying  all  these  experiments  and 


BIOGRAPHICAL  41 

endeavours  there  was  a  permanent  element  of 
active  intellectual  faith  by  which  Clifford  was 
akin  to  a  philosophic  scholar  in  most  external 
respects  exceedingly  unlike  him,  Mark  Pattison. 
This  faith  is  summed  up  by  Pattison  in  a 
saying  not  known  to  Clifford,  I  think,  in  its 
terms,  but  wholly  after  his  heart :  "  The  learning 
of  true  propositions,  dogmatically  delivered,  is 
not  science."  When  or  how  Clifford  first  came 
to  a  clear  perception  that  his  position  of  quasi- 
scientific  Catholicism  was  untenable  I  do  not 
exactly  know  ;  but  I  know  that  the  discovery 
cost  him  an  intellectual  and  moral  struggle, 
of  which  traces  may  be  found  here  and  there 
in  his  essays.  It  is  not  the  case,  however, 
that  there  was  any  violent  reaction  or  rushing 
to  an  opposite  extreme.  Some  time  elapsed 
before  his  philosophical  opinions  assumed  their 
final  consistency ;  and  in  truth  what  took 
place  was  not  a  reaction,  but  the  fuller  develop- 
ment of  principles  which  had  been  part  of 
his  thoughts  ever  since  he  began  to  think 
for  himself. 

Meanwhile  he  was  eagerly  assimilating  the 
ideas  which  had  been  established  as  an  assured 
possession  of  biological  science  by  Mr.  Darwin, 
and  the  kindred  ones  already  at  an  earlier  time 
applied  and  still  being  applied  to  the  framing 
of  a  constructive  science  of  psychology,  and  to 
the  systematic  grouping  and  gathering  together 


4*  INTRODUCTION 

of  human  knowledge,  by  Mr.  Herbert  Spencer  ; 
who  had,  in  Clifford's  own  words,  "  formed  the 
conception  of  evolution  as  the  subject  of  general 
propositions  applicable  to  all  natural  processes." 
Clifford  was  not  content  with  merely  giving  his 
assent  to  the  doctrine  of  evolution  :  he  seized 
on  it  as  a  living  spring  of  action,  a  principle 
to  be  worked  out,  practised  upon,  used  to  win 
victories  over  nature,  and  to  put  new  vigour 
into  speculation.  For  two  or  three  years  the 
knot  of  Cambridge  friends  of  whom  Clifford 
was  the  leading  spirit  were  carried  away  by 
a  wave  of  Darwinian  enthusiasm :  we  seemed 
to  ride  triumphant  on  an  ocean  of  new  life  and 
boundless  possibilities.  Natural  Selection  was 
to  be  the  master-key  of  the  universe  ;  we  ex- 
pected it  to  solve  all  riddles  and  reconcile  all 
contradictions.  Among  other  things  it  was  to 
give  us  a  new  system  of  ethics,  combining  the 
exactness  of  the  utilitarian  with  the  poetical 
ideals  of  the  transcendentalist.  We  were  not 
only  to  believe  joyfully  in  the  survival  of  the 
fittest,  but  to  take  an  active  and  conscious 
part  in  making  ourselves  fitter.  At  one  time 
Clifford  held  that  it  was  worth  our  while  to 
practise  variation  of  set  purpose  ;  not  only  to 
avoid  being  the  slaves  of  custom,  but  to  eschew 
fixed  habits  of  every  kind,  and  to  try  the 
greatest  possible  number  of  experiments  in 
living  to  increase  the  chances  of  a  really 


BIOGRAPHICAL  43 

valuable  one  occurring  and  being  selected  for 
preservation.  So  much  of  this  theory  as  he 
ever  gave  to  the  world  will  be  found  in  the 
discourse  "  On  Some  Conditions  of  Mental 
Development " ;  and  I  do  not  know  that  he 
would  ever  have  deliberately  committed  himself 
to  anything  more  than  is  there  propounded. 
One  practical  deduction  was  that  education 
ought  to  be  directed  not  to  mere  instruction, 
but  to  making  people  think  and  act  for  them- 
selves ;  and  this  Clifford  held  to  be  of  special 
importance  in  the  case  of  women,  where  the 
cultivation  of  independent  power  is  too  com- 
monly neglected  or  even  purposely  discouraged. 
"  It  seems  to  me,"  he  once  wrote,  "  that  the 
thing  that  is  wanting  in  the  education  of  women 
is  not  the  acquaintance  with  any  facts,  but 
accurate  and  scientific  habits  of  thought,  and 
the  courage  to  think  that  true  which  appears 
to  be  unlikely.  And  for  supplying  this  want 
there  is  a  special  advantage  in  geometry,  namely 
that  it  does  not  require  study  of  a  physically 
laborious  kind,  but  rather  that  rapid  intuition 
which  women  certainly  possess  ;  so  that  it  is 
fit  to  become  a  scientific  pursuit  for  them." 

The  duty  of  independence  and  spontaneous 
activity  conceived  by  Clifford  as  being  revealed 
by  the  philosophy  of  evolution  was  reinforced 
from  another  side  by  the  reading  of  Mazzini  ; 
and  the  result  was  a  conception  of  freedom 


44  INTRODUCTION 

as  the  one  aim  and  ideal  of  man.  This  freedom 
was  a  sort  of  transfigured  blending  of  all  powers 
of  activity  and  progress  ;  it  included  republi- 
canism as  opposed  to  the  compulsory  aspect 
of  government  and  traditional  authority  in 
general,  but  was  otherwise  not  bound  to  any 
particular  theory  in  politics.  Indeed  it  forbade 
binding  one's  self  irrevocably  to  any  theory 
whatever ;  and  the  one  commandment  of  freedom 
was  thus  expressed,  Thou  shalt  live  and  not 
formulise.  That  alone  was  right  which  was 
done  of  one's  own  inner  conviction  and  mere 
motion  ;  that  was  lifeless  and  evil  which  was 
done  out  of  obedience  to  any  external  authority. 
"There  is  one  thing  in  the  world,"  Clifford 
wrote  about  this  time,  "  more  wicked  than  the 
desire  to  command,  and  that  is  the  will  to 
obey."  Now  this  doctrine  of  individual  and 
independent  morality  may  look  on  the  face 
of  it  anarchical,  and  therefore  it  may  be  worth 
while  to  observe  that  the  Catholic  doctrine  of 
the  duty  of  following  conscience  is  essentially 
at  one  with  it.  The  conscience  may  or  may 
not  be  rightly  informed.  It  may  be  wrongly 
informed  without  one's  own  fault,  as  in  the 
case  of  invincible  ignorance,  or  with  it,  as  in 
the  case  of  culpable  ignorance  or  perversity. 
But  even  in  this  last  case  we  are  told  that 
the  sin  of  doing  an  absolutely  wrong  thing  in 
obedience  to  the  voice  of  conscience,  however 


BIOGRAPHICAL  45 

misguided,  is  infinitely  less  than  the  sin  of 
doing  the  absolutely  right  thing  against  one's 
conscience.  The  conscience  must  be  rightly 
informed  before  a  completely  right  action  is 
possible.1  Again,  Fichte  treats  the  sense  of 
will  and  duty  (from  which  he  deduces  not 
only  morality  but  the  existence  of  other  men 
and  of  the  world,  in  fact  all  knowledge  and 
reality  whatever)  as  absolutely  personal  and 
individual.  Clifford's  early  doctrine  of  freedom 
was  ardent  and  immature  ;  but  whoever  should 
call  it  immoral  would  find  himself  committed 
to  applying  the  same  language  to  some  of 
the  greatest  moralists  of  the  world.  The  social 
theory  of  morality  stated  and  partly  worked 
out  in  the  ethical  portion  of  Clifford's  essays 
is  quite  independent  of  this  earlier  phase.  At 
the  same  time  it  is  not  necessarily  inconsistent 
with  it ;  for  the  determination  of  social  morality 
is  apart  from  the  assignment  of  motives  for 
individual  morality,  and  leaves  untouched  the 
cultivation  of  individual  perfection.  Clifford, 
however,  does  in  his  later  writings  freely  and 
distinctly  recognise  the  validity  of  the  social, 
or,  as  he  sometimes  calls  it,  the  tribal  judgment, 

1  See  the  authorities  collected  in  Dr.  Newman's  Letter  to  the 
Duke  of  Norfolk,  pp.  65,  66: — "  Secundum  sententiam,  et  certam, 
asserentem  esse  peccatum  discordare  a  conscientia  erronea,  in- 
vincibili  aut  vincibili,  tenet  D.  Thomas,  quern  sequuntur  omnes 
Scholastici. "  "  In  no  manner  is  it  lawful  to  act  against  conscience, 
even  though  a  law  or  a  superior  commands  it."  Some  writers 
even  say  that  this  opinion  is  dejidt. 


46  INTRODUCTION 

on  the  moral  character  of  individual  acts  re- 
garded aS  an  external  quality  ;  and  there  was 
a  time  when  he  would  probably  have  hesitated 
to  allow  this. 

In  a  note-book  of  Clifford's  later  Cambridge 
time  there  are  some  speculations  on  the  com- 
pensating intellectual  pleasures  that  help  to 
break  the  shock  of  parting  with  old  beliefs. 
I  make  an  extract  from  one  of  these  pages. 
"Whosoever  has  learnt  either  a  language  or 
the  bicycle  can  testify  to  the  wonderful  sudden 
step  from  troublesome  acquirement  to  the 
mastery  of  new  powers,  whose  mere  exercise 
is  delightful,  while  it  multiplies  at  once  the 
intensity  and  the  objects  of  our  pleasures. 
This,  I  say,  is  especially  and  exceptionally  true 
of  the  pleasures  of  perception.  Every  time 
that  analysis  strips  from  nature  the  gilding 
that  we  prized,  she  is  forging  thereout  a  new 
picture  more  glorious  than  before,  to  be  suddenly 
revealed  by  the  advent  of  a  new  sense  whereby 
we  see  it — a  new  creation,  at  sight  of  which 
the  sons  of  God  shall  have  cause  to  shout 
for  joy. 

"What  now  shall  I  say  of  this  new-grown 
perception  of  Law,  which  finds  the  infinite  in 
a  speck  of  dust,  and  the  acts  of  eternity  in 
every  second  of  time?  Why,  that  it  kills 
our  sense  of  the  beautiful,  and  takes  all  the 
romance  out  of  nature.  And  moreover  that 


BIOGRAPHICAL  47 

it  is  nothing  more  than  a  combining  and  re- 
organising of  our  old  experiences,  never  can 
give  us  anything  really  new,  must  progress  in 
the  same  monotonous  way  for  ever.  But  wait 
a  moment.  What  if  this  combining  and 
organising  is  to  become  first  habitual,  then 
organic  and  unconscious,  so  that  the  sense  of 
law  becomes  a  direct  perception  ?  Shall  we 
not  then  be  really  seeing  something  new  ? 
Shall  there  not  be  a  new  revelation  of  a  great 
and  more  perfect  cosmos,  a  universe  freshborn, 
a  new  heaven  and  a  new  earth  ?  Mors  janua 
vita ;  by  death  to  this  world  we  enter  upon 
a  new  life  in  the  next.  A  new  Elysium  opens 
to  our  eager  feet,  through  whose  wide  fields 
we  shall  run  with  glee,  stopping  only  to  stare 
with  delight  and  to  cry,  '  See  there,  how  beauti- 
ful ! '  for  the  question,  '  Why  ? '  shall  be  very 
far  off,  and  for  a  time  shall  lose  its  meaning." 

"  For  a  time  ?  It  may  well  be  that  the 
new  world  also  shall  die.  Doubtless  there 
shall  by  and  by  be  laws  as  far  transcending 
those  we  know  as  they  do  the  simplest  obser- 
vation. The  new  incarnation  may  need  a 
second  passion ;  but  evermore  beyond  it  is 
the  Easter  glory." 

Even  at  the  time  of  these  half-poetical 
meditations  I  think  Clifford  must  have  felt 
them  to  be  too  poetical  for  scientific  use. 
Later  in  life,  as  we  have  seen  above  and 


48  INTRODUCTION 

may  see  in  the  Essays,  he  chose  to  make  sure 
of  a  solid  foundation  in  experience  at  the 
cost  of  sacrificing  ornament  and  rhetoric,  and 
his  admiration  of  Mazzini  became  compatible 
with  practical  empiricism  in  politics.  "  On  the 
whole  I  feel  confirmed,"  he  wrote  in  a  letter, 
"  that  the  English  distrust  .of  general  principles 
in  a  very  complex  affair  like  politics  is  a  sound 
scientific  instinct,  and  that  for  some  time  we 
must  go  blundering  on,  finding  out  by  ex- 
perience what  things  are  to  be  let  alone  and 
what  not." 

The  command,  "  thou  shalt  not  formulise," 
was  expressed  in  an  amusing  shape  in  a  review 
of  Problems  of  Life  and  Mind,  published  in 
1 874.  "  Rules  of  philosophising  are  admirable 
things  if  two  conditions  are  satisfied :  first, 
you  must  philosophise  before  you  make  your 
rules  ;  secondly,  you  should  publish  them  with 
a  fond  and  fervent  hope  that  no  philosophiser 
will  attend  to  them." 

As  to  Clifford's  ideas  on  metaphysics  proper 
I  have  not  much  to  say  beyond  what  is  dis- 
closed in  the  Essays  themselves.  His  interest 
in  philosophy  grew  up  rapidly  after  he  took 
his  degree,  as  is  generally  the  case  with  men 
who  have  any  bent  that  way.  I  remember 
many  long  talks  with  him  on  metaphysical 
questions,  but  not  much  of  the  substance  of 
them.  One  evening  in  the  Long  Vacation  of 


BIOGRAPHICAL  49 

1868,  when  we  were  up  for  the  Fellowship 
examination,  we  discussed  the  Absolute  for 
some  couple  of  hours,  and  at  last  defined  it 
to  our  own  exceeding  content  as  that  which 
is  in  necessary  relation  to  itself.  Probably 
we  laughed  at  our  definition  the  next  morning, 
or  soon  after ;  but  I  am  still  of  opinion  that, 
as  definitions  of  the  Absolute  go,  this  will  do 
quite  as  well  as  any  other.  Clifford's  philo- 
sophical reading  was  rather  select  than  wide. 
He  had  a  high  admiration  for  Berkeley,  next 
only  to  Hume,  and  even  more,  perhaps,  for 
the  Ethics  of  Spinoza.  The  interpretation  of 
Spinoza's  philosophy  which  I  have  put  forward 
on  one  or  two  occasions  was  common  to 
Clifford  and  myself,  and  on  that  subject  (as, 
indeed,  on  everything  we  discussed  together) 
I  owe  very  much  to  him.  He  was  to  have 
lectured  on  Spinoza  at  the  London  Institution 
in  1877,  but  his  health  would  not  allow  it. 
There  is  little  doubt  that  this  would  have 
been  one  of  his  most  brilliant  and  original 
discourses.  Students  of  Spinoza  will  easily 
trace  the  connection  between  his  theory  of 
mind  and  matter  and  the  doctrine  set  forth 
in  Clifford's  Essays  on  "  Body  and  Mind,"  and 
'  The  Nature  of  Things-in-themselves."  This 
was  arrived  at,  to  the  best  of  my  recollection, 
in  1871  or  1872;  certainly  before  1874,  in 
which  year  the  last-mentioned  paper  was  read 
VOL.  I  E 


5o  INTRODUCTION 

at  a  meeting  of  the  Metaphysical  Society. 
Briefly  put,  the  conception  is  that  mind  is 
the  one  ultimate  reality ;  not  mind  as  we 
know  it  in  the  complex  forms  of  conscious 
feeling  and  thought,  but  the  simpler  elements 
out  of  which  thought  and  feeling  are  built 
up.  The  hypothetical  ultimate  element  of 
mind,  or  atom  of  mind-stuff,  precisely  corre- 
sponds to  the  hypothetical  atom  of  matter, 
being  the  ultimate  fact  of  which  the  material 
atom  is  the  phenomenon.  Matter  and  the 
sensible  universe  are  the  relations  between 
particular  organisms,  that  is,  mind  organised 
into  consciousness,  and  the  rest  of  the  world. 
This  leads  to  results  which  would  in  a  loose 
and  popular  sense  be  called  materialist.  But 
the  theory  must,  as  a  metaphysical  theory, 
be  reckoned  on  the  idealist  side.  To  speak 
technically,  it  is  an  idealist  monism.  Indeed 
it  is  a  very  subtle  form  of  idealism,  and  by 
no  means  easy  of  apprehension  at  first  sight. 
Nevertheless  there  are  distinct  signs  of  a  con- 
vergence towards  it  on  the  part  of  recent 
inquirers  who  have  handled  philosophical  prob- 
lems in  a  scientific  spirit,  and  particularly  those 
who  have  studied  psychology  on  the  physio- 
logical side.  Perhaps  we  shall  be  told  that 
this  proves  the  doctrine  to  be  materialism  in 
disguise  ;  but  it  is  hardly  worth  while  to  dispute 
about  names  while  more  serious  things  remain 


BIOGRAPHICAL  51 

for  discussion.  And  the  idea  does  require 
much  more  working  out ;  involving,  as  it  does, 
extensive  restatement  and  rearrangement  of 
metaphysical  problems.  It  raises  not  only 
several  questions,  but  preliminary  (and  really 
fundamental)  problems  as  to  what  questions 
are  reasonable.  For  instance,  it  may  be  asked 
why,  on  this  hypothesis,  mind  should  become 
conscious  at  a  particular  degree  of  complexity, 
or  be  conscious  at  all.  I  should  myself  say 
that  I  do  not  know  and  do  not  expect  ever 
to  know,  and  I  believe  Clifford  would  have 
said  the  same.  But  I  can  conceive  some  one 
taking  up  the  theory  and  trying  to  make  it 
carry  further  refinements  and  explanations. 
Again,  a  more  subtle  objection,  but  in  my 
opinion  a  fallacious  one,  would  be  that  it  is 
not  really  a  monism  but  a  dualism,  putting 
mind  (as  the  undetermined  mind-stuff}  and 
consciousness  in  place  of  the  old-fashioned 
matter  and  mind.  This,  however,  is  not  the 
place  to  pursue  the  subject ;  and  I  do  not 
think  the  outline  of  the  hypothesis  can  be  made 
clearer  by  any  explanation  of  mine  than  Clifford 
has  already  made  it  Looking  back  on  this 
brilliant  piece  of  speculation  after  seven  years, 
I  suppose  my  sight  is  more  impartial.  I  alter 
nothing  of  what  I  wrote  in  the  first  edition, 
but  feel  bound  in  sincerity  to  add  that  I  cannot 
now  accept  mind-stuff.  The  atom  of  mind- 


52  INTRODUCTION 

stuff  is  a  "  thing  in  itself"  :  Clifford  so  described 
it.  But  the  purpose  of  modern  philosophy  is 
to  abolish  things  in  themselves.  Kant  proved 
them  unknowable  :  the  inevitable  step  onward 
is  to  cast  them  out  as  illusions,  though  Kant 
would  not  take  it.  By  no  amount  of  ingenious 
manipulation  can  psychology  henceforth  be 
made  to  serve  instead  of  metaphysics.  Mind 
per  se,  or  mind-stuff,  abstracted  by  Clifford's 
or  any  like  method  from  the  intelligible  world, 
is  no  more  intelligible  than  matter  per  se. 
We  have  simplified  a  scientific  statement,  not 
solved  a  philosophical  problem. 

After  all  I  have  wished  to  speak  of  the  man 
rather  than  his  opinions  ;  but  the  speculative 
interests  I  shared  with  him,  being  in  a  manner 
part  of  himself,  have  claimed  their  due,  and 
perhaps  obtained  rather  more.  Let  us  now 
gather  up  a  few  matters  of  personal  habit  and 
character  which  have  not  yet  been  noticed. 
The  predominance  of  light  as  a  figure  and  a 
symbol  in  Clifford's  writing  will  be  remarked  : 
he  associates  it  with  the  right  and  all  things 
good  so  constantly  and  naturally  that  it  is  one 
of  the  marks  of  his  style.  He  had  physically 
a  great  love  of  light,  and  chose  to  write,  when 
he  could,  in  a  clear  and  spacious  room,  with  the 
windows  quite  free  of  curtains.  Though  he  was 
not  for  most  ordinary  purposes  a  business-like 
man,  and  was  careless  of  his  own  attire,  he  was 


BIOGRAPHICAL  53 

neat  and  exact  in  his  literary  work.  He  would 
not  allow  books  to  be  misused  or  carelessly  cut, 
and  his  own  MS.  was  very  fair,  regular,  and 
free  from  erasures.  He  was  careful  about 
punctuation,  and  insisted  on  having  his  own 
way  in  it,  and  he  especially  disliked  superfluous 
commas.  At  the  same  time  he  was  fond  of 
handicraft,  and  his  thoughts  often  ran  upon 
mechanical  invention.  He  speculated  much  on 
the  practicability  of  constructing  a  flying  machine, 
and  began  experiments  at  sundry  times,  which, 
however,  never  led  to  anything  definite.  Indeed 
it  is  pretty  obvious  that  if  a  successful  flying 
machine  is  ever  made  (and  there  is  no  impossi- 
bility in  it),  the  inventor  will  be  some  one  who 
combines  theoretical  knowledge  of  mechanics 
with  familiar  knowledge  of  machinery  and  the 
strength  of  materials  and  ready  command  of 
the  various  resources  of  engineering.  At  one 
time  the  notion  of  the  flying  machine  turned 
Clifford's  attention  to  kites,  and  this  led  to  a 
ludicrous  accident.  It  was  in  the  Long  Vacation 
of  1877,  when  Clifford  and  his  wife  were  Mrs. 
Crawshay's  guests  in  Wales.  A  kite  of  unusual 
dimensions,  with  tail  in  proportion,  had  been 
made  ready  for  a  flight  which  was  to  exceed 
everything  achieved  by  kites  before.  It  was  to 
be  flown  with  a  great  length  of  string,  and  it 
cost  a  morning's  work  to  lay  out  the  string  in 
a  field  so  that  the  kite  might  rise  easily  when 


54  INTRODUCTION 

started.  Having  accomplished  this,  the  party 
went  in  to  luncheon,  and  were  presently  called 
out  by  the  announcement  that  a  flock  of  sheep 
had  been  turned  into  the  field.  Clifford  rushed 
out  to  prevent  the  disaster,  but  it  was  too  late. 
Shepherd  and  sheep  were  caught  as  in  a  snare, 
and  when  they  were  extricated  the  string  was 
left  hopelessly  entangled.  Another  piece  of 
engineering  undertaken  at  the  same  time  and 
place  was  the  construction  of  a  duck-pond  for 
the  benefit  of  a  family  of  ducklings  who  fre- 
quented a  narrow  ditch  by  the  roadside.  The 
little  stream  that  trickled  in  the  ditch  was 
dammed  according  to  the  rules  of  art,  and  in 
course  of  time  a  complete  pond  was  formed, 
and  the  ducks  were  happy  for  a  season  :  till 
one  day  some  over-zealous  minister  of  local 
authority,  conceiving  the  pond,  as  it  was 
supposed,  to  be  an  encroachment  on  the  high- 
way, restored  the  ancient  state  of  things  with  a 
few  strokes  of  the  spade.  Clifford  regretted  the 
duck-pond  even  more  than  the  kite.  Other 
amusing  and  characteristic  anecdotes  might  be 
added  ;  but  I  forbear. 

No  enumeration  of  tastes  and  occupations 
can  adequately  represent  the  variety  and  flexi- 
bility of  Clifford's  intellect,  and  still  less  the 
tender,  imaginative,  poetical  side  of  his  mind. 
Now  and  then  he  wrote  verses  in  which  this 
partly  found  expression.  They  were  mostly  of 


BIOGRAPHICAL  55 

a  private  or  occasional  nature,  or  else  too 
fragmentary  for  publication.  One  very  graceful 
song  is  to  be  found  in  the  volume  of  fairy  tales 
already  spoken  of.  But  the  real  expression  of 
Clifford's  varied  and  fascinating  qualities  was 
in  his  whole  daily  life  and  conversation,  per- 
ceived and  felt  at  every  moment  in  his  words 
and  looks,  and  for  that  very  reason  impossible 
to  describe.  Nor  can  portraits  go  very  far  to 
supply  that  part  of  it  which  fell  to  the  sight ; 
for  the  attractive  animation  and  brightness  of 
his  countenance  depended  on  very  slight,  subtle, 
and  rapidly  succeeding  changes.  His  com- 
plexion was  fair ;  his  figure  slight,  but  well- 
knit  and  agile  ;  the  hands  small,  and,  for  a  man, 
singularly  slender  and  finely  formed.  The 
features  were  of  a  massive  and  irregular  type 
which  may  be  called  Socratic  ;  in  a  bust  they 
might  have  looked  stern,  in  the  living  face  they 
had  an  aspect  not  only  of  intellectual  beauty 
but  of  goodwill  and  gentle  playfulness.  But  I 
began  with  declaring  my  task  impossible,  and 
at  the  end  I  feel  still  more  keenly  that  all 
words  fall  short  of  what  I  would  convey.  The 
part  has  fallen  to  me  of  doing  to  a  loved  and 
honoured  friend  such  honour  as  I  could  :  the 
will  at  least  will  be  accepted. 

Purpureos  spargam  flores  .  .  et  fungar  inani 
munere. 


PART  II 
SELECTIONS  FROM  LETTERS,  ETC. 

THE  following  is  a  selection  from  letters  written 
by  Clifford  at  various  times,  partly  to  my  mother 
and  partly  to  myself.  I  begin  with  some  philo- 
sophical passages. 

[To  F.  Pollock.} 

"Trinity  College,  Cambridge,  April  2,  1870. 

"  Several  new  ideas  have  come  to  me  lately  : 
first  I  have  procured  Lobatschewsky,  £tudes 
Gfomttrigues  sur  la  Thtorie  des  Paralleles  .  .  . 
a  small  tract,  of  which  Gauss,  therein  quoted, 
says,  c  L'auteur  a  trait6  la  matiere  en  main  de 
maitre  et  avec  le  veritable  esprit  geom£trique. 
Je  crois  devoir  appeler  votre  attention  sur  ce 
livre,  dont  la  lecture  ne  peut  manquer  de  vous 
causer  le  plus  vif  plaisir.'  It  is  quite  simple, 
merely  Euclid  without  the  vicious  assumption, 
but  the  way  the  things  come  out  of  one  another 
is  quite  lovely.  .  .  . 


SELECTIONS  FROM  LETTERS,  ETC.     57 

"  I  am  a  dogmatic  nihilist,  and  shall  say  the 
brain  is  conscious  if  I  like."  (This  in  reply  to 
some  verbal  criticism  of  mine.)  "  Only  I  do 
not  say  it  in  the  same  sense  as  that  in  which  I 
say  that  /  am  conscious.  It  seems  to  me  that 
not  even  Vogt,  however  you  fix  it,  can  talk 
about  matter  for  scientific  purposes  except  as  a 
phenomenon  ;  that  in  saying  the  brain  is  con- 
scious— or,  better,  that  you  are  conscious,  I  only 
affirm  a  correlation  of  two  phenomena,  and  am 
as  ideal  as  I  can  be  ;  that,  consequently,  a  true 
idealism  does  not  want  to  be  stated,  and,  con- 
versely, an  idealism  that  requires  to  be  stated 
must  have  something  wrong  about  it.  In  the 
same  way  to  say  that  there  is  God  apart  from 
the  universe  is  to  say  that  the  universe  is  not 
God,  or  that  there  is  no  real  God  at  all ;  it  may 
be  all  right,  but  it  is  atheism.  And  an  idealism 
which  can  be  denied  by  any  significant  aggrega- 
tion of  words  is  no  true  idealism." 

The  following  is  on  the  recent  edition  of 
Hume  by  Messrs.  Green  and  Grose : — 

[To  F.  Pollock^ 

"Exeter,  September  II,  1874. 

"...  I  hope  you  have  seen  Sidgwick's 
remarks  (I  think  in  the  Academy] ;  1  he  points 
out  that  to  prove  Hume  insufficient  is  not  to  do 

1  May  30,  1874,  vol.  v.  p.  608. 


58  INTRODUCTION 

much  in  the  present  day.  It  should,  I  think, 
be  brought  out  clearly  that  if  we  pay  attention 
only  to  the  scientific  or  empirical  school,  the 
theory  of  consciousness  and  its  relation  to  the 
nervous  system  has  progressed  in  exactly  the 
same  way  as  any  other  scientific  theory  ;  that 
no  position  once  gained  has  ever  been  lost,  and 
that  each  investigator  has  been  able  to  say  '  I 
don't  know '  of  the  questions  which  lay  beyond 
him  without  at  all  imperilling  his  own  con- 
clusions. Green,  for  instance,  points  out  that 
Hume  has  no  complete  theory  of  the  object^ 
which  is  of  course  a  very  complex  thing  from 
the  subjective  point  of  view,  because  of  the 
mixture  of  association  and  symbolic  substitution 
in  it ;  and  in  fact  I  suppose  this  piece  of  work 
has  not  yet  been  satisfactorily  done.  But  it 
seems  merely  perverse  to  say  that  the  scientific 
method  is  a  wrong  one,  because  there  is  yet 
something  for  it  to  do  ;  and  to  find  fault  with 
Hume  for  the  omission  is  like  blaming  Newton 
for  not  including  Maxwell's  Electricity  in  the 
Prtncipta" 

The    following    suggestions     on     education 
were  sent  from  Algiers  in  June  1876  : — 

[To  F.  Pollock.} 

"...  I  have  a  scheme  which  has  been  com- 
municated   in    part    to  Macmillan,  and  which 


SELECTIONS  FROM  LETTERS,  ETC.     59 

grows  like  a  snowball.  It  is  founded  on 
Pleasant  Pages,  the  book  I  was  taught  out  of; 
which  is  a  series  of  ten  minutes'  lessons  on  the 
Pestalozzian  plan  of  making  the  kids  find  out 
things  for  themselves  :  history  of  naughty  boys 
on  Monday,  animals  on  Tuesday,  bricks  on 
Wednesday,  Black  Prince  on  Thursday,  and  so 
on.  In  the  book  it  was  very  well  done,  by  a 
man  who  had  a  genius  for  it.  If  you  go  to  see 
Macmillan  in  Bedford  Street  he  will  show  you 
the  book,  which  he  got  on  my  recommendation 
— he  is  also  himself  newly  interested  in  the 
question.  His  partner  Jack  read  part  of  it  and 
was  struck.  Well,  I  first  want  that  brought  up 
to  to-day,  both  in  choice  of  subject  and  in 
accuracy  ;  adding,  e.g.  a  series  of  object  lessons 
on  man  (papa,  mamma,  house,  street,  clothes, 
shop,  policeman,  c  wild  and  field ').  Then  I 
want  it  taught  on  the  Russian  system,  in 
different  languages  on  successive  days ;  no 
direct  teaching  of  language  until  there  are  facts 
enough  to  make  Grimm's  law  intelligible,  for 
which  English,  German,  and  the  Latin  element 
in  French  would  be  enough  ;  no  grammar  at 
all  till  very  late,  and  then  as  analysis  of 
sentences  and  introductory  to  logic.  This  is 
the  difficult  part ;  it  would  require  a  French 
and  German  teacher,  both  trained  and  com- 
petent, besides  the  English  one.  So  far  as  the 
book  is  concerned,  it  would  of  course  be  easy  to 


60  INTRODUCTION 

print  it  in  the  three  languages.  Lastly,  I  have 
bought  twelve  volumes  of  the  Bibliotheque 
Nationale  for  three  francs — Rabelais,  five 
volumes,  and  Montesquieu,  Pascal,  Diderot,  and 
Vauvenargues.  They  are  twenty-five  centimes 
each,  admirable  for  the  pocket — and  of  course 
you  know  them.  There  are  two  or  three 
hundred  volumes.  Whereupon  we  must  of 
course  get  the  same  thing  done  for  English 
literature,  and  the  setting  forth  of  all  literature 
in  English  (e.g.  I  have  Les  Maximes  (tEpictete), 
but  more  particularly  we  must  get  published 
excellent  little  manuals  at  twopence  or  three- 
pence for  the  use  of  Board  and  other  primary 
schools.  I  do  not  even  know  that  penny 
schoolbooks  would  not  be  a  successful  move — 
the  size  of  a  Daily  News,  say,  printed  by  the 
million  in  a  Walter  press,  folded  and  sewed  by 
machinery  to  about  the  size  of  the  Bibliotheque. 

"  A  Daily  News  would  just  make  one  of 
these  volumes.  Fancy  the  Penstes  of  Pascal, 
with  the  notes  of  Voltaire,  Fontenelle,  and 
Condorcet,  a  good  life  at  the  beginning,  etc., 
all  well  printed  on  a  sheet  of  the  Daily  News  ! 
But  of  such  a  size  could  be  made  a  very  good 
elementary  schoolbook  of  arithmetic,  geometry, 
animals,  plants,  physics,  etc. — rather  larger 
than  Macmillan's  primers,  but  of  the  same 
sort." 

The    remaining    letters     and     extracts    are 


SELECTIONS  FROM  LETTERS,  ETC.     61 

chiefly  descriptive,  and  will  be  given  without 
further  remark,  except  such  brief  note  of  dates 
and  circumstances  as  may  seem  necessary. 

[To  Lady  Pollock] 

"Cambridge,  September  26,  1871. 

"...  My  ideal  theory  is  quite  different 
from  yours.  In  the  case  of  persons  I  worship 
the  actual  thing  always  ;  this  is  the  only  way 
to  be  trusted.  The  one  advantage  of  having 
indestructible  family  relations  is  that,  whatever 
you  do  and  whatever  anybody  thinks  of  you, 
there  are  always  one  or  two  people  who  will 
love  you  exactly  as  much  as  (if  not  more  than) 
if  you  were  blameless  and  universally  respected. 
I  used  to  recognise  an  exception,  viz.  that  in 
certain  cases  what  had  been  a  person  might 
cease  to  be  one,  and  become  a  thing,  towards 
which  one  could  have  no  moral  relations,  and 
which  might  be  set  aside  by  safe  means,  or  used 
as  the  occasion  served.  But  the  more  people 
I  know  and  the  better  I  know  each,  the  further 
off  this  possibility  seems  to  be.  I  want  to  take 
up  my  cross  and  follow  the  true  Christ, 
humanity ;  to  accept  the  facts  as  they  are, 
however  bitter  or  severe,  to  be  a  student  and 
a  lover,  but  never  a  lawgiver.  But  then  besides 
this  I  do  look  for  an  ideal  which  is  at  some 
time  to  be  created  or  awakened  out  of  potenti- 


62  INTRODUCTION 

alities — like  the  lady  that  Phantastes  set  free 
from  the  block  of  marble.  Meanwhile  I  chip 
various  blocks,  and  generally  set  free  something ; 
not  hitherto  I  think  quite  the  right  one  ;  when 
I  do  she  will  probably  go  straight  off  to  some- 
body else.  All  this,  by  the  way,  is  only  theory ; 
my  practice  is  just  like  other  people's." 

[To  Lady  Pollock.} 

"Florence,  December  1870. 

(Clifford  was  one  of  the  English  Eclipse 
expedition  :  the  Psyche,  with  the  expedition  on 
board,  struck  on  a  rock  near  Catania.  All 
hands  and  the  instruments  were  saved,  the  ship 
was  lost.) 

"No  ink,  no  paper,  no  nothing — Florence, 
Thursday  5th.  The  above 1  you  guess.  After 
that  somehow  to  Catania,  some  in  boats  and 
some  in  holy  carts  of  the  country,  all  over 
saints  in  bright  shawls — well,  if  ever  a  ship- 
wreck was  nicely  and  comfortably  managed, 
without  any  fuss — but  I  can't  speak  calmly 
about  it  because  I  am  so  angry  at  the  idiots 
who  failed  to  save  the  dear  ship — alas  !  my 
heart's  in  the  waters  close  by  Polyphemus's 
eye,  which  we  put  out.  At  Catania,  orange 
groves  and  telescopes  ;  thence  to  camp  at 
Augusta  ;  Jonadab,  son  of  Rechab,  great  fun, 

1  A  grotesque  fancy  sketch  of  the  shipwreck. 


SELECTIONS  FROM  LETTERS,  ETC.     63 

natives  kept  off  camp  by  a  white  cord  ;  200 
always  to  see  us  wash  in  the  morning — a  per- 
formance which  never  lost  its  charm — only  five 
seconds  totality  free  from  cloud,  found  polarisa- 
tion on  moon's  disk,  agree  with  Pickering,  other 
people  successful.  Then  by  Catania  to  Messina, 
no  steamers,  kept  five  days,  Mediterranean 
stormy,  we  also  at  last  to  Naples,  very  bad 
night,  everybody  ill  but  me,  and  I  have  been 
out  of  sorts  ever  since.  Called  on  Mrs. 
Somerville,  and  came  on  to  Rome  after  seeing 
Pompeii.  At  Rome  2-^  days,  pictures,  statues, 
Coliseum  by  moonlight.  Both  of  us  sneezed 
awfully  next  morning.  The  shops  are  in  the 
streets  where  the  Tiber  left  them — nice  for 
purchasing  but  not  so  convenient  for  walking 
about.  This  morning  arrive  in  Florence — 
Pitti  palace — spent  all  my  money,  and  shall 
get  stranded  between  Cologne  and  Ostend 
unless  I  can  live  on  one  egg  every  other  day, 
and  thereout  suck  no  small  advantage, — be 
better  off  in  Paris.  Addio." 

[To  Lady  Pollock] 

"Sunday,  July  2,  1876. 

"This  comes  from  Oran  in  the  west  of 
Algeria,  a  sad  place,  with  too  many  Spaniards 
in  it.  We  came  here  yesterday  after  a  long 
and  tiresome  journey  from  Blidah,  near  Algiers. 


64  INTRODUCTION 

The  train  is  somewhat  amusing  because  the 
carriages  are  open  at  the  ends  and  you  can  sit 
in  the  air  as  if  it  was  a  tram-car.  You  have 
then  to  be  careful  not  to  let  the  very  large 
grasshoppers  eat  you  up.  Playfair,  the  English 
Consul  at  Algiers,  told  us  to  go  to  Bougie  to 
see  the  gorge  of  the  Chabet ;  so  we  got  a 
Murray's  Guide  and  started  off  obediently.  It 
was  the  steamer  that  had  brought  us  from 
Marseilles,  and  the  captain,  who  is  very  fond 
of  us,  gave  us  the  ladies'  cabin  all  to  ourselves. 
There  was  on  board  a  little  Frenchman  who 
had  observed  us  in  a  restaurant  at  Algiers. 
He  made  great  love  to  us,  and  said  he  wanted  to 
marry  an  Englishwoman,  but  we  think  he  lied 
a  good  deal  about  his  town  and  country  house, 
and  his  carriage  and  his  good  family.  How- 
ever, he  woke  us  up  in  time  for  the  diligence  at 
Bougie,  and  there  is  no  harm  in  him,  though 
indeed  very  little  else.  All  this  expedition 
was  undertaken  for  the  sake  of  the  road  from 
Bougie  to  Setif,  and  it  was  well  worth  it 
There  is  a  narrow  rent  made  by  the  stream 
which  winds  in  and  out  for  miles  among  the 
hills  ;  these  are  splendidly  wooded,  and  rise  to 
an  enormous  height  on  either  side,  while  the 
torrent  roars  away  down  below.  The  road  is 
cut  in  one  side  of  the  gorge.  The  cochon  who 
drove  the  diligence  tried  every  ruse  to  get  us 
inside,  that  he  might  have  a  friend  of  his  on 


SELECTIONS  FROM  LETTERS,  ETC.     65 

the  front  seat ;  but  we  stuck  to  our  places  till 
the  scenery  was  finished,  and  then  a  great  rain 
came  and  drenched  both  of  them  well.  Setif 
is  a  complete  French  town,  stuck  in  the  middle 
of  an  African  plain  with  its  cafes  and  boulevards, 
just  as  if  it  had  never  lived  anywhere  else.  We 
saw  more  Arabs  there  than  anywhere  else,  and 
the  native  market  pleased  us  much.  On  the 
way  back  we  travelled  with  an  Arab  who  had 
a  gazelle  in  a  basket  which  he  was  taking  to 
somebody  at  Bougie  ;  he  said  you  might  buy 
them  occasionally  in  the  market  at  Se"tif  for 
twenty-five  francs  ;  we  pitied  the  sweet  little 
thing,  which  baaed  like  a  sheep  and  struggled 
hard  to  get  out,  but  he  was  pacified  with  some 
bread  and  some  flowers  which  I  had  picked, 
and  went  to  sleep  with  his  head  on  my  arm. 
On  waking  up  he  saw  Lucy's  straw  hat  near 
him  and  tried  to  eat  it.  We  saw  the  most 
exquisite  masses  of  maiden-hair  fern,  as  large 
as  the  side  of  a  room  (the  masses  I  mean,  not 
the  fern),  where  the  streams  came  down  near  the 
side  of  the  road.  Our  little  Frenchman  was 
still  at  Bougie  and  came  back  with  us  in  the 
boat.  The  next  day  but  one  we  had  an  amus- 
ing experience  in  the  Jardin  d'Acclimatation. 
We  were  taking  coffee  in  an  Arab  cafe,  and 
there  was  a  boy  there  with  an  instrument  of 
two  strings,  whose  sounding  board  was  made  of 
bladder  stretched  over  the  shell  of  a  tortoise 
VOL.  I  F 


66  INTRODUCTION 

— quite  the  Apollo.  We  asked  him  to  play 
something  to  us,  and  then  a  flute  painted  red 
and  blue  was  given  to  an  old  man  who  had 
been  smoking  quite  still.  I  couldn't  make  out 
the  music  because  the  little  Frenchman  kept 
on  chattering ;  but  the  old  man  gradually 
became  excited  ;  he  had  been  sitting  European 
fashion  with  his  feet  on  the  ground,  but  one  of 
his  great  toes  got  restive  and  then  all  the  others, 
until  his  shoe  was  too  much  for  that  foot ;  so 
he  dropped  the  shoe  and  laid  the  foot  on  his 
knee,  where  it  could  wriggle  comfortably. 
Then  the  other  foot  became  excited  and  went 
through  the  same  process.  When  his  agony 
grew  still  more  intense,  he  put  one  foot  down 
and  bent  the  shoe  about  with  it  to  get  more 
resistance.  All  this  time  the  upper  part  of  his 
body,  except  the  fingers  playing  on  the  pipe, 
was  perfectly  still,  and  his  face  had  a  rapt 
expression.  Meanwhile  a  pipe  of  kif  had  been 
got  ready  and  was  handed  round,  and  a  whiff 
of  that  seemed  to  calm  him.  I  tried  it  also, 
and  it  brought  the  tears  into  my  eyes,  I  was  so 
nearly  suffocated.  I  went  to  a  lecture  of  the 
Arabic  course  which  is  given  at  Algiers  in  the 
Museum.  It  consisted  in  the  translation  of  an 
article  from  a '  Constantinople  paper,  passages 
from  which  were  written  up  on  a  black  board, 
read  out,  and  translated.  The  point  of  interest 
was  the  quotation  from  a  passage  in  the  Koran 


SELECTIONS  FROM  LETTERS,  ETC.  67 

in  support  of  the  constitution,  to  the  effect  that 
'the  Government  shall  not  be  absolute  but 
consultative.'  The  lecturer  said  that  absolutism 
was  a  Turkish  institution,  not  Arabic,  and  that 
the  Caliphate  had  been  a  sort  of  republic,  with 
a  president  elected  for  life.  Also  that  when  a 
certain  Caliph  boasted  that  he  had  never 
swerved  from  the  path  of  justice,  a  soldier 
looked  up  and  said  '  Inshallah !  (or  words  to 
that  effect,  meaning,  By  Jove !)  our  swords 
would  have  speedily  brought  you  back.'  This 
appears  interesting  if  true.  Already  a  Parisian 
scent  is  sold  in  the  Moorish  bazaars  as  a  per- 
fume of  the  Sultana  Valide. 

"  We  felt  very  much  injured  at  only  seeing 
two  monkeys  in  the  woods  at  La  Chiffa  the 
day  before  yesterday,  but  there  were  some 
green  parrots  on  the  bushes  near  the  railway. 

"  To-morrow  we  go  by  a  Spanish  boat  to 
Almeira,  and  thence  by  diligence  or  another 
boat  to  Malaga.  The  Spanish  boat  will  be 
nasty,  but  it  is  only  twelve  hours  or  so.  I  am 
very  much  better,  and  shall  be  glad  of  a  rest 
at  Granada  after  this  gadding  about. 

"  P.  S. — I  wrote  to  Fred  about  the  education 
of  our  infants.  I  am  very  glad  we  have  both 
begun  with  girls,  because  it  will  be  so  good  for 
the  other  children  to  have  an  elder  sister.  How 
very  fond  those  kids  will  be  of  each  other  and 
of  Fred  and  me  !  because  girls  always  like  their 


68  INTRODUCTION 

fathers  best,  you  know.  I  have  thought  of  a 
way  to  make  them  read  and  write  shorthand 
by  means  of  little  sticks  (not  to  whop  them 
with  but  to  put  together  on  a  table  and  make 
the  shorthand  signs).  Ask  G.  whether  she 
thinks  they  had  better  learn  to  sing  on  the  sol- 
fa  system  ;  it  is  very  amusing  and  seems  to  me 
more  adapted  for  children  than  the  other.  Of 
course  I  can  teach  them  to  stand  on  their 
heads. 

"  We  have  seen  the  Spanish  boat,  which  is 
called  La  Encarnacion,  and  that  rightly  ;  for  it 
is  the  incarnation  of  everything  bad." 

[The  Encarnacion  aforesaid  more  than  justi- 
fied the  worst  expectations  :  the  engines  broke 
down  at  sea,  nobody  on  board  was  competent 
to  repair  them,  and  the  ship  lay  helpless  till  a 
vessel  was  hailed  which  had  a  French  engineer 
on  board.] 

[To  F.  Pollock.} 

"Malaga,  Saturday,  July  15,  1876. 

"...  As  for  this  country,  I  think  it  re- 
quires to  be  colonised  by  the  white  man. 
The  savages  would  gradually  die  out  in  his 
presence.  The  mark  of  a  degraded  race  is 
clear  upon  their  faces  ;  only  the  children  have 
a  look  of  honesty  and  intelligence,  a  fact  which 
is  also  observed  in  the  case  of  the  negro,  and  is 
a  case  of  Von  Bar's  law,  that  the  development 


SELECTIONS  FROM  LETTERS,  ETC.     69 

of  the  individual  is  an  epitome  of  that  of  the 
race.  It  is  instructive  also  to  contrast  the 
politeness  fossilised  in  their  language  with  the 
brutal  coarseness  of  their  present  manners,  of 
which  I  may  some  time  tell  you  what  I  will 
not  soil  paper  with.  I  think  it  possible  that 
one  Spaniard  may  have  told  me  the  truth  :  he 
had  lost  so  many  teeth  that  he  left  out  all  his 
consonants,  and  I  could  not  understand  a  word 
he  said.  When  we  went  on  board  the  Rosario 
at  1 1  P.M.  the  boatman  stood  in  the  way  to 
keep  us  from  the  ladder,  and  threatened  us  for 
the  sake  of  another  peseta  over  the  regular 
charge.  The  steward  tried  to  cheat  me  over 
the  passage -money,  but  I  appealed  to  the 
authorities  who  came  on  board  at  Malaga  and 
got  the  money  back  (there  are  many  strangers 
here).  Then  he  made  another  grab  in  the 
matter  of  our  breakfasts,  in  the  face  of  a  tariff 
hung  up  in  the  cabin.  It  is  tiring  to  have  to 
think  that  every  man  you  meet  is  ready  to  be 
your  enemy  out  of  pure  cussedness.  I  don't 
understand  why  one  is  expected  to  be  polite 
and  reticent  about  the  distinction  between  the 
Hebrew  piety  and  Roman  universalism  attri- 
buted to  Jesus  and  Paul,  and  the  ecclesiastical 
system  which  is  only  powerful  over  men's  lives 
in  Spain,  the  middle  and  south  of  Italy,  and 
Greece — countries  where  the  population  con- 
sists chiefly  of  habitual  thieves  and  liars,  who 


7o  INTRODUCTION 

are  willing  opportunely  to  become  assassins  for 
a  small  sum.  I  suppose  it  frightens  people  to 
be  told  that  historical  Christianity  as  a  social 
system  invariably  makes  men  wicked  when  it 
has  full  swing.  Then  I  think  the  sooner  they 
are  well  frightened  the  better." 


[To  F.  Pollock.} 

"  Washington  Irving  Hotel,  Granada, 
August  3,  1876. 

"  You  are  quite  right,  and  one  ought  not  to 
despair  of  the  Republic.  These  folks  are  kind 
and  rather  pleasant  when  one  is  en  rapport  with 
them,  and  they  have  a  deal  of  small  talk.  We 
found  a  jolly  old  couple  one  morning  when  we 
were  coming  back  from  a  hot  walk  in  the  Vega 
of  Almeira  (vega  =  cultivated  plain  surrounding 
a  town  which  feeds  it) ;  we  asked  for  some 
milk,  which  they  had  not,  but  they  gave  us  a 
rifresco  of  syrup  and  cold  water,  not  at  all  bad, 
and  the  old  woman  showed  Lucy  all  over  her 
house  while  the  man  smoked  a  cigarette  with 
me.  Lucy's  passport  is  the  baby's  portrait, 
with  which  she  gains  the  hearts  of  all  the 
women  and  most  of  the  men.  What  made  it 
more  surprising  was  that  they  took  us  for  Jews. 
Wilkinson,  our  Consul  at  Malaga,  who  has  been 
here  with  his  wife  and  daughter  (awfully  nice 


SELECTIONS  FROM  LETTERS,  ETC.     71 

people  and  cheered  us  up  no  end),  says  that 
the  country  people  are  better  than  those  in  the 
towns. 

"...  But  although  we  have  been  nearly  a 
fortnight  at  Granada,  only  one  murder  has  been 
even  attempted,  so  far  as  I  know,  within  a 
hundred  yards  of  the  hotel.  A.  had  been  mak- 
ing love  to  B.'s  wife,  and  so  she  was  instructed  to 
walkwithhim  one  eveningunderthese  lovely  trees. 
She  took  occasion  to  borrow  his  sword-stick,  and 
stuck  him  in  the  back  with  it  while  her  husband 
fired  at  his  head  with  a  revolver.  One  ball  grazed 
his  temple,  and  another  went  in  at  his  cheek 
and  out  of  his  mouth,  carrying  away  some 
teeth  and  lip.  He  came  round  to  the  Spanish 
hotel  opposite  and  was  tied  up  on  the  door- 
step ;  they  dared  not  let  him  come  in  because 
the  police  are  so  troublesome  about  these  affairs. 
The  defence  was  that  A.  was  a  Republican,  and 
had  been  a  Protestant ;  so  you  see  B.'s  love  of 
order  was  such  that  he  did  not  think  jealousy 
a  sufficient  justification.  Wilkinson  had  just 
received  a  report  of  the  last  quarter  of  1875  '•> 
in  those  three  months  there  had  been  only  a 
few  more  than  400  murder  cases  in  the  whole 
province  of  Granada.  The  hot  weather  seems 
to  try  them  ;  a  paragraph  in  the  Malaga  paper, 
headed  '  Estadistico  Criminal  de  Domingo,  30,' 
gives  1 5  cases  of  shooting  and  stabbing  last 
Sunday  in  Malaga,  but  only  five  appear  to  have 


72  INTRODUCTION 

been  fatal.  This  is  not  assassination,  but  is 
merely  an  accompaniment  of  their  somewhat 
boisterous  conviviality  ;  they  get  drunk  together 
and  then  draw  their  knives  and  go  in  for  a 
hacking  match.  It  is  not  even  quarrelling  in 
all  cases ;  in  Granada  the  other  day  three  men 
shut  themselves  up  and  fought  till  they  were 
all  dead.  They  might,  to  be  sure,  have  dis- 
liked each  other  mutually  all  round,  but  I  am 
inclined  to  think  it  was  a  party  of  pleasure 
rather  than  of  business.  They  do  not  attack 
strangers  in  this  way  (i.e.  with  knives  and 
revolvers),  unless,  of  course,  there  is  a  reason 
for  it ;  but  when  anything  offends  their  delicate 
sense  of  propriety  one  cannot  expect  them  not 
to  show  it  a  little.  Thus  they  threw  stones  in 
Seville  and  Cordova  at  a  lady  who  is  now  stay- 
ing here,  because  she  went  into  the  street  by 
herself,  and  they  do  not  approve  of  that.  I  am 
afraid  my  Norfolk  jacket  hurts  their  feelings 
in  some  way,  but  they  have  been  very  forbear- 
ing, and  have  only  stoned  me  once,  and  then 
did  not  hit  me.  Another  time  a  shopkeeper 
set  his  dog  at  me,  but  although  this  was  rather 
alarming,  with  temperature  92°  in  the  shade,  it 
must  have  been  meant  as  a  joke,  for  Spanish 
dogs  only  bite  cripples  of  their  own  species — 
except,  indeed,  the  great  mastiffs  that  are  kept 
to  bait  bulls  that  won't  fight.  Of  course  one 
is  not  so  insular  as  to  think  there  is  only  one 


SELECTIONS  FROM  LETTERS,  ETC.     73 

way  of  giving  a  welcome  to  the  stranger  ;  and 
the  '  'eave  'arf  a  brick  at  'im '  method  is  im- 
proved by  variety.  What  generally  happens 
is  this  :  the  grown  people  stop  suddenly  at  the 
sight  of  you,  and  wheel  round,  staring  with 
open  mouths  until  you  are  out  of  sight ;  while 
the  children,  less  weighted  with  the  cares  of  this 
world,  form  a  merry  party  and  follow  at  your 
heels.  When  you  go  into  a  shop  to  buy  any- 
thing, they  crowd  round  the  door  so  that  it  is 
rather  difficult  to  get  out.  The  beggars  come 
inside  and  pull  you  by  the  arm  while  you  are 
talking  to  the  shopman.  I  have  invented  a 
mode  of  dealing  with  the  crowd  of  children  ; 
it  is  to  sit  on  a  chair  in  the  shop  door  and 
tickle  their  noses  with  the  end  of  my  cane.  I 
fear  that  universal  sense  of  personal  dignity 
which  is  so  characteristic  of  this  country  is  in 
some  way  injured  by  my  familiarity  ;  the  more 
so  as  it  cannot  be  resented,  for  the  other  end 
of  my  cane  is  loaded,  and  I  do  not  try  it  on  in 
a  macadamised  street.  Anyhow  they  go  a 
little  way  off.  In  Malaga  the  people  seemed 
more  accustomed  to  the  sight  of  strangers,  and 
contented  themselves  with  shouting  abusive 
epithets.  .  .  .  Everybody  says  there  will  be  a 
revolution  before  long.  ...  If  Castelar  returns 
to  power,  I  hope  among  other  little  reforms 
that  he  will  prevent  the  post-office  officials  from 
stealing  letters  for  the  sake  of  the  stamps  on 


74  INTRODUCTION 

them  ;  it  is  a  great  interruption  to  business 
and  must  be  a  laborious  way  of  earning  money. 
One  of  them  was  caught  in  Malaga  because 
a  packet  of  letters  which  he  had  thrown  into 
the  sea  was  accidentally  fished  up  ;  but 
he  was  shielded  from  punishment  by  the 
authorities. 

"We  are  very  happy  here,  with  a  Swiss 
cook  and  an  Italian  landlord.  There  are  some 
English,  Germans,  and  Italians  staying  over  the 
way,  and  in  a  few  minutes  we  can  be  among 
the  memorials  of  a  better  time.  I  am  too  tired 
now  to  talk  about  the  Alhambra,  but  it  seems 
to  me  to  want  that  touch  of  barbarism  which 
hangs  about  all  Gothic  buildings.  One  thinks 
in  a  Cathedral  that  since  somebody  has  chosen 
to  make  it  it  is  no  doubt  a  very  fine  thing  in 
its  way  ;  but  that,  being  a  sane  man,  one  would 
not  make  anything  like  it  for  any  reasonable 
purpose.  But  the  Alhambra  gives  one  the 
feeling  that  one  would  wish  to  build  something 
very  like  it,  mutatis  mutandis,  and  the  more  like 
it  the  more  reasonable  the  purpose  was.  More- 
over, I  think  it  must  be  beautiful,  if  anything 
ever  was  ;  but  then  I  have  no  taste." 

Clifford's  verses,  as  has  been  said,  were 
mostly  fragmentary  or  intimate.  Two  songs, 
however,  may  here  be  given,  of  which  one  is 
unpublished  elsewhere. 


SELECTIONS  FROM  LETTERS,  ETC.     75 


Song  from  "  The  Little  People? 

THIS  is  the  song  that  Daisy  sang  j  and  it  is 
about  a  water-lily  bud  that  saw  a  reflection  of 
herself  in  the  surface  of  the  water  while  she  was 
under  it. 

You  grow  through  the  water  apace,  lily ; 

You'll  soon  be  as  tall  as  the  pond, 
There  is  fresh  hope  high  in  your  face,  lily, 
Your  white  face  so  firm  and  so  fond. 
Ah,  lily,  white  lily, 

What  can  you  see 
Growing  to  meet  lily 
Graciously  ? 

There's  a  face  looks  down  from  the  sky,  lily  ; 

It  grows  to  me  dim  from  above. 
If  I  ever  can  reach  me  so  high,  lily, 
I  shall  kiss — ah  !  the  face  of  my  love. 
Ah,  lily,  white  lily, 
That  can  I  see, 
Giving  me  light,  lily, 
Lovingly. 

The  lily-bud  met  with  her  mate,  ah  me  ! 

And  her  flower  came  through  to  the  air, 
And  her  bright  face  floated  in  state,  ah  me  ! 
But  the  shadow-love  never  was  there ! 
Ah,  lily,  great  lily, 

Queenly  and  free, 
Float  out  your  fate,  lily, 
Friendlessly. 


INTRODUCTION 


Verses  sent  to  George  Eliot  with  a  Copy  of 
"  The  Little  People?  J 

Baby  drew  a  little  house, 

Drew  it  all  askew ; 
Mother  saw  the  crooked  door 

And  the  window  too. 

Mother-heart,  whose  wide  embrace 

Holds  the  hearts  of  men, 
Grows  with  all  our  growing  hopes, 

Gives  them  birth  again. 

Listen  to  this  baby-talk  ; 

'Tisn't  wise  or  clear  ; 
But  what  baby-sense  it  has 

Is  for  you  to  hear. 

The  bibliographical  sketch  of  Clifford's  work 
which  formed  part  of  this  Introduction  in  the 
first  edition  is  considered  to  have  served  its 
turn,  and  is  not  now  reproduced.  The  editors 
have  not  received  any  later  information  capable 
of  giving  definite  results. 

1  Now  (1886)  first  printed. 


LECTURES  AND   ESSAYS 


ON  SOME  OF  THE  CONDITIONS  OF 
MENTAL  DEVELOPMENT1 

IF  you  will  carefully  consider  what  it  is  that 
you  have  done  most  often  during  this  day,  I 
think  you  can  hardly  avoid  being  drawn  to  this 
conclusion  :  that  you  have  really  done  nothing 
else  from  morning  to  night  but  change  your 
mind.  You  began  by  waking  up.  Now  that 
act  of  waking  is  itself  a  passage  of  the  mind 
from  an  unconscious  to  a  conscious  state,  which 
is  about  the  greatest  change  that  the  mind  can 
undergo.  Your  first  idea  upon  waking  was 
probably  that  you  were  going  to  rest  for  some 
time  longer ;  but  this  rapidly  passed  away,  and 
was  changed  into  a  desire  for  action,  which 
again  transformed  itself  into  volition,  and  pro- 
duced the  physical  act  of  getting  up.  From 
this  arose  a  series  of  new  sensations  ;  that  is  to 
say,  a  change  of  mind  from  the  state  of  not 
perceiving  or  feeling  these  things  to  the  state 
of  feeling  them.  And  so  afterwards.  Did  you 

1  Discourse  delivered  at  the  Royal  Institution,  March  6,  1868. 


8o  LECTURES  AND  ESSAYS 

perform  any  deliberate  action  ?  There  was  the 
change  of  mind  from  indecision  to  decision,  from 
decided  desire  to  volition,  from  volition  to  act. 
Did  you  perform  an  impulsive  action  ?  Here 
there  is  the  more  sudden  and  conspicuous 
change  marked  by  the  word  impulsive ;  as  if 
your  mind  were  a  shuttlecock,  which  has  its 
entire  state  of  motion  suddenly  changed  by  the 
impulse  of  the  battledore  :  conceive  the  shuttle- 
cock descending  quite  regularly  with  a  gentle 
corkscrew  motion — the  battledore  intervenes — 
instantaneously  the  shuttlecock  flies  off  in  a 
totally  unexpected  direction,  having  apparently 
no  relation  to  its  previous  motion  ;  and  you 
will  see  how  very  apt  and  expressive  a  simile 
you  use  when  you  speak  of  certain  people  as 
having  an  impulsive  temperament.  Have  you 
felt  happy  or  miserable  ?  It  was  a  change  in 
your  way  of  looking  at  things  in  general ;  a 
transition,  as  Spinoza  says,  from  a  lower  to  a 
higher  state  of  perfection,  or  vice  versa.  In  a 
word,  whatever  you  have  done,  or  felt,  or  thought, 
you  will  find  upon  reflection  that  you  could  not 
possibly  be  conscious  of  anything  else  than  a 
change  of  mind. 

But  then,  you  will  be  inclined  to  say,  this 
change  is  only  a  small  thing  after  all.  It  does 
not  penetrate  beyond  the  surface  of  the  mind, 
so  to  speak.  Your  character,  the  general  atti- 
tude which  you  take  up  with  regard  to  circum- 


CONDITIONS  OF  MENTAL  DEVELOPMENT      81 

stances  outside,  remains  the  same  throughout 
the  day  :  even  for  great  numbers  of  days.  You 
can  distinguish  between  individual  people  to 
such  an  extent  that  you  have  a  general  idea  of 
how  a  given  person  will  act  when  placed  in 
given  circumstances.  Now  for  this  to  be  the 
case,  it  is  clear  that  each  person  must  have 
retained  his  individual  character  for  a  consider- 
able period,  so  as  to  enable  you  to  take  note  of 
his  behaviour  in  different  cases,  to  frame  some 
sort  of  general  rules  about  it,  and  from  them  to 
calculate  what  he  would  do  in  any  supposed 
given  case.  But  is  it  true  that  this  character 
or  mark  by  which  you  know  one  person  from 
another  is  absolutely  fixed  and  unvarying  ?  Do 
you  not  speak  of  the  character  of  a  child  growing 
into  that  of  a  man  :  of  a  man  in  new  circum- 
stances being  quite  a  different  person  from  what 
he  was  before?  Is  it  not  regarded  as  the 
greatest  stroke  of  art  in  a  novelist  that  he 
should  be  able  not  merely  to  draw  a  character 
at  any  given  time,  but  also  to  sketch  the  growth 
of  it  through  the  changing  circumstances  of  life  ? 
In  fact,  if  you  consider  a  little  further,  you  will 
see  that  it  is  not  even  true  that  a  character 
remains  the  same  for  a  single  day  :  every  cir- 
cumstance, however  trivial,  that  in  any  way 
affects  the  mind,  leaves  its  mark,  infinitely  small 
it  may  be,  imperceptible  in  itself,  but  yet  more 
indelible  than  the  stone-carved  hieroglyphics  of 
VOL.  I  G 


82  LECTURES  AND  ESSAYS 

Egypt.  And  the  sum  of  all  these  marks  is 
precisely  what  we  call  the  character,  which  is 
thus  itself  a  history  of  the  entire  previous  life  of 
the  individual ;  which  is  therefore  continually 
being  added  to,  continually  growing,  continually 
in  a  state  of  change. 

Let  me  illustrate  this  relation  by  the  example 
of  the  motion  of  a  planet.  People  knew,  ages 
and  ages  ago,  that  a  planet  was  a  thing  con- 
stantly moving  about  from  one  place  to  another  ; 
and  they  made  continual  attempts  to  discover 
the  character  of  its  motion,  so  that  by  observ- 
ing the  general  way  in  which  it  went  on,  they 
might  be  able  to  tell  where  it  would  be  at  any 
particular  time.  And  they  invented  most  in- 
genious and  complicated  ways  of  expressing 
this  character : 

"  Cycle  on  epicycle,  orb  on  orb," 

till  a  certain  very  profane  king  of  Portugal,  who 
was  learning  astronomy,  said  that  if  he  had  been 
present  at  the  making  of  the  Solar  System,  he 
would  have  tendered  some  good  advice.  But 
the  fact  was  that  they  were  all  wrong,  and  the 
real  case  was  by  no  means  so  complicated  as 
they  supposed  it  to  be.  Kepler  was  the  first 
to  discover  what  was  the  real  character  of  a 
planetary  orbit ;  and  he  did  this  in  the  case  of 
the  planet  Mars.  He  found  that  this  planet 
moved  in  an  ellipse  or  oval  curve  round  the 


CONDITIONS  OF  MENTAL  DEVELOPMENT      83 

sun  which  was  situated  rather  askew  near  the 
middle.  But  upon  further  observation,  this  was 
found  to  be  not  quite  exact ;  the  orbit  itself  is 
revolving  slowly  round  the  sun,  it  is  getting  elon- 
gated and  then  flattened  in  turns,  and  even  the 
plane  in  which  the  motion  takes  place  sways 
slowly  from  side  to  side  of  its  mean  position. 
Thus  you  see  that  although  the  elliptic  character 
of  the  motion  does  represent  it  with  consider- 
able exactness  for  a  long  time  together,  yet  this 
character  itself  must  be  regarded  as  incessantly 
in  a  state  of  gradual  change.  But  the  great 
point  of  the  comparison — to  aid  in  the  concep- 
tion of  which,  in  fact,  I  have  used  the  compari- 
son at  all — is  this  :  that  for  no  two  seconds 
together  does  any  possible  ellipse  accurately 
represent  the  orbit.  It  is  impossible  for  the 
planet  to  move  a  single  inch  on  its  way,  without 
the  oval  having  slightly  turned  round,  become 
slightly  elongated  or  shortened,  and  swayed 
slightly  out  of  its  plane  ;  so  that  the  oval  which 
accurately  represented  the  motion  at  one  end 
of  the  inch  would  not  accurately  represent  the 
motion  at  the  other  end.  The  application  is 
obvious.  In  like  manner  it  is  true  that  the 
character  which  will  roughly  represent  the  law 
of  a  man's  actions  for  some  considerable  time, 
will  not  accurately  represent  that  law  for  two 
seconds  together.  No  action  can  take  place  in 
accordance  with  the  character  without  modify- 


84  LECTURES  AND  ESSAYS 

ing  the  character  itself;  just  as  no  motion  of  a 
planet  could  take  place  along  its  orbit  without 
a  simultaneous  change  in  the  orbit  itself. 

But  I  will  go  even  further.  Historians  are 
accustomed  to  say  that  at  any  given  point  of  a 
nation's  history  there  is  a  certain  general  type 
which  prevails  among  the  various  changes  of 
character  which  different  men  undergo.  There 
is  some  kind  of  law,  they  say,  which  regulates 
the  slow  growth  of  each  character  from  child- 
hood to  age  ;  so  that  if  you  compared  together 
all  the  biographies  you  would  find  a  sort  of 
family  likeness  suggesting  that  some  common 
force  had  acted  upon  them  all  to  make  these 
changes.  This  force  they  call  the  Spirit  of  the 
Age.  The  spirit,  then,  which  determines  all 
the  changes  of  character  that  take  place,  which 
is,  therefore,  more  persistent  than  character 
itself, — is  this,  at  last,  a  thing  absolutely  fixed, 
permanent,  free  from  fluctuations  ?  No  :  for 
the  entire  history  of  humanity  is  an  account  of 
its  continual  changes.  It  tells  how  there  were 
great  waves  of  change  which  spread  from 
country  to  country,  and  swept  over  whole 
continents,  and  passed  away  ;  to  be  succeeded 
by  similar  waves.  No  history  can  be  philo- 
sophical which  does  not  trace  the  origin  and 
course  of  these :  things  far  more  important 
than  all  the  kings  and  rulers  and  battles  and 
dates  which  some  people  imagine  to  be  history. 


CONDITIONS  OF  MENTAL  DEVELOPMENT     85 

To  recapitulate.  The  mind  is  changing  so 
constantly  that  we  only  know  it  by  its  changes. 
The  law  of  these  changes,  which  we  call  char- 
acter, is  also  a  thing  which  is  continually 
changing,  though  more  slowly.  And  that  law 
of  force  which  governs  all  the  changes  of 
character  in  a  given  people  at  a  given  time, 
which  we  call  the  Spirit  of  the  Age,  this  also 
changes,  though  more  slowly  still. 

Now  it  is  a  belief  which,  whether  true  or 
not,  we  are  all  of  us  constantly  acting  upon, 
that  these  changes  have  some  kind  of  fixed 
relation  to  the  surrounding  circumstances.  In 
every  part  of  our  conduct  towards  other  people 
we  proceed  constantly  upon  the  assumption 
that  what  they  will  do  is  to  a  certain  extent, 
and  in  some  way  or  other,  dependent  upon 
what  we  do.  If  I  want  a  man  to  treat  me 
with  kindness  and  respect,  I  have  to  behave  in 
a  certain  way  towards  him.  If  I  want  to  pro- 
duce a  more  special  and  defined  effect,  I  have 
recourse  to  threats  or  promises.  And  even  if 
I  want  to  produce  a  certain  change  of  mind  in 
myself,  I  proceed  upon  the  same  assumption 
that  in  some  way  or  other,  and  to  a  certain 
extent,  I  am  dependent  on  the  surrounding 
circumstances.  People  tie  knots  in  their  hand- 
kerchiefs to  make  themselves  remember  things  ; 
they  also  read  definite  books  with  a  view  of 
putting  themselves  into  definite  mental  states 


86  LECTURES  AND  ESSAYS 

or  moods  ;  and  attempts  are  constantly  made 
to  produce  even  a  further  and  more  permanent 
effect,  to  effect  an  alteration  in  character. 
What  else  is  the  meaning  of  schools,  prisons, 
reformatories,  and  the  like?  Some  have  actually 
gone  further  than  this :  there  have  not  been 
wanting  enterprising  and  far-seeing  statesmen 
who  have  attempted  to  control  and  direct  the 
Spirit  of  the  Age.  Now  in  all  these  cases  in 
which  we  use  means  to  an  end,  we  are  clearly 
proceeding  on  the  assumption  that  there  is 
some  fixed  relation  of  cause  and  effect,  in  virtue 
of  which  the  means  we  adopt  may  be  ante- 
cedently expected  to  bring  about  the  end  we 
are  in  pursuit  of.  We  are  all  along  assuming, 
in  fact,  that  changes  of  mind  are  connected  by 
some  fixed  laws  or  relations  with  surrounding 
circumstances.  Now  this  being  so,  since  every 
mind  is  thus  continually  changing  its  character 
for  better  or  worse,  and  since  the  character  of 
a  race  or  nation  is  subject  to  the  same  constant 
change  ;  since  also  these  changes  are  connected 
in  some  definite  manner  with  surrounding 
circumstances  ;  the  question  naturally  presents 
itself,  What  is  that  attitude  of  mind  which  is 
likely  to  change  for  the  better  ?  All  the  in- 
dividuals of  a  race  are  changing  in  character, 
all  changing  in  different  directions,  with  every 
possible  degree  of  divergence  ;  also  the  average 
character  itself,  the  Spirit  of  the  Age,  is  either 


CONDITIONS  OF  MENTAL  DEVELOPMENT     87 

changing  in  some  one  definite  direction,  or 
tending  to  split  into  two  different  characters  : 
an  individual,  therefore,  may  be  going  with  the 
race  or  dropping  out  of  it ;  a  portion  of  the 
race  may  be  going  right  or  wrong.  Let  us 
suppose  that  some  portion  of  the  race  is  going 
right  and  improving  :  the  question  is,  In  what 
way  are  we  to  distinguish  that  individual  who 
is  improving  with  the  race,  from  the  others  who 
are  either  dropping  out  of  the  march  altogether 
or  going  wrong  ? 

Now  what  I  have  proposed  to  myself  to  do 
to-night  is  this,  merely  to  suggest  a  method 
by  which  this  question  may  ultimately  be 
answered.  I  shall  also  endeavour  afterwards 
to  point  out  what  I  conceive  to  be  one  or  two 
results  of  this  method  :  but  this  part  will  be  of 
minor  importance  ;  the  results  depend  upon  my 
application  of  the  method,  can  be  only  partially 
true,  and  may  be  wholly  false  ;  the  method 
itself  I  believe  to  be  altogether  a  true  one,  and 
one  which  must  ultimately  lead  to  the  correct 
results. 

It  consists  in  observing  and  making  use  of 
a  certain  analogy,  namely,  the  analogy  between 
the  mind  and  the  visible  forms  of  organic  life. 
You  know  that  every  animal  and  every  plant 
is  constantly  going  through  a  series  of  changes. 
The  flower  closes  at  night  and  opens  in  the 
morning  ;  trees  are  bare  in  winter  and  covered 


88  LECTURES  AND  ESSAYS 

with  leaves  in  summer;  while  the  growth  of 
every  organism  from  birth  to  maturity  cannot 
fail  to  strike  you  as  a  forcible  illustration  of  the 
gradual  change  of  character  in  the  human  mind. 
In  fact,  it  is  the  peculiarity  of  living  things  not 
merely  that  they  change  under  the  influence  of 
surrounding  circumstances,  but  that  any  change 
which  takes  place  in  them  is  not  lost  but  re- 
tained, and,  as  it  were,  built  into  the  organism 
to  serve  as  the  foundation  for  future  actions. 
If  you  cause  any  distortion  in  the  growth  of  a 
tree  and  make  it  crooked,  whatever  you  may 
do  afterwards  to  make  the  tree  straight,  the 
mark  of  your  distortion  is  there  ;  it  is  absolutely 
indelible ;  it  has  become  part  of  the  tree's  nature, 
and  will  even  be  transmitted  in  some  small  de- 
gree to  the  seeds.  Suppose,  however,  that  you 
take  a  piece  of  inanimate  matter — a  lump  of 
gold,  say,  which  is  yellow  and  quite  hard — you 
melt  it,  and  it  becomes  liquid  and  green.  Here 
an  enormous  change  has  been  produced  ;  but 
let  it  cool ;  it  returns  to  the  solid  and  yellow 
condition,  and  looks  precisely  as  before — there 
is  no  trace  whatever  of  the  actions  that  have 
been  going  on.  No  one  can  tell  by  examining 
a  piece  of  gold  how  often  it  has  been  melted 
and  cooled  in  geologic  ages  by  changes  of  the 
earth's  crust,  or  even  in  the  last  year  by  the 
hand  of  man.  Any  one  who  cuts  down  an  oak 
can  tell  by  the  rings  in  its  trunk  how  many 


CONDITIONS  OF  MENTAL  DEVELOPMENT     89 

times  winter  has  frozen  it  into  widowhood  and 
summer  has  warmed  it  into  life.  A  living 
being  must  always  contain  within  itself  the 
history  not  merely  of  its  own  existence  but  of 
all  its  ancestors.  Seeing  then  that  in  its  con- 
tinual changes  and  in  the  preservation  of  the 
records  of  those  changes  every  organism  re- 
sembles the  mind,  so  that  to  this  extent  they 
belong  to  the  same  order  of  phenomena,  may 
we  not  reasonably  suppose  that  the  laws  of 
change  are  alike,  if  not  identical,  in  the  two 
cases  ?  This  is  of  course  a  mere  supposition, 
not  deducible  from  anything  which  we  have  yet 
observed,  which  requires  therefore  to  be  tested 
by  facts.  I  shall  endeavour  to  show  that  the 
supposition  is  well  founded  ;  that  such  laws  of 
change  as  have  been  observed  in  animals  and 
plants  do  equally  hold  good  in  the  case  of  the 
mind.  I  shall  then  endeavour  to  find  out  what 
we  mean  by  higher  and  lower  in  the  two  cases, 
and  to  show,  in  fact,  that  we  mean  much  the 
same  thing.  Supposing  all  this  to  have  been 
done,  the  question  will  have  been  stated  in  a 
form  which  it  is  possible  to  answer.  I  shall 
then  make  an  attempt  to  give  part  of  the 
answer  to  it. 

In  investigating  the  laws  of  change  of 
organic  beings  I  shall  make  use  of  what  is 
called  the  Evolution -hypothesis,  which,  as 
applied  to  this  subject,  is  much  the  same  thing 


90  LECTURES  AND  ESSAYS 

as  the  Darwinian  theory,  though  it  is  not  by 
any  means  tied  down  to  the  special  views  of 
Mr.  Darwin.  But  I  shall  use  this  merely  as 
an  hypothesis  ;  and  the  validity  of  the  method 
of  investigation  which  I  have  suggested  is 
entirely  independent  of  the  truth  of  that 
hypothesis.  If  you  will  pardon  me  for  a  short 
time,  I  should  like  to  illustrate  somewhat 
further  what  I  mean  by  this. 

When  Kepler  found  out  what  was  the  form 
of  the  orbit  described  by  the  planet  Mars,  he 
thought  that  the  planet  was  driven  by  some 
force  which  acted  in  the  direction  in  which  the 
planet  was  going.  I  have  known  people  who 
learned  a  certain  amount  of  astronomy  for 
nautical  purposes,  whose  ideas  were  very 
similar  to  those  of  Kepler.  They  thought 
that  the  sun's  rotation  was  what  caused  the 
planets  to  revolve  about  him,  just  as  if  you  spin 
a  teaspoon  in  the  middle  of  a  cup  of  tea,  it 
makes  the  bubbles  go  round  and  round.  But 
Newton  discovered  that  the  real  state  of  the 
case  was  far  different.  If  you  fasten  a  ball  on 
to  the  end  of  an  elastic  string,  and  then  swing 
it  round  and  round,  you  can  make  the  ball 
describe  an  orbit  very  similar  to  that  of  the 
planet,  so  that  your  hand  is  not  quite  in  the 
centre  of  it  Now  here  the  pulling  force  does 
not  act  in  the  direction  in  which  the  ball  is 
going,  but  always  in  the  direction  of  your  hand, 


CONDITIONS  OF  MENTAL  DEVELOPMENT     91 

and  yet  the  ball  revolves  about  your  hand  and 
never  actually  comes  to  it.  Newton  supposed 
that  the  case  of  the  planet  was  similar  to  that 
of  the  ball ;  that  it  was  always  pulled  in  the 
direction  of  the  sun,  and  that  this  attraction  or 
pulling  of  the  sun  produced  the  revolution  of 
the  planet,  in  the  same  way  that  the  traction 
or  pulling  of  the  elastic  string  produces  the 
revolution  of  the  ball.  What  there  is  between 
the  sun  and  the  planet  that  makes  each  of  them 
pull  the  other,  Newton  did  not  know  ;  nobody 
knows  to  this  day  ;  and  all  we  are  now  able  to 
assert  positively  is  that  the  known  motion  of  the 
planet  is  precisely  what  would  be  produced  if 
it  were  fastened  to  the  sun  by  an  elastic  string, 
having  a  certain  law  of  elasticity.  Now  observe 
the  nature  of  this  discovery,  the  greatest  in  its 
consequences  that  has  ever  yet  been  made  in 
physical  science : — 

I.  It  begins  with  an  hypothesis,  by  suppos- 
ing that  there  is  an  analogy  between  the  motion 
of  a  planet  and  the  motion  of  a  ball  at  the  end 
of  a  string. 

II.  Science    becomes    independent    of   the 
hypothesis,  for  we  merely  use  it  to  investigate 
the  properties  of  the  motion,  and  do  not  trouble 
ourselves  further  about  the  cause  of  it. 

I  will  take  another  example.  It  has  been 
supposed  for  a  long  time  that  light  consists  of 
waves  transmitted  through  an  extremely  thin 


92  LECTURES  AND  ESSAYS 

ethereal  jelly  that  pervades  all  space ;  it  is 
easy  to  see  the  very  rapid  tremor  which 
spreads  through  a  jelly  when  you  strike  it  at 
one  point  From  this  hypothesis  we  can 
deduce  laws  of  the  propagation  of  light,  and  of 
the  way  in  which  different  rays  interfere  with 
one  another,  and  the  laws  so  deduced  are 
abundantly  confirmed  by  experiment.  But 
here  also  science  kicks  down  the  ladder  by 
which  she  has  risen.  In  order  to  explain  the 
phenomena  of  light  it  is  not  necessary  to 
assume  anything  more  than  a  periodical  oscil- 
lation between  two  states  at  any  given  point 
of  space.  What  the  two  states  are  nobody 
knows  ;  and  the  only  thing  we  can  assert  with 
any  degree  of  probability  is  that  they  are  not 
states  of  merely  mechanical  displacement  like 
the  tremor  of  a  jelly  ;  for  the  phenomena  of 
fluorescence  appear  to  negative  this  supposi- 
tion. Here  again,  then,  the  same  two  remarks 
may  be  made.  The  scientific  discovery  appears 
first  as  the  hypothesis  of  an  analogy;  and  science 
tends  to  become  independent  of  the  hypothesis. 
The  theory  of  heat  is  another  example.  If 
you  hold  one  end  of  a  poker  in  the  fire,  the 
other  end  becomes  hot,  even  though  it  is  not 
exposed  to  the  rays  of  the  fire.  Fourier,  in 
trying  to  find  the  laws  of  this  spread  of  heat 
from  one  part  of  a  body  to  another  part,  made 
the  hypothesis  that  heat  was  a  fluid  which 


CONDITIONS  OF  MENTAL  DEVELOPMENT     93 

flowed  from  the  hot  end  into  the  cold  as  water 
flows  through  a  pipe.  From  this  hypothesis 
the  laws  of  conduction  were  deduced  ;  but  in 
the  process  it  was  found  that  the  very  same 
laws  would  flow  from  other  hypotheses.  In 
fact,  whatever  can  be  explained  by  the  motion 
of  a  fluid  can  be  equally  well  explained  either 
by  the  attraction  of  particles  or  by  the  strains 
of  a  solid  substance  ;  the  very  same  mathe- 
matical calculations  result  from  the  three 
distinct  hypotheses  ;  and  science,  though  com- 
pletely independent  of  all  three,  may  yet  choose 
one  of  them  as  serving  to  link  together  different 
trains  of  physical  inquiry. 

Now  the  same  two  remarks  which  may  be 
made  in  all  these  cases  apply  equally  to  the 
evolution -hypothesis.  It  is  grounded  on  a 
supposed  analogy  between  the  growth  of  a 
species  and  the  growth  of  an  individual.  It 
supposes,  for  instance,  that  the  race  of  crabs 
has  gone  through  much  the  same  sort  of 
changes  as  every  crab  goes  through  now,  in  the 
course  of  its  formation  in  the  egg  ;  changes 
represented  by  its  pristine  shape  utterly  unlike 
what  it  afterwards  attains,  and  by  its  gradual 
metamorphosis  and  formation  of  shell  and 
claws.  By  this  analogy  the  laws  of  change  are 
suggested,  and  these  are  afterwards  checked 
and  corrected  by  the  facts.  But  as  before, 
science  tends  to  become  independent  of  hypo- 


94  LECTURES  AND  ESSAYS 

thesis.  The  laws  of  change  are  established  for 
present  and  finitely  distant  times ;  but  they 
give  us  no  positive  information  about  the  origin 
of  things.  So,  therefore,  if  I  make  use  of  this 
hypothesis  to  represent  to  you  the  laws  of 
change  that  are  deduced  from  it,  you  will  see 
that  the  truth  of  those  laws  and  the  conclusions 
which  may  be  drawn  from  them  are  in  no 
way  dependent  on  the  truth  of  the  hypothesis. 

There  are  certain  errors  current  about  the 
nature  of  the  evolution-theory  which  I  wish 
particularly  to  guard  against.  In  the  first  place 
it  is  very  commonly  supposed  that  all  existing 
animals  can  be  arranged  in  one  continuous 
chain,  from  the  highest  to  the  lowest ;  that  the 
transition  is  gradual  all  through,  and  that  nature 
makes  no  jumps.  This  idea  was  worked  out 
into  a  system  of  classification  by  Linnaeus,  and 
survived  among  naturalists  until  the  time  of 
Cuvier.  "  They  were  bent,"  says  Agassiz, 
"  upon  establishing  one  continual  uniform  series 
to  embrace  all  animals,  between  the  links  of 
which  it  was  supposed  there  were  no  unequal 
intervals."  ..."  They  called  their  system  la 
chatne  des  etres."  The  holders  of  the  Darwinian 
theory  are  then  supposed  to  believe  that  all 
these  forms  grew  out  of  one  another,  beginning 
with  the  lowest  and  ending  with  the  highest ; 
so  that  any  one  animal  of  the  series  has  in  the 
course  of  its  evolution  passed  through  all  the 


• 
•  •.»  JfcDTOTtolji*-      •  ,  •  * 

f  »»*  ReptJia. 
VERTEBRATA 

lvibia.     \        Pisces 


Amphibia. 


Pteropoda        •  Cephalopoda. 

*  •  ^Gasteropoda 

•  •    dicecia. 
Gasurcpoda.  « 
nwrioecia      •  • 

MOLLU    SCA 


Arachnida. 


Artjiculata 


ANN    U,'  L   O    S    A 


A'nnuloida 
Echitwdernfaa.  **% 


JMolluscoida 
'l\Poly* 


.•  Gregarinida 


•      /       Sponaida      Jnfiiseria 
Hydroxaa.     *  »  •    •  •    •  • 


CCELENTE    RA  TA 


96  LECTURES  AND  ESSAYS 

lower  forms.  And  as  the  species  is  thus  sup- 
posed to  have  grown  up  through  the  chain,  and 
the  lower  species  to  be  continually  growing  into 
the  higher,  so  it  is  imagined  that  every  individual 
creature,  in  the  course  of  its  production,  passes 
through  the  lower  adult  forms  ;  that  a  chicken, 
for  instance,  while  it  is  being  formed  in  the  egg, 
becomes  in  succession  a  snail,  an  insect,  a  fish, 
and  a  reptile,  before  it  becomes  a  bird.  Now 
that  all  these  ideas  are  entirely  wrong,  I  need 
hardly  remind  you  ;  and  I  have  mentioned 
them  in  order  that  there  may  be  no  mistake 
about  the  theory  which  I  am  using  as  an 
analogy.  So  far  is  it  from  being  possible  to 
arrange  existing  organisms  in  a  single  line  or 
chain,  that  they  cannot  be  adequately  repre- 
sented even  in  the  manner  which  is  attempted 
in  the  preceding  diagram,  taken  from  Spencer's 
Principles  of  Biology ',  vol.  i.  p.  303. 

In  the  next  place,  no  existing  organism 
could  possibly  grow  into  any  other.  What  is 
really  supposed  is  this  : — that  if  you  went  back 
a  million  years  or  so,  and  made  a  picture  like 
this  one,  representing  the  forms  that  existed 
then,  no  single  spot  which  is  covered  in  one 
figure  would  be  covered  in  the  other ;  but  the 
general  arrangement  would  be  very  similar, 
except  that  all  the  groups  would  be  nearer  to 
the  centre  or  radiant  point,  and  therefore  nearer 
to  each  other.  And  if  you  made  a  third 


CONDITIONS  OF  MENTAL  DEVELOPMENT     97 

picture,  representing  the  state  of  things  another 
million  years  or  so  further  back,  then  they 
would  be  still  nearer  together  ;  and  at  a 
distance  of  time  too  vast  to  be  represented, 
they  would  all  converge  into  this  radiant  point. 
So  the  theory  is  that  at  that  stupendous  distance 
of  time  all  species  were  alike,  mere  specks  of 
jelly  ;  that  they  gradually  diverged  from  each 
other  and  got  more  and  more  different,  till  at 
last  they  attained  the  almost  infinite  variety 
that  we  now  find.  If  you  will  imagine  a  tree 
with  spreading  branches,  like  an  oak  ;  then  the 
outside  leaves  at  any  time  may  be  taken  to 
represent  all  the  existing  species  at  a  given 
time.  It  is  quite  impossible  to  arrange  them 
in  any  serial  order.  As  the  tree  grows,  the 
outer  leaves  diverge,  and  get  further  from  the 
trunk  and  from  each  other  ;  and  two  extremities 
that  have  once  diverged  never  converge  and 
grow  together  again.  But  even  this  simile  is 
insufficient ;  for  species  may  diverge  in  a  far 
greater  variety  of  directions  than  the  branches 
of  a  tree.  Space  has  not  dimensions  enough 
to  represent  the  true  state  of  the  case. 

Von  Baer's  doctrine  of  development  is  illus- 
trated by  the  same  figure.  If  you  took  embryos 
of  polypes,  and  snails,  and  cuttle-fish,  and 
insects,  and  crabs,  and  fish,  and  frogs,  and  if 
you  could  watch  their  gradual  growth  into 
these  several  animals  :  at  first  they  would  be 
VOL.  I  H 


98  LECTURES  AND  ESSAYS 

all  absolutely  alike  and  indistinguishable. 
Then,  after  a  little  while,  you  would  find  that 
they  might  be  sorted  off  into  these  four  great 
classes.  Afterwards  these  groups  might  be 
divided  into  smaller  groups,  representing  orders  ; 
then  these  into  families  and  genera  ;  last  of  all 
would  appear  those  differences  which  would 
separate  them  into  species. 

The  evolution -hypothesis,  then,  represents  a 
race  of  animals  or  plants  as  a  thing  slowly 
changing  :  and  it  also  represents  these  changes 
as  connected  by  fixed  laws  with  the  action  of 
the  surrounding  circumstances,  or,  as  it  is 
customary  to  say,  the  environment.  Now  the 
action  of  the  environment  on  a  race  is  of  two 
kinds,  direct  and  indirect  That  part  which  is 
called  direct  action  is  very  easily  understood. 
There  is  no  difficulty  in  seeing  how  changes  of 
climate  might  produce  changes  in  the  colour  of 
the  skin,  or  how  new  conditions  which  neces- 
sitated the  greater  use  of  any  organ  would  lead 
to  the  increase  of  that  organ,  as  we  know  that 
muscles  may  be  made  to  swell  with  exercise  ; 
and  changes  thus  made  habitual  would  in  time 
be  inherited.  But  the  indirect  action  of  the 
environment,  which  is  called  natural  selection, 
is  still  more  important.  The  mode  of  its  opera- 
tion may  be  seen  from  an  example.  There  are 
two  butterflies  in  South  America,  nearly  resem- 
bling one  another  in  form,  but  one  of  which 


CONDITIONS  OF  MENTAL  DEVELOPMENT     99 

has  a  very  sweet  taste  and  is  liked  by  the  birds, 
while  the  other  is  bitter  and  distasteful  to  them. 
Now  suppose  that,  for  some  reason  or  other, 
sweet  butterflies  were  occasionally  produced 
with  markings  similar  to  the  bitter  ones,  these, 
being  mistaken  by  the  birds  for  bitter  ones, 
would  run  less  chance  of  being  eaten,  and  there- 
fore more  chance  of  surviving  and  leaving  off- 
spring. If  this  peculiarity  of  marking  is  at  all 
inheritable,  then  the  number  of  sweet  butterflies 
with  bitter  marks  will  in  the  next  generation 
be  greater  in  proportion  to  the  whole  number 
than  before  ;  and,  as  this  process  goes  on,  the 
sweet  butterflies  which  retain  their  distinguish- 
ing marks  will  be  all  weeded  out  by  the  birds, 
and  the  entire  species  will  have  copied  the 
markings  of  the  bitter  species.  This  has 
actually  taken  place  :  the  one  species  has 
mimicked  the  markings  of  the  other.  Here  we 
see  the  working  of  Natural  Selection.  Any 
variation  in  an  individual  which  gives  him  an 
advantage  in  the  struggle  for  life  is  more  likely 
to  be  transmitted  to  offspring  than  any  other 
variation,  because  the  individual  is  more  likely 
to  survive ;  so  that  nature  gradually  weeds  out 
all  those  forms  which  are  not  suited  to  the 
environment,  and  thus  tends  to  produce  equili- 
brium between  the  species  and  its  surrounding 
circumstances.  Changes,  then,  are  produced  in 
a  species  by  the  selection  of  advantageous 


ioo  LECTURES  AND  ESSAYS 

changes  which  happen  to  be  made  in  in- 
dividuals. Now  there  are  three  kinds  of 
change  that  are  produced  in  individuals  :  change 
of  size,  or  growth  ;  change  of  structure,  that  is 
to  say,  change  in  the  shape  and  arrangement 
of  the  parts,  as  when  the  cartilaginous  skeleton 
of  an  infant  becomes  hardened  into  bone  ;  and 
change  of  function,  that  is  to  say,  change  in  the 
use  which  is  made  of  any  part  of  the  organism. 
I  have  one  or  two  remarks  to  make  about  the 
first  of  these,  namely,  growth,  or  change  of  size. 
Every  organism  is  continually  taking  in  matter 
through  the  external  surface  to  feed  the  inside. 
A  certain  quantity  of  this  is  needed  to  make 
up  for  the  waste  that  is  continually  going  on. 
But  let  us  suppose,  to  begin  with,  that  an 
organism  has  more  surface  than  it  absolutely 
wants  to  make  up  for  waste,  then  a  certain 
portion  of  the  assimilated  matter,  or  food,  will 
remain  over,  and  the  organism  will  increase  in 
size.  But,  you  say,  if  this  is  all  that  is  meant 
by  growth  why  does  it  not  go  on  for  ever  ? 
The  explanation  is  very  simple.  I  take  this 
cube,  which  has  six  sides,  each  a  square  inch  ; 
let  us  suppose  it  to  represent  an  animal,  and 
imagine,  to  begin  with,  that  two  of  the  sides  by 
themselves  are  capable  of  feeding  the  whole 
mass,  then  the  nutrition  taken  in  by  the  other 
four  sides  is  left  over,  and  the  mass  must 
increase  in  size.  Imagine  it  now  grown  to 


CONDITIONS  OF  MENTAL  DEVELOPMENT      101 

twice  the  linear  dimensions,  that  is  to  say,  to  a 
cube  every  side  of  which  is  two  inches.  The 
mass  to  be  fed  is  now  eight  times  what  it  was, 
while  the  surface  is  only  four  times  as  great ; 
of  the  twenty -four  square  inches  of  surface 
sixteen  are  taken  up  with  feeding  the  mass, 
while  only  eight,  or  one-third,  are  left  to  supply 
the  materials  for  growth.  Still  there  is  an 
overplus,  and  the  organism  will  grow.  Let  it 
now  acquire  three  times  its  original  height  and 
breadth  and  thickness,  the  mass  is  twenty-seven 
times  as  great,  and  the  surface  only  nine  times  : 
that  is  to  say,  while  there  are  twenty-seven 
cubic  inches  to  be  fed,  there  are  just  fifty-four 
square  inches  to  feed  them.  There  is  no  longer 
any  overplus  ;  the  organism  will  stop  growing. 
And  it  is  a  general  rule  that,  in  any  case,  when 
a  thing  grows  its  mass  increases  much  faster 
than  its  surface.  However  much,  therefore,  the 
feeding  power  of  the  surface  may  be  in  excess 
to  begin  with,  the  mass  must  inevitably  catch 
it  up,  and  the  growth  will  stop. 

Now  the  changes  of  an  individual  mind  may 
be  reduced  to  the  same  three  types  : — 

Growth. 

Change  of  structure. 

Change  of  function. 

First,  then,  what  is  the  growth  of  the  mind  ? 
It  is  the  acquisition  of  new  knowledge ;  not 
merely  of  that  which  is  required  to  make  up  for 


loa  LECTURES  AND  ESSAYS 

our  wonderful  power  of  forgetting,  for  oblivion 
is  really  a  far  more  marvellous  thing  than 
memory  ;  but  of  a  certain  overplus  which  goes 
to  increase  the  entire  mass  of  our  mental 
experiences.  Now  I  do  not  know  whether 
there  is  any  race  between  surface  and  mass 
here  as  in  the  case  of  an  organism  ;  but  it  is 
certainly  true  that  whereas  in  childhood  the 
amount  we  forget  is  very  little,  and  our  powers 
of  acquisition  preponderate  immensely  over  our 
powers  of  oblivion  ;  as  we  grow  up,  the  powers 
of  oblivion  gain  rapidly  upon  the  acquisitive 
ones,  and  finally  catch  them  up  ;  the  growth 
ceases  as  soon  as  this  balance  is  attained.  So 
that  in  this  first  law,  you  see,  there  is  an  entire 
analogy  between  the  two  cases. 

In  the  next  place,  the  mind  experiences 
changes  of  structure  ;  that  is  to  say,  changes  in 
the  shape  and  arrangement  of  its  parts.  Ideas 
which  were  only  feebly  connected  become 
aggregated  into  a  close  and  compact  whole. 
The  ideas  of  several  different  qualities,  for 
instance,  which  we  never  thought  of  as  connected 
with  each  other,  are  brought  together  by  the 
qualities  being  found  to  exist  in  the  same  object. 
In  this  way  we  form  conceptions  of  things, 
which  gradually  get  so  compact  that  we  cannot 
even  in  thought  separate  them  into  their  com- 
ponent parts.  Portions  of  our  knowledge  which 
we  held  as  distinct  are  connected  together 


CONDITIONS  OF  MENTAL  DEVELOPMENT      103 

by  scientific  theories  ;  images  which  were 
scattered  all  about  are  bound  up  into  living 
bundles  by  the  artist,  and  so  we  find  them 
rearranged. 

Lastly,  changes  of  function  take  place. 
Everybody  knows  how  the  mental  faculties  open 
out  and  become  visible  as  a  child  grows  up. 
Men  acquire  faculties  by  practice.  And  without 
any  conscious  seeking,  you  must  know  how 
often  we  wake  up  as  it  were  and  find  ourselves 
gifted  with  new  powers.  We  have  found  evi- 
dence then  of  the  existence  of  our  three  types 
of  change, — growth,  structure,  and  function. 

The  actions  therefore  which  go  on  between 
the  environment  and  the  individual  may  be 
reduced  to  the  same  three  types  in  the  case  of 
the  mind  as  in  the  case  of  any  visible  organism. 
Being  somewhat  encouraged  by  this  result,  let 
us  go  back  to  our  original  question.  What  is 
that  attitude  of  mind  which  is  likely  to  change 
for  the  better  ?  What  is  the  meaning  of  better  ? 

Although  it  is  quite  impossible  to  arrange 
all  existing  organisms  in  a  serial  chain,  yet  we 
certainly  have  a  general  notion  of  higher  and 
lower.  A  bird  we  regard  as  higher  than  a  fish, 
and  a  dog  is  higher  than  a  snake.  And  if  we 
return  to  our  illustration  of  the  tree,  we  shall 
see  that  at  every  point,  at  any  given  time,  there 
is  a  definite  direction  of  development.  So  that 
though  we  might  not  be  able  to  say  which  of 


io4  LECTURES  AND  ESSAYS 

two  co-existing  organisms  was  the  higher,  yet, 
by  comparing  a  species  with  itself  at  a  slightly 
later  time,  we  might  say  whether  it  had  de- 
generated or  improved.  Now  by  examining 
various  cases,  we  shall  find  that  there  are  six 
marks  of  improvement : — 

The  parts  of  the  organism  get  more  different. 

The  parts  of  the  organism  get  more  con- 
nected. 

The  organism  gets  more  different  from  the 
environment 

The  organism  gets  more  connected  with  the 
environment 

The  organism  gets  more  different  from  other 
individuals. 

The  organism  gets  more  connected  with 
other  individuals. 

The  processes  in  fact  which  result  in  develop- 
ment are  made  up  of  differentiation  and  integra- 
tion ;  differentiation  means  the  making  things 
to  be  different,  integration  means  the  binding 
them  together  into  a  whole  ;  these  are  applied 
to  the  parts  of  the  organism,  the  organism  and 
surrounding  nature,  the  organism  and  other 
organisms.  Differentiation  of  parts  is  illustrated 
by  the  figure  on  the  following  page.  [Spencer's 
Principles  of  Biology,  vol.  ii.  p.  187.] 

Integration  of  parts  means  the  connected 
play  of  them  ;  so  that  one  being  affected  the 
rest  are  affected.  Differentiation  from  the 


CONDITIONS  OF  MENTAL  DEVELOPMENT      105 

environment  takes  place  in  weight,  composition, 
and  temperature.     A  polype  is  little  else  than 


sea-water,  which  it  inhabits  ;  a  fish  is  several 
degrees  of  temperature  above  it,  and  made  of 
quite  different  materials  ;  till  at  last  a  mammal 
is  70°  or  80°  above  the  surrounding  matter, 


io6  LECTURES  AND  ESSAYS 

and  made  of  still  more  different  materials. 
Integration  with  the  environment  means  close 
correspondence  with  it ;  actions  of  the  environ- 
ment are  followed  by  corresponding  actions 
of  the  animal.  Differentiation  from  other 
organisms  means  individuality ;  integration 
with  them  sociality. 

In  a  similar  way  we  have  a  sort  of  general 
notion  of  higher  and  lower  stages  of  mental 
development.  I  will  endeavour  to  show  that 
this  general  notion  resolves  itself  into  a  measure 
of  the  extent  to  which  the  same  six  processes 
have  gone  on,  namely  : — 

Separation  of  parts, 

Connection  of  parts, 

Separation  from  the  environment, 

Closer  correspondence  with  the  environment, 

Separation  from  other  individuals, 

Sociality. 

The  only  conception  we  can  form  of  a  purely 
unconscious  state  is  one  in  which  all  is  exactly 
alike,  or  rather,  in  which  there  is  no  difference. 

There  is  not  one  thing  with  another, 
But  Evil  saith  to  Good :  My  brother, 

My  brother,  I  am  one  with  thee  : 
They  shall  not  strive  nor  cry  for  ever : 
No  man  shall  choose  between  them  :  never 

Shall  this  thing  end  and  that  thing  be. 

The  first  indication  of  consciousness  is  a 
perception  of  difference.  The  child's  eyes 


CONDITIONS  OF  MENTAL  DEVELOPMENT      107 

follow  the  light.  Immediately  this  colourless, 
homogeneous  universe  splits  up  into  two  parts, 
the  light  part  and  the  dark  part.  A  line  is 
drawn  across  it,  it  is  made  heterogeneous,  and 
the  first  thing  that  exists  is  a  distinction.  Then 
other  lines  are  drawn ;  appearance  is  separated 
into  white,  black,  blue,  red,  and  so  on.  This  is 
the  first  process,  the  differentiation  of  the  parts 
of  consciousness.  But  by  and  by  a  number  of 
these  lines  of  distinction  are  found  to  enclose 
a  definite  space ;  they  assume  relations  to  one 
another ;  the  lines  white,  round,  light,  capable 
of  being  thrown  at  people,  include  the  con- 
ception of  a  ball ;  this  gains  coherence,  becomes 
one,  a  thing,  holding  itself  together  not  only 
separated  from  the  rest  of  consciousness,  but 
connected  in  itself  into  a  distinct  whole,  in- 
tegrated. Here  we  have  the  second  process. 
And  throughout  our  lives  the  same  two  pro- 
cesses go  hand  in  hand  ;  whatever  we  perceive 
is  a  line  of  demarcation  between  two  different 
things  ;  we  can  be  conscious  of  nothing  but  a 
separation,  a  change  in  passing  from  one  thing 
to  another.  And  these  different  lines  of  de- 
marcation are  constantly  connecting  themselves 
together,  marking  out  portions  of  our  conscious- 
ness as  complete  wholes,  and  making  them 
cohere.  Just  as  a  sculptor  clears  away  from 
a  block  of  marble  now  this  piece  and  now  that, 
making  every  time  a  separation  between  what 


io8  LECTURES  AND  ESSAYS 

is  to  be  kept  and  what  is  to  be  chipped  off,  till 
at  last  all  these  chippings  manifest  the  connec- 
tion that  ran  through  them,  and  the  finished 
statue  stands  out  as  a  complete  whole,  a  positive 
thing  made  up  of  contradictory  negations  :  so 
is  a  conception  formed  in  the  mind. 

And  this  conception,  when  it  is  thus  made 
into  a  whole,  integrated,  by  an  act  of  the  mind, 
what  does  it  immediately  appear  to  be  ?  Why, 
something  outside  of  ourselves,  a  real  thing, 
different  from  us.  This  is  the  third  process, 
the  process  of  differentiation  from  the  environ- 
ment. This  is  beautifully  described  by  Cuvier, 
who  pictures  the  first  man  wandering  about  in 
ecstasies  at  the  discovery  of  so  many  new  parts 
of  himself;  till  gradually  he  learns  that  they 
are  not  himself,  but  things  outside.  This 
notion,  then,  of  a  thing  being  real,  existing 
external  to  ourselves,  is  due  to  the  active  power 
of  the  mind  which  regards  it  as  one,  which 
binds  together  all  its  boundaries.  And  this 
goes  on  as  long  as  we  live.  Constantly  we 
frame  to  ourselves  more  complicated  combina- 
tions of  ideas,  and  by  giving  them  unity  make 
them  real.  And,  at  the  same  time,  the  con- 
verse process  is  equally  active.  While  more 
and  more  of  our  ideas  are  put  outside  of  us 
and  made  real,  our  minds  are  continually  growing 
more  and  more  into  accordance  with  the  nature 
of  external  things ;  our  ideas  become  truer, 


CONDITIONS  OF  MENTAL  DEVELOPMENT       109 

more  conformable  to  the  facts  ;  and  at  the  same 
time  they  answer  more  surely  and  completely 
to  changes  in  the  environment ;  a  new  experi- 
ence is  more  rapidly  and  more  completely 
connected  with  the  sum  of  previous  experiences. 
But  there  is  more  than  this.  The  action  of 
these  two  laws  taken  together  does  in  fact 
amount  to  the  creation  of  new  senses.  Men  of 
science,  for  example,  have  to  deal  with  extremely 
abstract  and  general  conceptions.  By  constant 
use  and  familiarity,  these,  and  the  relations 
between  them,  become  just  as  real  and  external 
as  the  ordinary  objects  of  experience  ;  and  the 
perception  of  new  relations  among  them  is  so 
rapid,  the  correspondence  of  the  mind  to 
external  circumstances  so  great,  that  a  real 
scientific  sense  is  developed,  by  which  things 
are  perceived  as  immediately  and  truly  as  I  see 
you  now.  Poets  and  painters  and  musicians 
also  are  so  accustomed  to  put  outside  of  them 
the  idea  of  beauty,  that  it  becomes  a  real 
external  existence,  a  thing  which  they  see  with 
spiritual  eyes,  and  then  describe  to  you,  but  by 
no  means  create,  any  more  than  we  seem  to 
create  these  ideas  of  table  and  forms  and  light, 
which  we  put  together  long  ago.  There  is  no 
scientific  discoverer,  no  poet,  no  painter,  no 
musician,  who  will  not  tell  you  that  he  found 
ready-made  his  discovery  or  poem  or  picture — 
that  it  came  to  him  from  outside,  and  that  he 


no  LECTURES  AND  ESSAYS 

did  not  consciously  create  it  from  within.  And 
there  is  reason  to  think  that  these  senses  or 
insights  are  things  which  actually  increase 
among  mankind.  It  is  certain,  at  least,  that 
the  scientific  sense  is  immensely  more  developed 
now  than  it  was  three  hundred  years  ago  ;  and 
though  it  may  be  impossible  to  find  any 
absolute  standard  of  art,  yet  it  is  acknowledged 
that  a  number  of  minds  which  are  subject  to 
artistic  training  will  tend  to  arrange  themselves 
under  certain  great  groups,  and  that  the 
members  of  each  group  will  give  an  independent 
yet  consentient  testimony  about  artistic  ques- 
tions. And  this  arrangement  into  schools,  and 
the  definiteness  of  the  conclusions  reached  in 
each,  are  on  the  increase,  so  that  here,  it  would 
seem,  are  actually  two  new  senses,  the  scientific 
and  the  artistic,  which  the  mind  is  now  in  the 
process  of  forming  for  itself.  There  are  two 
remaining  marks  of  development :  differentiation 
from  surrounding  minds,  which  is  the  growth  of 
individuality ;  and  closer  correspondence  with 
them,  wider  sympathies,  more  perfect  under- 
standing of  others.  These,  you  will  instantly 
admit,  are  precisely  the  twin  characteristics  of 
a  man  of  genius.  He  is  clearly  distinct  from 
the  people  that  surround  him,  that  is  how  you 
recognise  him  ;  but  then  this  very  distinction 
must  be  such  as  to  bind  him  still  closer  to  them, 
extend  and  intensify  his  sympathies,  make  him 


CONDITIONS  OF  MENTAL  DEVELOPMENT      m 

want  their  wants,  rejoice  over  their  joys,  be  cast 
down  by  their  sorrows.  Just  as  the  throat  is 
a  complicated  thing,  quite  different  from  the 
rest  of  the  body,  but  yet  is  always  ready  to  cry 
when  any  other  part  is  hurt. 

We  have  thus  got  a  tolerably  definite  notion 
of  what  mental  development  means.  It  is  a 
process  of  simultaneous  differentiation  and 
integration  which  goes  on  in  the  parts  of  con- 
sciousness, between  the  mind  and  external  things, 
between  the  mind  and  other  minds.  And  the 
question  I  want  answered  is,  What  attitude  of 
mind  tends  to  further  these  processes  ? 

I  have  now  done  all  that  it  was  my  business 
to  do,  namely,  I  have  stated  the  question  in  a 
form  in  which  it  is  possible  to  answer  it. 
There  is  no  doubt  that  by  a  careful  study  of 
the  operations  of  nature  we  shall  be  able  to 
find  out  what  actions  of  an  organism  are 
favourable  to  its  higher  development.  Having 
formulated  these  into  a  law,  we  shall  be  able 
to  interpret  this  law  with  reference  to  the  mind. 

But  now  I  am  going  to  venture  on  a  partial 
answer  to  this  question.  What  I  am  going  to 
say  is  mere  speculation,  and  requires  to  be 
verified  by  facts. 

The  changes  which  take  place  in  an  organism 
are  of  two  kinds.  Some  are  produced  by  the 
direct  action  of  things  outside,  and  these  are  to 
a*great  extent  similar  to  the  changes  which  we 


U2  LECTURES  AND  ESSAYS 

observe  in  inanimate  things.  When  a  tree  is 
bent  over  by  the  wind  and  gets  ultimately  fixed 
in  this  position,  the  change  is  in  no  way  different 
from  that  which  takes  place  when  we  bend  a 
wire  and  it  does  not  entirely  return  to  its  former 
straightness.  Other  changes  are  produced  by 
the  spontaneous  action  of  that  store  of  force 
which  by  the  process  of  growth  is  necessarily 
accumulated  within  the  organism.  Such  are 
all  those  apparently  disconnected  motions 
which  make  up  the  great  distinction  between 
living  things  and  dead.  Now  my  speculation 
is,  that  advantageous  permanent  changes  are 
always  produced  by  the  spontaneous  action  of 
the  organism,  and  not  by  the  direct  action  of 
the  environment  This,  I  think,  is  most  clear 
when  we  take  an  extreme  case.  Let  us  suppose 
a  race  of  animals  that  never  had  any  changes 
produced  by  their  spontaneous  activity.  The 
race  must  at  a  certain  time  have  a  definite 
amount  of  plasticity,  that  is,  a  definite  power  of 
adapting  itself  to  altered  circumstances  by 
changing  in  accordance  with  them.  Every 
permanent  effect  of  the  environment  upon  them 
is  a  crystallisation  of  some  part  which  before 
was  plastic ;  for  the  part  must  have  been 
plastic  for  the  effect  to  be  produced  at  all ; 
and  as  the  effect  is  permanent,  the  part  has  to 
that  extent  lost  in  plasticity.  As  this  goes  on, 
the  race  of  animals  will  bind  up  in  itself  more 


CONDITIONS  OF  MENTAL  DEVELOPMENT      113 

and  more  of  its  history,  but  will  in  that  process 
lose  the  capability  of  change  which  it  once  had ; 
at  last  it  will  be  quite  fixed,  crystallised,  in- 
capable of  change.  Then  it  must  inevitably 
die  out  in  time ;  for  the  environment  must 
change  sooner  or  later,  and  then  the  race,  in- 
capable of  changing  in  accordance  with  it,  must 
be  killed  off.  On  the  other  hand,  any  addition 
to  the  organism  which  is  made  by  its  spon- 
taneous activity  is  an  addition  of  something 
which  has  not  yet  been  acted  upon  by  the 
environment,  which  is  therefore  plastic,  capable 
of  indefinite  modification,  in  fact,  an  increase 
of  power.  The  bending  of  a  tree  by  the  wind 
is  a  positive  disadvantage  to  it  if  the  wind 
should  ever  happen  to  blow  from  the  other 
side.  But  when  a  plant,  for  no  apparent 
reason,  grows  long  hairs  to  its  seed — the 
material  for  which  may  have  been  accidentally 
supplied  by  the  environment,  while  its  use  in 
this  way  is  a  spontaneous  action  of  the  plant 
— this  is  a  definite  increase  of  power ;  for  the 
new  organ  may  be  modified  in  any  conceivable 
way  to  suit  the  exigencies  of  the  environment, 
may  cling  to  the  sides  of  beasts,  and  so 
help  the  distribution  of  the  seed,  or  effect  the 
same  object  by  being  caught  by  the  wind. 
Activity,  in  fact,  is  the  first  condition  of  de- 
velopment. A  very  good  example  of  this 
occurs  in  Professor  Huxley's  lizards,  of  which 
VOL.  I  I 


U4  LECTURES  AND  ESSAYS 

you  heard  two  or  three  weeks  ago.1  About 
the  time  marked  by  the  Primary  strata  it 
appears  that  there  was  a  race  of  lizards,  thirty 
feet  high,  that  walked  on  their  hind  legs, 
balancing  themselves  by  their  long  tails,  and 
having  three  toes  like  birds.  This  race  di- 
verged in  three  directions.  Some  of  them 
yielded  to  the  immediate  promptings  of  the 
environment,  found  it  convenient  to  go  on  all 
fours  and  eat  fish ;  they  became  crocodiles. 
Others  took  to  exercising  their  forelegs  vio- 
lently, developed  three  long  fingers,  and  became 
birds.  The  rest  were  for  a  long  while  un- 
decided whether  they  would  use  their  arms  or 
their  legs  most ;  at  length  they  diverged,  and 
some  became  pterodactylesand  others  kangaroos. 
For  Mr.  Seeley,  of  Cambridge,  has  discovered 
marsupial  bones  in  pterodactyles  ;  that  is  to 
say,  bones  like  those  which  were  supposed 
peculiar  to  the  order  of  mammals  to  which  the 
kangaroo  belongs. 

Assuming  now  that  this  law  is  true,  and 
that  the  development  of  an  organism  proceeds 
from  its  activities  rather  than  its  passivities, 
let  us  apply  it  to  the  mind.  What,  in  fact, 
are  the  conditions  which  must  be  satisfied  by 
a  mind  in  process  of  upward  development,  so 
far  as  this  law  gives  them  ? 

1  ["  On  the  animals  which  are  most  nearly  intermediate  between 
birds  and  reptiles,"  Roy.  Inst.  Proc.   I'.  1869,  p.  278.] 


CONDITIONS  OF  MENTAL  DEVELOPMENT      115 

They  are  two ;  one  positive,  the  other 
negative.  The  positive  condition  is  that  the 
mind  should  act  rather  than  assimilate,  that  its 
attitude  should  be  one  of  creation  rather  than 
of  acquisition.  If  scientific,  it  must  not  rest  in 
the  contemplation  of  existing  theories,  or  the 
learning  of  facts  by  rote  ;  it  must  act,  create, 
make  fresh  powers,  discover  new  facts  and  laws. 
And,  if  the  analogy  is  true,  it  must  create 
things  not  immediately  useful.  I  am  here  put- 
ting in  a  word  for  those  abstruse  mathematical 
researches  which  are  so  often  abused  for  having 
no  obvious  physical  application.  The  fact  is 
that  the  most  useful  parts  of  science  have  been 
investigated  for  the  sake  of  truth,  and  not  for 
their  usefulness.  A  new  branch  of  mathematics, 
which  has  sprung  up  in  the  last  twenty  years, 
was  denounced  by  the  Astronomer-Royal  before 
the  University  of  Cambridge  as  doomed  to  be 
forgotten,  on  account  of  its  uselessness.  Now 
it  turns  out  that  the  reason  why  we  cannot 
go  further  in  our  investigations  of  molecular 
action  is  that  we  do  not  know  enough  of  this 
branch  of  mathematics.  If  the  mind  is  artistic, 
it  must  not  sit  down  in  hopeless  awe  before  the 
monuments  of  the  great  masters,  as  if  heights 
so  lofty  could  have  no  heaven  beyond  them. 
Still  less  must  it  tremble  before  the  conven- 
tionalism of  one  age,  when  its  .mission  may  be 
to  form  the  whole  life  of  the  age  succeeding. 


ii6  LECTURES  AND  ESSAYS 

No  amount  of  erudition  or  technical  skill  or 
critical  power  can  absolve  the  mind  from  the 
necessity  of  creating,  if  it  would  grow.  And 
the  power  of  creation  is  not  a  matter  of  static 
ability,  so  that  one  man  absolutely  can  do  these 
things  and  another  man  absolutely  cannot ;  it  is 
a  matter  of  habits  and  desires.  The  results  of 
things  follow  not  from  their  state  but  from  their 
tendency.  The  first  condition  then  of  mental 
development  is  that  the  attitude  of  the  mind 
should  be  creative  rather  than  acquisitive:  or,  as 
it  has  been  well  said,  that  intellectual  food  should 
go  to  form  mental  muscle  and  not  mental  fat. 

The  negative  condition  is  plasticity :  the 
avoidance  of  all  crystallisation  as  is  immediately 
suggested  by  the  environment.  A  mind  that 
would  grow  must  let  no  ideas  become  per- 
manent except  such  as  lead  to  action.  Towards 
all  others  it  must  maintain  an  attitude  of 
absolute  receptivity  ;  admitting  all,  being 
modified  by  all,  but  permanently  biassed  by 
none.  To  become  crystallised,  fixed  in  opinion 
and  mode  of  thought,  is  to  lose  the  great  char- 
acteristic of  life,  by  which  it  is  distinguished 
from  inanimate  nature  :  the  power  of  adapting 
itself  to  circumstances. 

This  is  true  even  more  of  the  race.  There 
are  nations  in  the  East  so  enslaved  by  custom 
that  they  seem  to  have  lost  all  power  of  change 
except  the  capability  of  being  destroyed. 


CONDITIONS  OF  MENTAL  DEVELOPMENT      117 

Propriety,  in  fact,  is  the  crystallisation  of  a 
race.  And  if  we  consider  that  a  race,  in  pro- 
portion as  it  is  plastic  and  capable  of  change, 
may  be  regarded  as  young  and  vigorous,  while 
a  race  which  is  fixed,  persistent  in  form,  unable 
to  change,  is  as  surely  effete,  worn  out,  in  peril 
of  extinction  ;  we  shall  see,  I  think,  the  im- 
mense importance  to  a  nation  of  checking  the 
growth  of  conventionalities.  It  is  quite  possible 
for  conventional  rules  of  action  and  conventional 
habits  of  thought  to  get  such  power  that  pro- 
gress is  impossible,  and  the  nation  only  fit  to 
be  improved  away.  In  the  face  of  such  a 
danger  it  is  not  right  to  be  proper. 

NOTE. — The  following  letter,  published  in 
the  Pall  Mall  Gazette  of  June  24,  1 868,  should 
be  read  in  connection  with  this  Discourse. 

"  Sir — I  ask  for  a  portion  of  your  space  to 
say  something  about  a  lecture,  '  On  some  of 
the  Conditions  of  Mental  Development,'  which  I 
delivered  at  the  Royal  Institution  in  March  last. 

"In  that  lecture  I  attempted  to  state  and 
partially  answer  the  question,  'What  is  that 
attitude  of  mind  which  is  most  likely  to  change 
for  the  better  ? '  I  proposed  to  do  this  by 
applying  the  hypothesis  of  the  variability  of 
species  to  the  present  condition  of  the  human 
race.  I  put  forward  also  for  this  purpose  a 
certain  biological  law,  viz.  that  permanent 


Ii8  LECTURES  AND  ESSAYS 

advantageous  changes  in  an  organism  are  due 
to  its  spontaneous  activity,  and  not  to  the 
direct  action  of  the  environment. 

"  In  the  short  account  of  the  evolution- 
hypothesis  which  I  prefixed,  I  followed  Mr. 
Herbert  Spencer's  Principles  of  Biology,  not 
knowing,  at  the  time,  how  much  of  the  theory 
was  due  to  him  personally,  but  imagining  that 
the  greater  part  of  it  was  the  work  of  previous 
biologists.  On  this  account  I  omitted  to  make 
such  references  to  my  special  sources  of  informa- 
tion as  I  should  otherwise  have  made.  I  was 
also  ignorant  of  the  developments  and  applica- 
tions of  the  theory  which  he  has  made  in  his 
other  works,  in  which  a  great  portion  of  my 
remarks  had  been  anticipated.  These  omissions 
I  desire  now  to  rectify. 

"  Mr.  Spencer's  theory  is  to  the  ideas  which 
preceded  it  even  more  than  the  theory  of 
gravitation  was  to  the  guesses  of  Hooke  and 
the  facts  of  Kepler. 

"  Finding  only  a  vague  notion  of  progress 
from  lower  to  higher,  he  has  affixed  the  specific 
meaning  to  the  word  higher  of  which  I  gave  an 
account,  defining  the  processes  by  which  this 
progress  is  effected.  He  has,  moreover,  formed 
the  conception  of  evolution  as  the  subject  of 
general  propositions  applicable  to  all  natural 
processes,  a  conception  which  serves  as  the 
basis  of  a  complete  system  of  philosophy. 


CONDITIONS  OF  MENTAL  DEVELOPMENT      119 

In  particular,  he  has  applied  this  theory  to  the 
evolution  of  mind,  developing  the  complete 
accordance  between  the  laws  of  mental  growth 
and  of  the  growth  of  other  organic  functions. 
In  fact,  even  if  the  two  points  which  I  put 
forward  as  my  own — viz.  the  formal  application 
of  the  biological  method  to  a  certain  special 
problem,  and  the  biological  law  which  serves  as 
a  partial  solution  of  it — have  not  before  been 
explicitly  developed  (and  of  this  I  am  not  sure), 
yet  they  are  consequences  so  immediate  of  the 
general  theory  that  in  any  case  the  credit  of 
them  should  entirely  belong  to  the  philosopher 
on  whose  domains  I  have  unwittingly  trespassed. 
The  mistake,  of  course,  affects  me  only,  and 
could  in  no  way  injure  the  fame  of  one  whose 
philosophical  position  is  so  high  and  so  assured. 
"  I  may  perhaps  be  excused  for  anticipating 
here  what  I  hope  to  say  more  at  length  at 
another  time,1  that  in  my  belief  the  further 
deductions  to  be  made  from  this  theory,  with 
reference  to  modern  controversies,  will  lead  to 
results  at  once  more  conservative,  and  in  a 
certain  sense  more  progressive,  than  is  com- 
monly supposed. 

"  I  remain,  Sir,  yours,  etc., 

"  W.  K.  CLIFFORD." 

1  This  intention  was  never  carried  out,   so  far  as  the  editors 
are  aware. 


ON  THEORIES  OF  THE    PHYSICAL 
FORCES  l 

[REFERRING  to  the  passage  in  Faust, 

"  Geschrieben  steht :   Im  Anfang  war  das  Wort. 
Hier  stock"  ich  schon  !     Wer  hilft  mir  welter  fort  ? 
Ich  kann  das  Wort  so  hoch  unmoglich  schatzen, 
Ich  muss  es  anders  iibersetzen, 
Wenn  ich  vom  Geiste  recht  erleuchtet  bin. 
Geschrieben  steht :   Im  Anfang  war  der  Sinn. 
Bedenke  wohl  die  erste  Zeile, 
Dass  deine  Feder  sich  nicht  iibereile  ! 
1st  es  der  Sinn,  der  alles  wirkt  und  schafft  ? 
Es  sollte  stehn  :   Im  Anfang  war  die  Kraft ! 
Doch,  auch  indem  ich  dieses  niederschreibe, 
Schon  warnt  mich  was,  dass  ich  dabei  nicht  bleibe. 
Mir  hilft  der  Geist !     Auf  einmal  seh'  ich  Rath, 
Und  schreibe  getrost :   Im  Anfang  war  die  That !  " 

the  speaker  regarded  it  as  a  description  of  four 
views  or  stages  of  opinion  through  which  a 
man  looking  for  himself  on  the  face  of  things 

1  Discourse  delivered  at  the  Royal  Institution,  February  18, 
1870.  This  discourse  is  reprinted  as  it  stands  in  the  Proceedings 
of  the  Royal  Institution.  The  opening  paragraphs,  being  reported 
in  the  third  person  and  apparently  abridged,  are  enclosed  in  square 
brackets. 


THEORIES  OF  THE  PHYSICAL  FORCES       121 

is  likely  to  pass  ;  through  which  also  successive 
generations  of  the  men  who  look  for  themselves 
on  the  face  of  things  are  likely  to  pass.  He 
considered  that  by  far  the  larger  portion  of 
scientific  thought  at  the  present  day  is  in  the 
third  stage — that,  namely,  in  which  Force  is 
regarded  as  the  great  fact  that  lies  at  the  bottom 
of  all  things  ;  but  that  this  is  so  far  from  being 
the  final  one,  that  even  now  the  fourth  stage  is 
on  its  heels.  In  the  fourth  stage  the  concep- 
tion of  Force  disappears,  and  whatever  happens 
is  regarded  as  a  deed.  The  object  of  the  dis- 
course was  to  explain  the  nature  of  this  transi- 
tion, and  to  introduce  certain  conceptions  which 
might  serve  to  prepare  the  way  for  it. 

There  are,  then,  to  be  considered  two  different 
answers  to  the  question,  "  What  is  it  that  lies 
at  the  bottom  of  things  ?  "  The  two  answers 
correspond  to  two  different  ways  of  stating  the 
question  ;  namely,  first,  "  Why  do  things  hap- 
pen ?  "  and,  secondly,  "  What  is  it  precisely  that 
does  happen  ?  "  The  speaker  maintained  that 
the  first  question  is  external  to  the  province  of 
science  altogether,  and  science  has  nothing  to 
do  with  it ;  but  that  the  second  is  exactly  the 
question  to  which  science  is  always  trying  to 
find  the  answer.  It  may  be  doubted  whether 
the  first  question  is  within  the  province  of 
human  knowledge  at  all.  For  it  is  as  necessary 
that  a  question  should  mean  something^  in  order 


122  LECTURES  AND  ESSAYS 

to  be  a  real  question,  as  that  an  answer  should 
mean  something,  in  order  to  be  a  real  answer. 
And  it  is  quite  possible  to  put  words  together 
with  a  note  of  interrogation  after  them  without 
asking  any  real  question  thereby.  Whether  the 
phrase,  "  Why  do  things  happen  ?  "  as  applied 
to  physical  phenomena,  is  a  phrase  of  this  kind 
or  no,  is  not  here  to  be  considered.  But  that 
to  the  scientific  inquirer  there  is  not  any  "why" 
at  all,  and  that  if  he  ever  uses  the  word  it  is 
always  in  the  sense  of  what,  the  speaker  regarded 
as  certain.  In  order  to  show  what  sort  of  way 
an  exact  knowledge  of  the  facts  would  super- 
sede the  inquiry  after  the  cause  of  them,  he  then 
made  use  of  the  hypothesis  of  continuity  ;  show- 
ing, in  the  following  manner,  that  it  involves 
such  an  interdependence  of  the  facts  of  the  uni- 
verse as  forbids  us  to  speak  of  one  fact  or  set  of 
facts  as  the  cause  of  another  fact  or  set  of  facts.] 

The  hypothesis  of  the  continuity  of  space  and 
time  is  explained,  and  the  alternative  hypothesis 
is  formulated. 

From  the  hypothesis  of  the  complete  continuity 
of  time-changes,  a  knowledge  of  the  entire  history 
of  a  single  particle  is  shown  to  be  involved  in  a 
complete  knowledge  of  its  state  at  any  moment. 

Things  frequently  move.  Some  things  move 
faster  than  others.  Even  the  same  thing  moves 
faster  at  one  time  than  it  does  at  another  time. 
When  you  say  that  you  are  walking  four  miles 


THEORIES  OF  THE  PHYSICAL  FORCES       123 

an  hour,  you  do  not  mean  that  you  actually 
walk  exactly  four  miles  in  any  particular  hour  ; 
you  mean  that  if  anybody  did  walk  for  an 
hour,  keeping  all  the  time  exactly  at  the  rate 
at  which  you  are  walking,  he  would  in  that 
hour  walk  four  miles.  But  now  suppose  that 
you  start  walking  four  miles  an  hour,  and 
gradually  quicken  your  pace,  until  you  are 
walking  six  miles  an  hour.  Then  this  question 
may  be  asked  :  Suppose  that  anybody  chose  a 
particular  number  between  four  and  six,  say 
four  and  five-eighths,  is  it  perfectly  certain  that 
at  some  instant  or  other  during  that  interval 
you  were  walking  at  the  rate  of  four  miles  and 
five-eighths  in  the  hour  ?  Or,  to  put  it  more 
accurately,  suppose  that  we  have  a  vessel  con- 
taining four  pints  of  water  exactly,  and  that 
somebody  adds  to  it  a  casual  quantity  of  water 
less  than  two  pints.  Then  is  it  perfectly  certain 
that  between  these  two  times,  when  you  were 
walking  at  four  miles  an  hour,  and  when  you 
were  walking  six  miles  an  hour,  there  was  some 
particular  instant  at  which  you  were  walking 
exactly  as  many  miles  and  fractions  of  a  mile 
an  hour  as  there  are  pints  and  fractions  of  a 
pint  of  water  in  the  vessel  ?  The  hypothesis  of 
continuity  says  that  the  answer  to  this  question 
is  yes  ;  and  this  is  the  answer  which  everybody 
gives  no  wad  ays ;  which  everybody  has  given  mostly 
since  the  invention  of  the  differential  calculus. 


I24  LECTURES  AND  ESSAYS 

But  this  is  a  question  of  fact,  and  not  of 
calculation.  Let  us,  therefore,  try  and  imagine 
what  the  contrary  hypothesis  would  be  like. 

You  know  what  a  "  wheel  of  life  "  is.  There 
is  a  cylinder  with  slits  in  its  side,  which  can  be 
spun  round  rapidly  ;  and  you  look  through  the 
slits  at  the  pictures  opposite.  The  result  is 
that  you  see  the  pictures  moving  ;  moreover, 
you  see  them  move  faster  or  slower  according 
as  you  turn  the  cylinder  faster  or  slower.  This 
is  what  you  see,  and  what  appears  to  happen  ; 
but  now  let  us  consider  what  actually  does 
happen.  I  remember  in  particular  a  picture  of 
a  man  rolling  a  ball  down  an  inclined  plane 
towards  you  ;  he  was  standing  at  the  farther 
edge  of  the  inclined  plane,  as  it  were  behind  a 
counter,  and  he  picked  up  the  balls  one  by  one 
and  rolled  them  towards  you.  But  now  when 
you  took  out  the  strips  of  paper  on  which  the 
pictures  were  drawn,  you  found  that  they  were 
really  pictures  of  this  man  and  his  ball  in  a 
graduated  series  of  positions.  Each  picture,  of 
course,  was  perfectly  still  in  itself,  a  mere  draw- 
ing on  the  paper.  The  first  one  represented 
him  with  his  hand  below  the  counter,  just  pick- 
ing up  the  ball  ;  in  the  next,  he  had  the  ball  in 
his  hand,  drawn  back  ready  to  roll  down  ;  in 
the  next,  the  hand  was  thrown  forwards  with 
the  ball  in  it ;  in  the  next,  the  ball  had  just 
left  his  hand  and  rolled  a  little  way  down  ;  in 


THEORIES  OF  THE  PHYSICAL  FORCES       125 

the  next  farther,  and  so  on.  Now,  these  pic- 
tures being  put  in  the  inside  of  the  cylinder 
which  is  turning  round,  come  opposite  you  one 
by  one.  But  you  do  not  look  directly  at  them ; 
there  are  slits  interposed.  The  effect  of  that 
is,  that  if  you  look  straight  at  a  certain  portion 
of  the  opposite  picture  you  can  only  see  it  for 
a  very  small  interval  of  time  ;  that,  namely, 
during  which  the  slit  is  passing  in  front  of  your 
eye.  Now  let  us  carefully  examine  what  hap- 
pens. When  the  slit  passes,  it  goes  so  quickly 
that  you  get,  as  it  were,  almost  an  instantaneous 
photograph  on  your  eye  of  the  opposite  picture; 
say  of  the  man  with  his  hand  below  the  counter. 
Then  this  is  effaced,  and  you  see  absolutely 
nothing  until  the  next  slit  passes.  But  by  the 
time  the  next  slit  comes,  another  picture  has 
got  opposite  to  you  ;  so  that  you  get  an 
instantaneous  photograph  this  time  of  the  man 
with  his  hand  drawn  back  and  the  ball  in  it. 
Then  this  in  its  turn  is  effaced,  for  a  time  you 
see  nothing,  and  then  you  are  given  an  in- 
stantaneous glimpse  of  the  hand  thrown  forward. 
In  this  way,  what  you  really  see  is  darkness 
relieved  by  regularly-recurring  glimpses  of  the 
figure  in  different  positions.  Now,  this  experi- 
ence that  you  get  is  obviously  consistent  with 
the  hypothesis  that  the  man  goes  on  moving 
all  the  time  when  he  is  hidden  from  you  ;  so 
as  to  be  in  exactly  that  series  of  positions  when 


126  LECTURES  AND  ESSAYS 

you  do  catch  a  glimpse  of  him.  And,  in  fact, 
you  do  instinctively,  by  an  inevitable  habit, 
admit  this  hypothesis,  not  merely  into  your 
mind  as  a  speculation,  but  into  your  very  sensa- 
tion as  an  observed  phenomenon.  You  simply 
see  the  man  move  ;  and,  except  for  a  certain 
weariness  in  the  eyes,  there  is  nothing  to  dis- 
tinguish this  perception  of  movement  from  any 
other  perception  of  movement.  At  the  same 
time  we  do  know  very  distinctly,  and  beyond 
the  shadow  of  a  doubt,  that  there  is  no  con- 
tinuity in  the  picture  at  all  :  that,  in  fact,  you 
do  not  see  the  same  picture  twice  following,  but 
a  new  one  every  time  till  the  cycle  is  completed  ; 
and  that  the  picture  never  is  in  any  position  in- 
termediate between  two  successive  ones  of  those 
which  you  see.  Here  then  is  an  apparently 
continuous  motion  which  is  really  discontinuous ; 
and  moreover  there  is  an  apparently  continuous 
perception  of  it  which  is  really  discontinuous — 
that  is,  it  seems  to  be  gradually  changed,  while 
it  really  goes  by  little  jumps. 

I  suppose  very  few  people  have  looked  at 
this  toy  without  wondering  whether  it  is  not 
actually  and  truly  a  wheel  of  life,  without  any 
joke  at  all.  I  mean,  that  it  is  very  natural  for 
the  question  to  present  itself,  Do  I  ever  really 
see  anything  move  ?  May  not  all  my  appar- 
ently continuous  perceptions  be  ultimately  made 
up  of  little  jumps,  which  I  run  together  by  this 


THEORIES  OF  THE  PHYSICAL  FORCES       127 

same  inevitable  instinct  ?  There  is  another 
way  in  which  this  is  sometimes  suggested.  If 
you  move  your  hand  quickly,  you  can  see  a 
continuous  line  of  light,  because  the  image  of 
every  position  of  your  hand  lingers  a  little  while 
upon  the  retina.  But  now,  if  you  do  this  in  a 
room  lighted  only  by  an  electric  spark  which  is 
not  going  very  fast,  so  that  the  general  result 
is  darkness  broken  by  nearly  instantaneous 
flashes  at  regular  intervals  ;  then,  instead  of 
seeing  a  continuous  line  of  light,  you  will  see  a 
distinct  series  of  different  hands,  perhaps  about 
an  inch  apart,  if  the  electric  spark  is  going  very 
slowly,  and  you  move  your  hand  very  quickly. 
But  now  make  the  spark  go  quicker,  or  your 
hand  slower;  the  distances  between  these  several 
hands  will  gradually  diminish,  till — you  do  not 
know  how — the  continuous  line  of  light  is  re- 
stored. And  the  question  inevitably  presents 
itself — is  not  every  case  of  apparently  con- 
tinuous perception  really  a  case  of  successive 
distinct  images  very  close  together? 

That  is  to  say,  for  instance,  if  I  move  my 
hand  so  in  front  of  me,  and  apparently  see  it 
take  up  in  succession  every  possible  position  on 
its  path  between  the  two  extreme  positions  ; 
do  I  really  see  this,  or  do  I  only  see  my  hand 
in  a  certain  very  large  number  of  distinct 
positions,  and  not  at  all  in  the  intervening 
spaces  ? 


i«8  LECTURES  AND  ESSAYS 

I  have  no  doubt  whatever  myself,  that  the 
latter  alternative  is  the  true  one,  and  that  the 
wheel  of  life  is  really  an  illustration  and  type 
of  every  moment  of  our  existence.  But  I  am 
not  going  to  give  my  reasons  for  this  opinion, 
because  it  is  quite  a  different  question  from  the 
one  I  am  trying  to  get  at.  The  question, 
namely,  is  this.  What  I  see,  or  fancy  I  see,  is 
quite  consistent  with  the  hypothesis  that  my 
hand  really  does  go  on  moving  continuously  all 
the  time,  and  takes  up  an  infinite  number  of 
positions  between  the  two  extreme  ones.  But 
if  this  hypothesis  is  not  true,  what  is  true  ?  and 
how  are  we  to  imagine  any  other  state  of 
things  than  that  supposed  by  the  hypothesis  of 
continuity  ? 

I  draw  here  two  rows  of  points.  The  upper 
row  of  points  is  to  represent  a  series  of  positions 
in  space  which  it  is  conceivable  that  a  certain 
thing  might  take  up.  The  lower  row  of  points 
is  to  represent  a  series  of  instants  in  time  at 
which  it  is  conceivable  that  the  same  thing 
might  exist.  Suppose  now  that  at  the  instant 
of  time  represented  by  the  first  point  of  the 
lower  row,  the  thing  held  the  position  in  space 
represented  by  the  first  point  of  the  upper  row. 
Suppose  that  it  only  existed  there  for  that  in- 
stant, and  then  disappeared  utterly,  so  that  at 
these  succeeding  instants  where  the  lower  points 
have  no  points  directly  above  them  the  thing 


THEORIES  OF  THE  PHYSICAL  FORCES       129 

is  nowhere  at  all.  Lastly,  suppose  that  at  this 
instant  of  time  which  has  a  space-dot  above  it, 
the  thing  existed  in  that  space-position  ;  and 
so  on  all  through,  the  thing  only  existing  at 
those  instants  whose  representative  points  have 
a  space-dot  exactly  above  them,  and  being  then 
in  the  space -position  signified  by  such  dots. 
Then  we  may  call  this  a  discontinuous  motion  ; 
a  motion  because  the  thing  is  in  different  places 
at  different  times,  though  it  is  not  at  all  times 
that  it  exists  at  all ;  and  a  discontinuous  motion 
because  the  thing  passes  from  one  position  to 
another  distant  from  it  without  going  through 
any  intermediate  position. 

Now  imagine  that  in  each  of  these  two 
series  the  dots  are  very  close  together  indeed, 
and  very  great  in  number  ;  so  that,  however 
small  one  made  them  on  the  paper,  the  lines 
would  look  as  if  they  were  continuous  lines. 
And  let  the  thing  be  a  white  speck  travelling 
along  the  upper  line  in  the  manner  I  have 
described ;  namely,  existing  only  when  there 
is  one  dot  exactly  over  another  ;  only  that  as 
the  lower  dots  represent  instants  of  time,  we 
may  make  some  definite  supposition  and 
assume  that  one  inch  of  them  represents  a 
second. 

Then  it  is  clear  that  if  the  dots  were  taken 
close  enough  together,  and  enough  of  them,  the 
appearance  would  be  precisely  what  we  ordi- 
VOL.  I  K 


130  LECTURES  AND  ESSAYS 

narily  see  when  a  white  speck  moves  along  a 
line.  That  is  to  say,  we  have  got  some  sort 
of  representation  of  what  we  might  have  to 
suppose,  if  we  did  not  assume  the  truth  of  the 
law  of  continuity. 

You  must  here  notice  in  particular  that  I 
suppose  the  series  of  positions  denoted  by  the 
upper  dots  to  be  all  the  positions  that  are 
between  the  two  end  ones  ;  that  is,  I  suppose 
the  path  from  one  of  these  end  ones  to  the 
other  to  be  made  up  of  a  series  of  discrete 
positions.  And  similarly  I  suppose  the  series 
of  instants  denoted  by  the  lower  dots  to  be  all 
the  time  that  elapses  between  the  two  end  ones ; 
that  is,  I  suppose  the  interval  of  time  to  contain 
a  perfectly  definite  number  of  instants,  these 
being  further  indivisible.  Or  we  may  say  that 
on  this  alternative  hypothesis  space  and  time 
are  discontinuous  ;  that  is,  they  are  in  separate 
parts  which  do  not  hold  together.  Now  I  must 
beg  you  to  remember  for  a  little  while  what 
the  hypothesis  of  continuity  is  not,  for  I  shall 
have  to  refer  to  this  point  again  subsequently. 
In  this  kind  of  jumping  motion  that  we  have 
been  imagining,  the  rate  of  motion  of  a  thing 
could  only  be  measured  by  the  size  of  one  of 
its  jumps  ;  that  is,  by  the  number  of  positions 
it  passed  over  between  two  existences  compared 
with  the  number  of  instants  passed  over.  And 
this  rate  might  obviously  change  by  jumps  as 


THEORIES  OF  THE  PHYSICAL  FORCES       131 

violent  and  sudden  as  those  of  the  thing  itself ; 
at  any  instant  when  the  thing  was  non-existent 
its  rate  would  be  non-existent,  and  whenever 
the  thing  came  into  existence  its  rate  would 
suddenly  have  a  value  depending  on  how  far 
off  its  last  position  was.  In  this  case,  there- 
fore, our  question  about  the  intermediate  rate 
— whether  between  walking  four  miles  an  hour 
and  walking  six  miles  an  hour  you  must 
necessarily  walk  at  all  intermediate  paces — 
must  be  answered  in  the  negative.  Now  then, 
at  last,  let  us  investigate  some  consequences  of 
supposing  that  motion  is  really  continuous  as 
it  seems  to  be. 

First,  how  to  measure  the  rate  at  which  a 
thing  is  moving?  This  was  done  experi- 
mentally by  Galileo  in  the  case  of  falling 
bodies,  and  I  shall  have  to  speak  again  of  the 
results  which  he  obtained.  But  at  present  I 
want  to  speak  not  of  an  experimental  method 
of  finding  the  rate,  but  of  a  theoretical  method 
of  representing  it,  invented  by  Newton,  and 
called  the  curve  of  velocities. 

Suppose  that  a  point  N  is  going  along  the 
line  O  Y,  sometimes  fast  and  sometimes  slow  ; 
and  that  a  point  M  is  going  along  the  line  O  x 
always  at  the  same  rate.  Also  somebody 
always  holds  a  stick  N  P  so  as  to  move  with 
the  point  N,  and  be  horizontal ;  and  somebody 
holds  a  stick  M  P  so  as  to  move  with  the 


132  LECTURES  AND  ESSAYS 

point  M,  and  be  vertical  ;  and  a  third  person 
keeps  a  pencil  pressed  in  the  corner  where 
the  two  sticks  cross  at  P.  Then 
.  when  the  points  M  and  N  move, 
Jf  the  point  P  will  move  too  ;  and 
its  motion  will  depend  on  that 
of  the  two  other  points.  For 
instance,  if  the  point  N  moves 
always  exactly  as  fast  as  the 
point  M,  then  the  point  P  will  go  along  the  line 
O  P  midway  between  the  lines  o  X  O  Y.  If  N 
moves  twice  as  fast  as  M  always,  the 
point  P  will  go  along  a  line  nearer  N 
O  Y ;  and  if  N  moves  only  half  as 
fast  as  M,  then  P  will  go  along  a  line 
nearer  o  x.  And  in  general,  the  faster 
N  moves,  the  more  the  line  will  be  tilted  up  ; 
and  if  the  rate  at  which  N  goes  is  changeable, 
the  direction  of  P's  motion  will  be  changeable, 
and  P  will  then  describe  a  curve,  which  will  be 
very  steep  when  N  is  going  fast,  and  more  flat 
when  N  is  going  slow.  So  that  the  steepness 
of  this  curve  is  now  a  visible  measure  of  the 
rate  at  which  N  is  going,  and  the  curvature  of 
it  is  a  visible  expression  of  the  fact  that  the 
rate  is  changeable.  Now  the  hypothesis  of 
continuity  in  the  motion  of  N  asserts  not 
merely  that  N  itself  moves  without  any  jumps, 
but  that  the  rate  at  which  N  is  going  changes 
gradually  without  any  jumps,  and  consequently 


THEORIES  OF  THE  PHYSICAL  FORCES       133 

that  the  direction  of  P's  motion  changes  gradu- 
ally ;  or  that  the  curve  described  by  P  cannot 
have  a  sharp  point  like  this.  But  it  asserts  a 
great  deal  more  besides  this,  which  I  shall  now 
endeavour  to  explain.  Let  us  imagine  a  new 
point  N1?  so  moving  that  whenever  the  old  N  is 
going  at  four  inches  a  second,  Nx  shall  be  four 
inches  from  O  ;  and  when  N  is  going  at  two 
inches  a  second,  Na  shall  be  two  inches  from 
O,  and  so  on,  the  distance  of  Nx  from  0  being 
always  exactly  as  far  as  N  would  go  in  a  second 
if  it  went  at  the  rate  at  which  it  was  moving  at 
that  instant.  Then  the  distance  O  N:  measures 
the  rate  at  which  N  is  going,  or  the  velocity  of 
N.  If,  for  example,  there  was  a  thing  like  a 
thermometer  hung  up  in  a  train,  so  that  the 
height  of  the  mercury  always  indicated  how 
fast  the  train  was  going ;  when  the  train  was 
going  17  miles  an  hour,  the  mercury  stood  at 
17  inches,  and  so  on  ;  then  the  top  of  the 
mercury  would  behave  towards  the  train  exactly 
as  I  want  the  point  NJ  to  behave  towards  the 
point  N.  It  is  to  indicate  by  its  height  how 
fast  N  is  going. 

If,  then,  the  velocity  of  N  is  changeable,  the 
point  NX  will  move  up  and  down  ;  and  the  rate 
at  which  Nx  moves  up  or  down  is  clearly  the 
rate  at  which  the  velocity  of  N  is  increasing  or 
diminishing.  This  rate  at  which  the  velocity 
of  N  changes  is  called  its  acceleration.  To 


134  LECTURES  AND  ESSAYS 

return  to  our  gauge  instead  of  a  train,  if  in  the 
course  of  a  minute  it  went  up  from  1 7  to  19, 
the  train  would  be  said  to  have  an  acceleration 
of  two  miles  an  hour  per  minute. 

Now  I  shall  take  another  point  N2,  which  is 
to  behave  towards  Nx  exactly  as  Nt  behaves 
towards  N  ;  namely,  the  distance  of  N2  from  O2 
is  to  be  always  equal  to  the  number  of  inches 
which  NX  is  going  in  a  second.  And  then  I 
shall  take  a  point  N3,  related  in  just  this  same 
way  to  N2,  and  so  on,  until  I  come  to  a  point 
that  does  not  move  at  all ;  and  that  I  might 
never  come  to,  so  that  I  should  have  to  go  on 
taking  new  points  for  ever.  But  suppose  now 
that  I  have  got  this  series  of  points,  and  that 
they  are  all  moving  together.  Then  first  of  all 
there  is  my  point  N,  which  moves  anyhow. 
Next  there  is  NI}  such  that  Ol  Nx  is  the  velocity 
of  N,  or  the  rate  of  change  of  N'S  position. 
Next  there  is  N2,  such  that  O2  N2  is  the  acceler- 
ation of  N,  or  the  rate  of  change  of  the  rate 
of  change  of  N'S  position.  Then  again  O3  N3 
is  the  change  of  the  acceleration  of  N,  or 
the  rate  of  change  of  the  rate  of  change  of 
the  rate  of  change  of  N'S  position,  and  so 
on.  We  may,  if  we  like,  agree  to  call  the 
velocity  of  N  the  change  of  the  first  order,  the 
velocity  of  Nx  the  change  of  the  second  order, 
and  so  on. 

Then  the  hypothesis  of  the  perfect  continuity 


THEORIES  OF  THE  PHYSICAL  FORCES       135 

of  N'S  motion  asserts  that  all  these  points  move 
continuously  without  any  jumps.  Now,  a  jump 
made  by  any  one  of  these  points,  being  a  finite 
change  made  in  no  time,  would  be  a  change 
made  at  an  infinite  rate  ;  the  next  point,  there- 
fore, and  all  after  it,  would  go  right  away  from 
O,  and  disappear  altogether.  We  may  thus 
express  the  law  of  continuity  also  in  this  form  ; 
that  there  is  no  infinite  change  of  any  order. 

Now,  observe  further  that  the  rate  at  which 
anything  is  going  is  a  property  of  the  thing  at 
that  instant,  and  exists  whether  the  thing  goes 
any  more  or  not.  If  I  drop  a  marble  on  the 
floor,  it  goes  faster  and  faster  till  it  gets  there, 
and  then  stops  ;  but  at  the  instant  when  it  hit 
the  floor  it  was  going  at  a  perfectly  definite  rate, 
which  can  be  calculated,  though  it  did  not 
actually  go  any  more. 

In  the  same  way  the  configuration  of  all 
these  points  which  depend  on  the  point  N  is  a 
property  of  its  motion  at  any  given  instant, 
quite  independent  of  the  continuance  of  that 
motion.  I  want  you  to  take  particular  notice 
of  this  fact,  that  as  the  point  N  moves  about, 
the  whole  set  of  points  connected  with  it  moves 
too  ;  and  that  you  may  regard  them  as  con- 
nected by  some  machine,  which  you  may  stop 
at  any  moment  to  contemplate  the  simultane- 
ous positions  of  all  these  points  ;  and  that  this  set 
of  simultaneous  positions  belongs  just  simply 


136  LECTURES  AND  ESSAYS 

to  that  one  position  of  the  point  N,  and  there- 
fore to  one  instant  of  time. 

Now  I  am  going  to  state  to  you  dogmatic- 
ally a  certain  mathematical  theorem,  called 
Taylor's  theorem ;  whereby  you  will  see  the 
very  remarkable  consequences  of  this  hypothesis 
that  we  have  made. 

Namely,  there  is  a  certain  rule  whereby  when 
the  positions  of  all  these  points  are  known  for 
any  particular  instant  of  time,  then  their  posi- 
tions at  any  other  instant  of  time  may  be 
calculated  from  these ;  and  it  is  impossible 
that  they  should  have  at  that  other  instant  any 
other  positions  than  those  so  calculated.  Pro- 
vided always  that  there  is  no  infinite  change  of 
any  order  ;  that  is  to  say,  that  no  one  of  the 
points  has  taken  a  sudden  jump  and  sent  all 
the  points  after  it  away  to  an  infinite  distance 
from  O  at  any  instant  between  the  one  for 
which  the  positions  are  given  and  the  one  for 
which  they  are  calculated. 

Remember  that  the  positions  of  all  the  de- 
rivative points  are  mere  properties  of  the  motion 
of  the  point  N  at  any  instant  ;  that  in  fact  we 
must  know  them  all  in  order  to  know  com- 
pletely the  state  of  the  point  N  at  that  instant. 
And  then  observe  the  result  that  we  have 
arrived  at.  From  the  knowledge  of  the  com- 
plete state  at  any  instant  of  a  thing  whose 
motion  obeys  the  law  of  continuity,  we  can 


THEORIES  OF  THE  PHYSICAL  FORCES       137 

calculate  where  it  was  at  any  past  time,  and 
where  it  will  be  at  any  future  time.  Now 
the  hypothesis  of  continuity,  of  which  we  have 
only  got  disjointed  fragments  hitherto,  is  this  ; 
that  the  motion  of  every  particle  of  the  whole 
universe  is  entirely  continuous.  It  follows  from 
this  hypothesis  that  the  state  at  this  moment 
of  any  detached  fragment — say  a  particle  of 
matter  at  the  tip  of  my  tongue — is  an  infallible 
record  of  the  eternal  past,  an  infallible  prediction 
of  the  eternal  future. 

This  is  not  the  same  as  the  statement  that 
a  complete  knowledge  of  the  position  and 
velocity  of  every  body  in  the  universe  at  a 
given  moment  would  suffice  to  determine  the 
position  at  any  previous  or  subsequent  moment. 
That  depends  on  an  entirely  different  hypothesis, 
and  relates  to  the  whole,  while  this  proposition 
that  I  am  now  expounding  relates  to  every 
several  part  however  small.  Now  reflect  upon 
the  fact  that  for  a  single  particle — quite  irre- 
spective of  everything  else — the  history  of 
eternity  is  contained  in  every  second  of  time  ; 
and  then  try  if  you  can  find  room  in  this  one 
stifling  eternal  fact  for  any  secondary  causes 
and  the  question  why  ?  Why  does  the  moon 
go  round  the  earth  ?  When  the  Solar  system 
was  nebulous,  anybody  who  knew  all  about  some 
one  particle  of  nebulous  vapour  might  have 
predicted  that  it  would  at  this  moment  form 


138  LECTURES  AND  ESSAYS 

part  of  the  moon's  mass,  and  be  rotating  about 
the  earth  exactly  as  it  does.  But  why  with 
an  acceleration  inversely  as  the  square  of  the 
distance  ?  There  is  no  why  ;  the  fact  is  prob- 
ably equivalent  to  saying  that  the  continuous 
motion  of  one  body  is  such  as  not  to  interfere 
with  the  continuous  motion  of  another.  If 
once  so,  then  always  ;  the  cause  is  only  the 
fact  that  at  some  moment  the  thing  is  so, — 
or  rather,  the  facts  of  one  time  are  not  the 
cause  of  the  facts  of  another,  but  the  facts  of 
all  time  are  included  in  one  statement,  and 
rigorously  bound  up  together. 

Parallel,  however,  with  this  hypothesis  of 
temporal  continuity,  there  is  another  hypo- 
thesis, not  so  universally  held,  of  a  continuity  in 
space ;  for  which  indeed  I  hope  to  make  more 
room  presently.  And  out  of  this  it  appears 
that  as  the  history  of  eternity  is  written  in 
every  second  of  time,  so  the  state  of  the  uni- 
verse is  written  in  every  point  of  space. 


ON  THE  AIMS  AND  INSTRUMENTS 
OF  SCIENTIFIC  THOUGHT1 

IT  may  have  occurred  (and  very  naturally  too) 
to  such  as  have  had  the  curiosity  to  read  the 
title  of  this  lecture,  that  it  must  necessarily  be 
a  very  dry  and  difficult  subject ;  interesting  to 
very  few,  intelligible  to  still  fewer,  and,  above 
all,  utterly  incapable  of  adequate  treatment 
within  the  limits  of  a  discourse  like  this.  It  is 
quite  true  that  a  complete  setting-forth  of  my 
subject  would  require  a  comprehensive  treatise 
on  logic,  with  incidental  discussion  of  the  main 
questions  of  metaphysics  ;  that  it  would  deal 
with  ideas  demanding  close  study  for  their 
apprehension,  and  investigations  requiring  a 
peculiar  taste  to  relish  them.  It  is  not  my  in- 
tention now  to  present  you  with  such  a  treatise. 
The  British  Association,  like  the  world  in 
general,  contains  three  classes  of  persons.  In 
the  first  place,  it  contains  scientific  thinkers  ; 

1  A  Lecture  delivered  before  the  members  of  the  British  Associa- 
tion, at  Brighton,  on  August  19,  1872. 


140  LECTURES  AND  ESSAYS 

that  is  to  say,  persons  whose  thoughts  have 
very  frequently  the  characters  which  I  shall 
presently  describe.  Secondly,  it  contains  persons 
who  are  engaged  in  work  upon  what  are  called 
scientific  subjects,  but  who  in  general  do  not, 
and  are  not  expected  to,  think  about  these 
subjects  in  a  scientific  manner.  Lastly,  it 
contains  persons  who  suppose  that  their  work 
and  their  thoughts  are  unscientific,  but  who 
would  like  to  know  something  about  the  busi- 
ness of  the  other  two  classes  aforesaid.  Now, 
to  any  one  who  belonging  to  one  of  these  classes 
considers  either  of  the  other  two,  it  will  be 
apparent  that  there  is  a  certain  gulf  between 
him  and  them  ;  that  he  does  not  quite  under- 
stand them,  nor  they  him  ;  and  that  an  oppor- 
tunity for  sympathy  and  comradeship  is  lost 
through  this  want  of  understanding.  It  is  this 
gulf  that  I  desire  to  bridge  over,  to  the  best  of 
my  power.  That  the  scientific  thinker  may 
consider  his  business  in  relation  to  the  great 
life  of  mankind  ;  that  the  noble  army  of 
practical  workers  may  recognise  their  fellowship 
with  the  outer  world,  and  the  spirit  which  must 
guide  both  ;  that  this  so-called  outer  world  may 
see  in  the  work  of  science  only  the  putting  in 
evidence  of  all  that  is  excellent  in  its  own  work, 
— may  feel  that  the  kingdom  of  science  is 
within  it :  these  are  the  objects  of  the  present 
discourse.  And  they  compel  me  to  choose 


AIMS  OF  SCIENTIFIC  THOUGHT  141 

such  portions  of  my  vast  subject  as  shall  be 
intelligible  to  all,  while  they  ought  at  least  to 
command  an  interest  universal,  personal,  and 
profound. 

In  the  first  place,  then,  what  is  meant  by 
scientific  thought  ?  You  may  have  heard  some 
of  it  expressed  in  the  various  Sections  this 
morning.  You  have  probably  also  heard  ex- 
pressed in  the  same  places  a  great  deal  of 
unscientific  thought ;  notwithstanding  that  it 
was  about  mechanical  energy,  or  about  hydro- 
carbons, or  about  eocene  deposits,  or  about 
malacopterygii.  For  scientific  thought  does 
not  mean  thought  about  scientific  subjects  with 
long  names.  There  are  no  scientific  subjects. 
The  subject  of  science  is  the  human  universe  ; 
that  is  to  say,  everything  that  is,  or  has  been, 
or  may  be  related  to  man.  Let  us  then,  taking 
several  topics  in  succession,  endeavour  to  make 
out  in  what  cases  thought  about  them  is 
scientific,  and  in  what  cases  not. 

Ancient  astronomers  observed  that  the 
relative  motions  of  the  sun  and  moon  recurred 
all  over  again  in  the  same  order  about  every 
nineteen  years.  They  were  thus  enabled  to 
predict  the  time  at  which  eclipses  would  take 
place.  A  calculator  at  one  of  our  observatories 
can  do  a  great  deal  more  than  this.  Like 
them,  he  makes  use  of  past  experience  to 
predict  the  future  ;  but  he  knows  of  a  great 


142  LECTURES  AND  ESSAYS 

number  of  other  cycles  besides  that  one  of  the 
nineteen  years,  and  takes  account  of  all  of  them  ; 
and  he  can  tell  about  the  solar  eclipse  of  six 
years  hence  exactly  when  it  will  be  visible,  and 
how  much  of  the  sun's  surface  will  be  covered 
at  each  place,  and,  to  a  second,  at  what  time  of 
day  it  will  begin  and  finish  there.  This  pre- 
diction involves  technical  skill  of  the  highest 
order  ;  but  it  does  not  involve  scientific  thought, 
as  any  astronomer  will  tell  you. 

By  such  calculations  the  places  of  the  planet 
Uranus  at  different  times  of  the  year  had  been 
predicted  and  set  down.  The  predictions  were 
not  fulfilled.  Then  arose  Adams,  and  from 
these  errors  in  the  prediction  he  calculated  the 
place  of  an  entirely  new  planet,  that  had  never 
yet  been  suspected  ;  and  you  all  know  how  the 
new  planet  was  actually  found  in  that  place. 
Now  this  prediction  does  involve  scientific 
thought,  as  any  one  who  has  studied  it  will  tell 
you. 

Here  then  are  two  cases  of  thought  about 
the  same  subject,  both  predicting  events  by  the 
application  of  previous  experience,  yet  we  say 
one  is  technical  and  the  other  scientific. 

Now  let  us  take  an  example  from  the 
building  of  bridges  and  roofs.  When  an 
opening  is  to  be  spanned  over  by  a  material 
construction,  which  must  bear  a  certain  weight 
without  bending  enough  to  injure  itself,  there 


AIMS  OF  SCIENTIFIC  THOUGHT  143 

are  two  forms  in  which  this  construction  can 
be  made,  the  arch  and  the  chain.  Every  part 
of  an  arch  is  compressed  or  pushed  by  the  other 
parts  ;  every  part  of  a  chain  is  in  a  state  of 
tension,  or  is  pulled  by  the  other  parts.  In 
many  cases  these  forms  are  united.  A  girder 
consists  of  two  main  pieces  or  booms,  of  which 
the  upper  one  acts  as  an  arch  and  is  compressed, 
while  the  lower  one  acts  as  a  chain  and  is 
pulled  ;  and  this  is  true  even  when  both  the 
pieces  are  quite  straight.  They  are  enabled  to 
act  in  this  way  by  being  tied  together,  or 
braced,  as  it  is  called,  by  cross  pieces,  which 
you  must  often  have  seen.  Now  suppose  that 
any  good  practical  engineer  makes  a  bridge  or 
roof  upon  some  approved  pattern  which  has 
been  made  before.  He  designs  the  size  and 
shape  of  it  to  suit  the  opening  which  has  to  be 
spanned  ;  selects  his  material  according  to  the 
locality;  assigns  the  strength  which  must  be 
given  to  the  several  parts  of  the  structure 
according  to  the  load  which  it  will  have  to 
bear.  There  is  a  great  deal  of  thought  in  the 
making  of  this  design,  whose  success  is  predicted 
by  the  application  of  previous  experience  ;  it 
requires  technical  skill  of  a  very  high  order ; 
but  it  is  not  scientific  thought.  On  the  other 
hand,  Mr.  Fleeming  Jenkin l  designs  a  roof 

1  On  Braced  Arches   and  Suspension  Bridges.       Edinburgh  : 
Neill,  1870. 


144  LECTURES  AND  ESSAYS 

consisting  of  two  arches  braced  together,  instead 
of  an  arch  and  a  chain  braced  together  ;  and 
although  this  form  is  quite  different  from  any 
known  structure,  yet  before  it  is  built  he  assigns 
with  accuracy  the  amount  of  material  that  must 
be  put  into  every  part  of  the  structure  in  order 
to  make  it  bear  the  required  load,  and  this 
prediction  may  be  trusted  with  perfect  security. 
What  is  the  natural  comment  on  this?  Why, 
that  Mr.  Fleeming  Jenkin  is  a  scientific  engineer. 

Now  it  seems  to  me  that  the  difference 
between  scientific  and  merely  technical  thought, 
not  only  in  these  but  in  all  other  instances 
which  I  have  considered,  is  just  this  :  Both  of 
them  make  use  of  experience  to  direct  human 
action  ;  but  while  technical  thought  or  skill 
enables  a  man  to  deal  with  the  same  circum- 
stances that  he  has  met  with  before,  scientific 
thought  enables  him  to  deal  with  different  cir- 
!  cumstances  that  he  has  never  met  with  before. 
But  how  can  experience  of  one  thing  enable  us 
to  deal  with  another  quite  different  thing  ?  To 
answer  this  question  we  shall  have  to  consider 
more  closely  the  nature  of  scientific  thought. 

Let  us  take  another  example.  You  know 
that  if  you  make  a  dot  on  a  piece  of  paper,  and 
then  hold  a  piece  of  Iceland  spar  over  it,  you 
will  see  not  one  dot  but  two.  A  mineralogist, 
by  measuring  the  angles  of  a  crystal,  can  tell 
you  whether  or  no  it  possesses  this  property 


AIMS  OF  SCIENTIFIC  THOUGHT  145 

without  looking  through  it.  He  requires  no 
scientific  thought  to  do  that.  But  Sir  William 
Rowan  Hamilton,  the  late  Astronomer-Royal 
of  Ireland,  knowing  these  facts  and  also  the 
explanation  of  them  which  Fresnel  had  given, 
thought  about  the  subject,  and  he  predicted 
that  by  looking  through  certain  crystals  in  a 
particular  direction  we  should  see  not  two  dots 
but  a  continuous  circle.  Mr.  Lloyd  made  the 
experiment,  and  saw  the  circle,  a  result  which 
had  never  been  even  suspected.  This  has 
always  been  considered  one  of  the  most  signal 
instances  of  scientific  thought  in  the  domain  of 
physics.  It  is  most  distinctly  an  application  of 
experience  gained  under  certain  circumstances 
to  entirely  different  circumstances. 

Now  suppose  that  the  night  before  coming 
down  to  Brighton  you  had  dreamed  of  a 
railway  accident  caused  by  the  engine  getting 
frightened  at  a  flock  of  sheep  and  jumping 
suddenly  back  over  all  the  carriages  ;  the  result 
of  which  was  that  your  head  was  unfortunately 
cut  off,  so  that  you  had  to  put  it  in  your  hat- 
box  and  take  it  back  home  to  be  mended. 
There  are,  I  fear,  many  persons  even  at  this 
day,  who  would  tell  you  that  after  such  a  dream 
it  was  unwise  to  travel  by  railway  to  Brighton. 
This  is  a  proposal  that  you  should  take  experi- 
ence gained  while  you  are  asleep,  when  you 
have  no  common  sense, — experience  about  a 
VOL.  I  L 


i46  LECTURES  AND  ESSAYS 

phantom  railway,  and  apply  it  to  guide  you 
when  you  are  awake  and  have  common  sense, 
in  your  dealings  with  a  real  railway.  And  yet 
this  proposal  is  not  dictated  by  scientific  thought. 
Now  let  us  take  the  great  example  of  Bio- 
logy. I  pass  over  the  process  of  classification, 
which  itself  requires  a  great  deal  of  scientific 
thought ;  in  particular  when  a  naturalist  who 
has  studied  and  monographed  a  fauna  or  a 
flora  rather  than  a  family  is  able  at  once  to 
pick  out  the  distinguishing  characters  required 
for  the  subdivision  of  an  order  quite  new  to 
him.  Suppose  that  we  possess  all  this  minute 
and  comprehensive  knowledge  of  plants  and 
animals  and  intermediate  organisms,  their 
affinities  and  differences,  their  structures  and 
functions  ; — a  vast  body  of  experience,  collected 
by  incalculable  labour  and  devotion.  Then 
comes  Mr.  Herbert  Spencer  :  he  takes  that  ex- 
perience of  life  which  is  not  human,  which  is 
apparently  stationary,  going  on  in  exactly  the 
same  way  from  year  to  year,  and  he  applies 
that  to  tell  us  how  to  deal  with  the  changing 
characters  of  human  nature  and  human  society. 
How  is  it  that  experience  of  this  sort,  vast  as 
it  is,  can  guide  us  in  a  matter  so  different  from 
itself?  How  does  scientific  thought,  applied 
to  the  development  of  a  kangaroo  foetus  or  the 
movement  of  the  sap  in  exogens,  make  predic- 
tion possible  for  the  first  time  in  that  most 


AIMS  OF  SCIENTIFIC  THOUGHT  147 

important  of  all  sciences,  the  relations  of  man 
with  man  ? 

In  the  dark  or  unscientific  ages  men  had 
another  way  of  applying  experience  to  altered 
circumstances.  They  believed,  for  example, 
that  the  plant  called  Jew's-ear,  which  does  bear 
a  certain  resemblance  to  the  human  ear,  was  a 
useful  cure  for  diseases  of  that  organ.  This 
doctrine  of  "  signatures,"  as  it  was  called,  exer- 
cised an  enormous  influence  on  the  medicine  of 
the  time.  I  need  hardly  tell  you  that  it  is 
hopelessly  unscientific  ;  yet  it  agrees  with  those 
other  examples  that  we  have  been  considering 
in  this  particular  ;  that  it  applies  experience 
about  the  shape  of  a  plant — which  is  one  cir- 
cumstance connected  with  it — to  dealings  with 
its  medicinal  properties,  which  are  other  and 
different  circumstances.  Again,  suppose  that 
you  had  been  frightened  by  a  thunder-storm 
on  land,  or  your  heart  had  failed  you  in  a  storm 
at  sea ;  if  any  one  then  told  you  that  in 
consequence  of  this  you  should  always  cultivate 
an  unpleasant  sensation  in  the  pit  of  your 
stomach,  till  you  took  delight  in  it,  that  you 
should  regulate  your  sane  and  sober  life  by  the 
sensations  of  a  moment  of  unreasoning  terror : 
this  advice  would  not  be  an  example  of  scientific 
thought.  Yet  it  would  be  an  application  of 
past  experience  to  new  and  different  circum- 
stances. 


148  LECTURES  AND  ESSAYS 

But  you  will  already  have  observed  what  is 
the  additional  clause  that  we  must  add  to  our 
definition  in  order  to  describe  scientific  thought 
and  that  only.  The  step  between  experience 
about  animals  and  dealings  with  changing 
humanity  is  the  law  of  evolution.  The  step 
from  errors  in  the  calculated  places  of  Uranus 
to  the  existence  of  Neptune  is  the  law  of 
gravitation.  The  step  from  the  observed  be- 
haviour of  crystals  to  conical  refraction  is 
made  up  of  laws  of  light  and  geometry.  The 
step  from  old  bridges  to  new  ones  is  the  laws 
of  elasticity  and  the  strength  of  materials. 

The  step,  then,  from  past  experience  to  new 
circumstances  must  be  made  in  accordance  with 
an  observed  uniformity  in  the  order  of  events. 
This  uniformity  has  held  good  in  the  past  in 
certain  places  ;  if  it  should  also  hold  good  in 
the  future  and  in  other  places,  then,  being  com- 
bined with  our  experience  of  the  past,  it  enables 
us  to  predict  the  future,  and  to  know  what  is 
going  on  elsewhere ;  so  that  we  are  able  to 
regulate  our  conduct  in  accordance  with  this 
knowledge. 

The  aim  of  scientific  thought,  then,  is  to 
apply  past  experience  to  new  circumstances  ; 
the  instrument  is  an  observed  uniformity  in  the 
course  of  events.  By  the  use  of  this  instru- 
ment it  gives  us  information  transcending  our 
experience,  it  enables  us  to  infer  things  that  we 


AIMS  OF  SCIENTIFIC  THOUGHT  149 

have  not  seen  from  things  that  we  have  seen  ; 
and  the  evidence  for  the  truth  of  that  infor- 
mation depends  on  our  supposing  that  the 
uniformity  holds  good  beyond  our  experience. 
I  now  want  to  consider  this  uniformity  a  little 
more  closely  ;  to  show  how  the  character  of 
scientific  thought  and  the  force  of  its  inferences 
depend  upon  the  character  of  the  uniformity  of 
Nature.  I  cannot  of  course  tell  you  all  that  is 
known  of  this  character  without  writing  an 
encyclopaedia ;  but  I  shall  confine  myself  to 
two  points  of  it  about  which  it  seems  to  me 
that  just  now  there  is  something  to  be  said.  I 
want  to  find  out  what  we  mean  when  we  say 
that  the  uniformity  of  Nature  is  exact ;  and 
what  we  mean  when  we  say  that  it  is  reasonable. 
When  a  student  is  first  introduced  to  those 
sciences  which  have  come  under  the  dominion 
of  mathematics,  a  new  and  wonderful  aspect  of 
Nature  bursts  upon  his  view.  He  has  been 
accustomed  to  regard  things  as  essentially  more 
or  less  vague.  All  the  facts  that  he  has  hitherto 
known  have  been  expressed  qualitatively,  with 
a  little  allowance  for  error  on  either  side.  Things 
which  are  let  go  fall  to  the  ground.  A  very 
observant  man  may  know  also  that  they  fall 
faster  as  they  go  along.  But  our  student  is 
shown  that,  after  falling  for  one  second  in  a 
vacuum,  a  body  is  going  at  the  rate  of  thirty- 
two  feet  per  second,  that  after  falling  for  two 


i$o  LECTURES  AND  ESSAYS 

seconds  it  is  going  twice  as  fast,  after  going 
two  and  a  half  seconds  two  and  a  half  times  as 
fast  If  he  makes  the  experiment,  and  finds  a 
single  inch  per  second  too  much  or  too  little  in 
the  rate,  one  of  two  things  must  have  happened  : 
either  the  law  of  falling  bodies  has  been  wrongly 
stated,  or  the  experiment  is  not  accurate — there 
is  some  mistake.  He  finds  reason  to  think 
that  the  latter  is  always  the  case  ;  the  more 
carefully  he  goes  to  work,  the  more  of  the  error 
turns  out  to  belong  to  the  experiment.  Again, 
he  may  know  that  water  consists  of  two  gases, 
oxygen  and  hydrogen,  combined  ;  but  he  now 
learns  that  two  pints  of  steam  at  a  temperature 
of  150°  Centigrade  will  always  make  two  pints 
of  hydrogen  and  one  pint  of  oxygen  at  the 
same  temperature,  all  of  them  being  pressed  as 
much  as  the  atmosphere  is  pressed.  If  he 
makes  the  experiment  and  gets  rather  more  or 
less  than  a  pint  of  oxygen,  is  the  law  disproved  ? 
No  ;  the  steam  was  impure,  or  there  was  some 
mistake.  Myriads  of  analyses  attest  the  law  of 
combining  volumes  ;  the  more  carefully  they 
are  made,  the  more  nearly  they  coincide  with 
it  The  aspects  of  the  faces  of  a  crystal  are 
connected  together  by  a  geometrical  law,  by 
which,  four  of  them  being  given,  the  rest  can 
be  found.  The  place  of  a  planet  at  a  given 
time  is  calculated  by  the  law  of  gravitation  ;  if 
it  is  half  a  second  wrong,  the  fault  is  in  the 


AIMS  OF  SCIENTIFIC  THOUGHT  151 

instrument,  the  observer,  the  clock,  or  the  law  ; 
now,  the  more  observations  are  made,  the  more 
of  this  fault  is  brought  home  to  the  instrument, 
the  observer,  and  the  clock.  It  is  no  wonder, 
then,  that  our  student,  contemplating  these  and 
many  like  instances,  should  be  led  to  say,  "  I 
have  been  short-sighted  ;  but  I  have  now  put 
on  the  spectacles  of  science  which  Nature  had 
prepared  for  my  eyes  ;  I  see  that  things  have 
definite  outlines,  that  the  world  is  ruled  by 
exact  and  rigid  mathematical  laws ;  xal  <rv, 
0eo9,  7ecoytteTpe4<?."  It  is  our  business  to  con- 
sider whether  he  is  right  in  so  concluding.  Is 
the  uniformity  of  Nature  absolutely  exact,  or 
only  more  exact  than  our  experiments  ? 

At  this  point  we  have  to  make  a  very  im- 
portant distinction.  There  are  two  ways  in 
which  a  law  may  be  inaccurate.  The  first  way 
is  exemplified  by  that  law  of  Galileo  which  I 
mentioned  just  now :  that  a  body  falling  in 
vacuo  acquires  equal  increase  in  velocity  in 
equal  times.  No  matter  how  many  feet  per 
second  it  is  going,  after  an  interval  of  a  second 
it  will  be  going  thirty-two  more  feet  per  second. 
We  now  know  that  this  rate  of  increase  is  not 
exactly  the  same  at  different  heights,  that  it 
depends  upon  the  distance  of  the  body  from 
the  centre  of  the  earth  ;  so  that  the  law  is  only 
approximate  ;  instead  of  the  increase  of  velocity 
being  exactly  equal  in  equal  times,  it  itself 


152  LECTURES  AND  ESSAYS 

increases  very  slowly  as  the  body  falls.  We 
know  also  that  this  variation  of  the  law  from 
the  truth  is  too  small  to  be  perceived  by  direct 
observation  on  the  change  of  velocity.  But 
suppose  we  have  invented  means  for  observing 
this,  and  have  verified  that  the  increase  of 
velocity  is  inversely  as  the  squared  distance 
from  the  earth's  centre.  Still  the  law  is  not 
accurate  ;  for  the  earth  does  not  attract  ac- 
curately towards  her  centre,  and  the  direction 
of  attraction  is  continually  varying  with  the 
motion  of  the  sea  ;  the  body  will  not  even  fall 
in  a  straight  line.  The  sun  and  the  planets, 
too,  especially  the  moon,  will  produce  deviations ; 
yet  the  sum  of  all  these  errors  will  escape  our 
new  process  of  observation,  by  being  a  great 
deal  smaller  than  the  necessary  errors  of  that 
observation.  But  when  these  again  have  been 
allowed  for,  there  is  still  the  influence  of  the 
stars.  In  this  case,  however,  we  only  give  up 
one  exact  law  for  another.  It  may  still  be 
held  that  if  the  effect  of  every  particle  of  matter 
in  the  universe  on  the  falling  body  were  calcu- 
lated according  to  the  law  of  gravitation,  the 
body  would  move  exactly  as  this  calculation 
required.  And  if  it  were  objected  that  the 
body  must  be  slightly  magnetic  or  diamagnetic, 
while  there  are  magnets  not  an  infinite  way  off; 
that  a  very  minute  repulsion,  even  at  sensible 
distances,  accompanies  the  attraction  ;  it  might 


AIMS  OF  SCIENTIFIC  THOUGHT  153 

be  replied  that  these  phenomena  are  themselves 
subject  to  exact  laws,  and  that  when  all  the 
laws  have  been  taken  into  account,  the  actual 
motion  will  exactly  correspond  with  the  calcu- 
lated motion. 

I  suppose  there  is  hardly  a  physical  student 
(unless  he  has  specially  considered  the  matter) 
who  would  not  at  once  assent  to  the  statement 
I  have  just  made  ;  that  if  we  knew  all  about  it, 
Nature  would  be  found  universally  subject  to 
exact  numerical  laws.  But  let  us  just  consider 
for  another  moment  what  this  means. 

The  word  "  exact "  has  a  practical  and  a 
theoretical  meaning.  When  a  grocer  weighs 
you  out  a  certain  quantity  of  sugar  very  care- 
fully, and  says  it  is  exactly  a  pound,  he  means 
that  the  difference  between  the  mass  of  the 
sugar  and  that  of  the  pound  weight  he  employs 
is  too  small  to  be  detected  by  his  scales.  If  a 
chemist  had  made  a  special  investigation,  wish- 
ing to  be  as  accurate  as  he  could,  and  told  you 
this  was  exactly  a  pound  of  sugar,  he  would 
mean  that  the  mass  of  the  sugar  differed  from 
that  of  a  certain  standard  piece  of  platinum  by 
a  quantity  too  small  to  be  detected  by  his 
means  of  weighing,  which  are  a  thousandfold 
more  accurate  than  the  grocer's.  But  what 
would  a  mathematician  mean,  if  he  made  the 
same  statement  ?  He  would  mean  this. 
Suppose  the  mass  of  the  standard  pound  to  be 


154  LECTURES  AND  ESSAYS 

represented  by  a  length,  say  a  foot,  measured 
on  a  certain  line ;  so  that  half  a  pound  would 
be  represented  by  six  inches,  and  so  on.  And 
let  the  difference  between  the  mass  of  the  sugar 
and  that  of  the  standard  pound  be  drawn  upon 
the  same  line  to  the  same  scale.  Then,  if  that 
difference  were  magnified  an  infinite  number  of 
times,  it  would  still  be  invisible.  This  is  the 
theoretical  meaning  of  exactness  ;  the  practical 
meaning  is  only  very  close  approximation  ; 
how  close,  depends  upon  the  circumstances. 
The  knowledge  then  of  an  exact  law  in  the 
theoretical  sense  would  be  equivalent  to  an 
infinite  observation.  I  do  not  say  that  such 
knowledge  is  impossible  to  man  ;  but  I  do  say 
that  it  would  be  absolutely  different  in  kind 
from  any  knowledge  that  we  possess  at  present 
I  shall  be  told,  no  doubt,  that  we  do  possess 
a  great  deal  of  knowledge  of  this  kind,  in  the 
form  of  geometry  and  mechanics  ;  and  that  it 
is  just  the  example  of  these  sciences  that  has 
led  men  to  look  for  exactness  in  other  quarters. 
If  this  had  been  said  to  me  in  the  last  century, 
I  should  not  have  known  what  to  reply.  But 
it  happens  that  about  the  beginning  of  the 
present  century  the  foundations  of  geometry 
were  criticised  independently  by  two  mathe- 
maticians, Lobatschewsky l  and  the  immortal 

1  Geometrische  Untersuchvngen  sur  Theorie  der  Parallellinien 
Berlin.  1840.     Translated  by  Hottel.     Gauthier-Villars,  1866. 


AIMS  OF  SCIENTIFIC  THOUGHT  155 

Gauss  ; l  whose  results  have  been  extended  and 
generalised  more  recently  by  Riemann 2  and 
Helmholtz.3  And  the  conclusion  to  which 
these  investigations  lead  is  that,  although  the 
assumptions  which  were  very  properly  made  by 
the  ancient  geometers  are  practically  exact — 
that  is  to  say,  more  exact  than  experiment 
can  be — for  such  finite  things  as  we  have  to 
deal  with,  and  such  portions  of  space  as  we  can 
reach  ;  yet  the  truth  of  them  for  very  much 
larger  things,  or  very  much  smaller  things,  or 
parts  of  space  which  are  at  present  beyond  our 
reach,  is  a  matter  to  be  decided  by  experiment, 
when  its  powers  are  considerably  increased.  I 
want  to  make  as  clear  as  possible  the  real  state 
of  this  question  at  present,  because  it  is  often 
supposed  to  be  a  question  of  words  or  meta- 
physics, whereas  it  is  a  very  distinct  and  simple 
question  of  fact.  I  am  supposed  to  know  then 
that  the  three  angles  of  a  rectilinear  triangle 
are  exactly  equal  to  two  right  angles.  Now 
suppose  that  three  points  are  taken  in  space, 
distant  from  one  another  as  far  as  the  Sun  is 
from  a  Centauri,  and  that  the  shortest  distances 
between  these  points  are  drawn  so  as  to  form 
a  triangle.  And  suppose  the  angles  of  this 

1  Letter  to  Schumacher,  Nov.  28,  1846  (refers  to  1792). 

2  Ueber  die  Hypothesen  -wekhe  der  Geometrie  zu  Grunde  liegen. 
Gottingen,  Abhandl.,    1866-67.     Translated  by  Hotiel  in  Annali 
di  Matematica,  Milan,  vol.  iii. 

3  The  Axioms  of  Geometry,  Academy,  vol.  i.  p.  128  (a  popular 
exposition).     [And  see  now  his  article  in  Mind,  No.  III.] 


IJ6  LECTURES  AND  ESSAYS 

triangle  to  be  very  accurately  measured  and 
added  together ;  this  can  at  present  be  done  so 
accurately  that  the  error  shall  certainly  be  less 
than  one  minute,  less  therefore  than  the  five- 
thousandth  part  of  a  right  angle.  Then  I  do 
not  know  that  this  sum  would  differ  at  all  from 
two  right  angles  ;  but  also  I  do  not  know  that 
the  difference  would  be  less  than  ten  degrees, 
or  the  ninth  part  of  a  right  angle.1  And  I  have 
reasons  for  not  knowing. 

This  example  is  exceedingly  important  as 
showing  the  connection  between  exactness  and 
universality.  It  is  found  that  the  deviation  if 
it  exists  must  be  nearly  proportional  to  the 
area  of  the  triangle.  So  that  the  error  in  the 
case  of  a  triangle  whose  sides  are  a  mile  long 
would  be  obtained  by  dividing  that  in  the  case 
I  have  just  been  considering  by  four  hundred 
quadrillions  ;  the  result  must  be  a  quantity  in- 
conceivably small,  which  no  experiment  could 
detect.  But  between  this  inconceivably  small 
error  and  no  error  at  all,  there  is  fixed  an 
enormous  gulf  ;  the  gulf  between  practical  and 
theoretical  exactness,  and,  what  is  even  more 
important,  the  gulf  between  what  is  practically 
universal  and  what  is  theoretically  universal. 
I  say  that  a  law  is  practically  universal  which 

1  Assuming  that  parallax  observations  prove  the  deviation  less 
than  half  a  second  for  a  triangle  whose  vertex  is  at  the  star  and  base 
a  diameter  of  the  earth's  orbit 


AIMS  OF  SCIENTIFIC  THOUGHT  157 

is  more  exact  than  experiment  for  all  cases 
that  might  be  got  at  by  such  experiments  as 
we  can  make.  We  assume  this  kind  of  univer- 
sality, and  we  find  that  it  pays  us  to  assume 
it.  But  a  law  would  be  theoretically  universal 
if  it  were  true  of  all  cases  whatever  ;  and  this 
is  what  we  do  not  know  of  any  law  at  all. 

I  said  there  were  two  ways  in  which  a  law 
might  be  inexact.  There  is  a  law  of  gases 
which  asserts  that  when  you  compress  a  perfect 
gas  the  pressure  of  the  gas  increases  exactly  in 
the  proportion  in  which  the  volume  diminishes. 
Exactly  ;  that  is  to  say,  the  law  is  more  accurate 
than  the  experiment,  and  experiments  are 
corrected  by  means  of  the  law.  But  it  so 
happens  that  this  law  has  been  explained ;  we 
know  precisely  what  it  is  that  happens  when  a 
gas  is  compressed.  We  know  that  a  gas 
consists  of  a  vast  number  of  separate  molecules, 
rushing  about  in  all  directions  with  all  manner 
of  velocities,  but  so  that  the  mean  velocity  of 
the  molecules  of  air  in  this  room,  for  example, 
is  about  twenty  miles  a  minute.  The  pressure 
of  the  gas  on  any  surface  with  which  it  is  in 
contact  is  nothing  more  than  the  impact  of 
these  small  particles  upon  it.  On  any  surface 
large  enough  to  be  seen  there  are  millions  of 
these  impacts  in  a  second.  If  the  space  in 
which  the  gas  is  confined  be  diminished,  the 
average  rate  at  which  the  impacts  take  place 


158  LECTURES  AND  ESSAYS 

will  be  increased  in  the  same  proportion  ;  and 
because  of  the  enormous  number  of  them,  the 
actual  rate  is  always  exceedingly  close  to  the 
average.  But  the  law  is  one  of  statistics  ;  its 
accuracy  depends  on  the  enormous  numbers 
involved  ;  and  so,  from  the  nature  of  the  case, 
its  exactness  cannot  be  theoretical  or  absolute. : 

Nearly  all  the  laws  of  gases  have  received 
these  statistical  explanations  ;  electric  and 
magnetic  attraction  and  repulsion  have  been 
treated  in  a  similar  manner  ;  and  an  hypothesis 
of  this  sort  has  been  suggested  even  for  the  law 
of  gravity.  On  the  other  hand  the  manner  in 
which  the  molecules  of  a  gas  interfere  with  each 
other  proves  that  they  repel  one  another  in- 
versely as  the  fifth  power  of  the  distance  ; x  so 
that  we  here  find  at  the  basis  of  a  statistical 
explanation  a  law  which  has  the  form  of 
theoretical  exactness.  Which  of  these  forms  is 
to  win  ?  It  seems  to  me  again  that  we  do  not 
know,  and  that  the  recognition  of  our  ignorance 
is  the  surest  way  to  get  rid  of  it. 

The  world  in  general  has  made  just  the 
remark  that  I  have  attributed  to  a  fresh  student 
of  the  applied  sciences.  As  the  discoveries  of 
Galileo,  Kepler,  Newton,  Dalton,  Cavendish, 
Gauss,  displayed  ever  new  phenomena  following 
mathematical  laws,  the  theoretical  exactness  of 

1  [This  statement  of  the  law  has  since  been  abandoned  :  see 
"The  Unseen  Universe,"  below.] 


AIMS  OF  SCIENTIFIC  THOUGHT  159 

the  physical  universe  was  taken  for  granted. 
Now,  when  people  are  hopelessly  ignorant  of  a 
thing,  they  quarrel  about  the  source  of  their 
knowledge.  Accordingly  many  maintained 
that  we  know  these  exact  laws  by  intuition. 
These  said  always  one  true  thing,  that  we  did 
not  know  them  from  experience.  Others  said 
that  they  were  really  given  in  the  facts,  and 
adopted  ingenious  ways  of  hiding  the  gulf 
between  the  two.  Others  again  deduced  from 
transcendental  considerations  sometimes  the  laws 
themselves,  and  sometimes  what  through  im- 
perfect information  they  supposed  to  be  the 
laws.  But  more  serious  consequences  arose 
when  these  conceptions  derived  from  Physics 
were  carried  over  into  the  field  of  Biology. 
Sharp  lines  of  division  were  made  between 
kingdoms  and  classes  and  orders  ;  an  animal 
was  described  as  a  miracle  to  the  vegetable 
world  ;  specific  differences  which  are  practically 
permanent  within  the  range  of  history  were 
regarded  as  permanent  through  all  time ;  a 
sharp  line  was  drawn  between  organic  and 
inorganic  matter.  Further  investigation,  how- 
ever, has  shown  that  accuracy  had  been  pre- 
maturely attributed  to  the  science,  and  has  filled 
up  all  the  gulfs  and  gaps  that  hasty  observers 
had  invented.  The  animal  and  vegetable 
kingdoms  have  a  debateable  ground  between 
them,  occupied  by  beings  that  have  the  char- 


160  LECTURES  AND  ESSAYS 

acters  of  both  and  yet  belong  distinctly  to 
neither.  Classes  and  orders  shade  into  one 
another  all  along  their  common  boundary. 
Specific  differences  turn  out  to  be  the  work  of 
time.  The  line  dividing  organic  matter  from 
inorganic,  if  drawn  to-day,  must  be  moved 
to-morrow  to  another  place  ;  and  the  chemist 
will  tell  you  that  the  distinction  has  now  no 
place  in  his  science  except  in  a  technical  sense 
for  the  convenience  of  studying  carbon  com- 
pounds by  themselves.  In  Geology  the  same 
tendency  gave  birth  to  the  doctrine  of  distinct 
periods,  marked  out  by  the  character  of  the 
strata  deposited  in  them  all  over  the  sea ; 
a  doctrine  than  which,  perhaps,  no  ancient 
cosmogony  has  been  further  from  the  truth, 
or  done  more  harm  to  the  progress  of 
science.  Refuted  many  years  ago  by  Mr. 
Herbert  Spencer,1  it  has  now  fairly  yielded  to 
an  attack  from  all  sides  at  once,  and  may  be 
left  in  peace. 

When  then  we  say  that  the  uniformity  which 
we  observe  in  the  course  of  events  is  exact  and 
universal,  we  mean  no  more  than  this  :  that  we 
are  able  to  state  general  rules  which  are  far 
more  exact  than  direct  experiment,  and  which 
apply  to  all  cases  that  we  are  at  present  likely 
to  come  across.  It  is  important  to  notice, 

1  "Illogical  Geology,"  in  Essays,  vol.  i.     Originally  published 
in  1859. 


AIMS  OF  SCIENTIFIC  THOUGHT  161 

however,  the  effect  of  such  exactness  as  we 
observe  upon  the  nature  of  inference.  When  a 
telegram  arrived  stating  that  Dr.  Livingstone 
had  been  found  by  Mr.  Stanley,  what  was  the 
process  by  which  you  inferred  the  finding  of 
Dr.  Livingstone  from  the  appearance  of  the 
telegram  ?  You  assumed  over  and  over  again 
the  existence  of  uniformity  in  nature.  That 
the  newspapers  had  behaved  as  they  generally 
do  in  regard  to  telegraphic  messages  ;  that  the 
clerks  had  followed  the  known  laws  of  the 
action  of  clerks  ;  that  electricity  had  behaved 
in  the  cable  exactly  as  it  behaves  in  the 
laboratory ;  that  the  actions  of  Mr.  Stanley 
were  related  to  his  motives  by  the  same 
uniformities  that  affect  the  actions  of  other 
men  ;  that  Dr.  Livingstone's  handwriting  con- 
formed to  the  curious  rule  by  which  an  ordinary 
man's  handwriting  may  be  recognised  as  having 
persistent  characteristics  even  at  different  periods 
of  his  life.  But  you  had  a  right  to  be  much 
more  sure  about  some  of  these  inferences  than 
about  others.  The  law  of  electricity  was  known 
with  practical  exactness,  and  the  conclusions 
derived  from  it  were  the  surest  things  of  all. 
The  law  about  the  handwriting,  belonging  to 
a  portion  of  physiology  which  is  unconnected 
with  consciousness,  was  known  with  less,  but 
still  with  considerable  accuracy.  But  the  laws 
of  human  action  in  which  consciousness  is 
VOL.  I  M 


162  LECTURES  AND  ESSAYS 

concerned  are  still  so  far  from  being  completely 
analysed  and  reduced  to  an  exact  form  that 
the  inferences  which  you  made  by  their  help 
were  felt  to  have  only  a  provisional  force.  It 
is  possible  that  by  and  by,  when  psychology 
has  made  enormous  advances  and  become  an 
exact  science,  we  may  be  able  to  give  to 
testimony  the  sort  of  weight  which  we  give  to 
the  inferences  of  physical  science.  It  will  then 
be  possible  to  conceive  a  case  which  will  show 
how  completely  the  whole  process  of  inference 
depends  on  our  assumption  of  uniformity. 
Suppose  that  testimony,  having  reached  the 
ideal  force  I  have  imagined,  were  to  assert  that 
a  certain  river  runs  uphill.  You  could  infer 
nothing  at  all.  The  arm  of  inference  would  be 
paralysed,  and  the  sword  of  truth  broken  in  its 
grasp  ;  and  reason  could  only  sit  down  and 
wait  until  recovery  restored  her  limb,  and 
further  experience  gave  her  new  weapons. 

I  want  in  the  next  place  to  consider  what 
we  mean  when  we  say  that  the  uniformity 
which  we  have  observed  in  the  course  of  events 
is  reasonable  as  well  as  exact. 

No  doubt  the  first  form  of  this  idea  was  sug- 
gested by  the  marvellous  adaptation  of  certain 
natural  structures  to  special  functions.  The 
first  impression  of  those  who  studied  comparative 
anatomy  was  that  every  part  of  the  animal 
frame  was  fitted  with  extraordinary  complete- 


AIMS  OF  SCIENTIFIC  THOUGHT  163 

ness  for  the  work  that  it  had  to  do.  I  say 
extraordinary,  because  at  the  time  the  most 
familiar  examples  of  this  adaptation  were 
manufactures  produced  by  human  ingenuity  ; 
and  the  completeness  and  minuteness  of  natural 
adaptations  were  seen  to  be  far  in  advance  of 
these.  The  mechanism  of  limbs  and  joints 
was  seen  to  be  adapted,  far  better  than  any 
existing  ironwork,  to  those  motions  and  com- 
binations of  motion  which  were  most  useful  to 
the  particular  organisms.  The  beautiful  and 
complicated  apparatus  of  sensation  caught  up 
indications  from  the  surrounding  medium, 
sorted  them,  analysed  them,  and  transmitted 
the  results  to  the  brain  in  a  manner  with  which, 
at  the  time  I  am  speaking  of,  no  artificial 
contrivance  could  compete.  Hence  the  belief 
grew  amongst  physiologists  that  every  structure 
which  they  found  must  have  its  function  and 
subserve  some  useful  purpose  ;  a  belief  which 
was  not  without  its  foundation  in  fact,  and 
which  certainly  (as  Dr.  Whewell  remarks)  has 
done  admirable  service  in  promoting  the  growth 
of  physiology.  Like  all  beliefs  found  successful 
in  one  subject,  it  was  carried  over  into  another, 
of  which  a  notable  example  is  given  in  the 
speculations  of  Count  Rumford  about  the 
physical  properties  of  water.  Pure  water  attains 
its  greatest  density  at  a  temperature  of  about 
39^°  Fahrenheit ;  it  expands  and  becomes 


1 64  LECTURES  AND  ESSAYS 

lighter  whether  it  is  cooled  or  heated,  so  as  to 
alter  that  temperature.  Hence  it  was  concluded 
that  water  in  this  state  must  be  at  the  bottom 
of  the  sea,  and  that  by  such  means  the  sea  was 
kept  from  freezing  all  through ;  as  it  was 
supposed  must  happen  if  the  greatest  density 
had  been  that  of  ice.  Here  then  was  a  sub- 
stance whose  properties  were  eminently  adapted 
to  secure  an  end  essential  to  the  maintenance 
of  life  upon  the  earth.  In  short,  men  came  to 
the  conclusion  that  the  order  of  nature  was 
reasonable  in  the  sense  that  everything  was 
adapted  to  some  good  end. 

Further  consideration,  however,  has  led  men 
out  of  that  conclusion  in  two  different  ways. 
First,  it  was  seen  that  the  facts  of  the  case  had 
been  wrongly  stated.  Cases  were  found  of 
wonderfully  complicated  structures  that  served 
no  purpose  at  all ;  like  the  teeth  of  that  whale 
of  which  you  heard  in  Section  D  the  other 
day,  or  of  the  Dugong,  which  has  a  horny 
palate  covering  them  all  up  and  used  instead  of 
them  ;  like  the  eyes  of  the  unborn  mole,  that 
are  never  used,  though  perfect  as  those  of  a 
mouse  until  the  skull  opening  closes  up,  cutting 
them  off  from  the  brain,  when  they  dry  up  and 
become  incapable  of  use  ;  like  the  outsides  of 
your  own  ears,  which  are  absolutely  of  no  use 
to  you.  And  when  human  contrivances  were 
more  advanced  it  became  clear  that  the  natural 


AIMS  OF  SCIENTIFIC  THOUGHT  165 

adaptations  were  subject  to  criticism.  The  eye, 
regarded  as  an  optical  instrument  of  human 
manufacture,  was  thus  described  by  Helmholtz 
— the  physiologist  who  learned  physics  for  the 
sake  of  his  physiology,  and  mathematics  for  the 
sake  of  his  physics,  and  is  now  in  the  first 
rank  of  all  three.  He  said,  "If  an  optician 
sent  me  that  as  an  instrument,  I  should  send  it 
back  to  him  with  grave  reproaches  for  the  care- 
lessness of  his  work,  and  demand  the  return  of 
my  money." 

The  extensions  of  the  doctrine  into  Physics 
were  found  to  be  still  more  at  fault  That 
remarkable  property  of  pure  water,  which  was 
to  have  kept  the  sea  from  freezing,  does  not 
belong  to  salt  water,  of  which  the  sea  itself  is 
composed.  It  was  found,  in  fact,  that  the  idea 
of  a  reasonable  adaptation  of  means  to  ends, 
useful  as  it  had  been  in  its  proper  sphere,  could 
yet  not  be  called  universal,  or  applied  to  the 
order  of  nature  as  a  whole. 

Secondly,  this  idea  has  given  way  because  it 
has  been  superseded  by  a  higher  and  more 
general  idea  of  what  is  reasonable,  which  has 
the  advantage  of  being  applicable  to  a  large 
portion  of  physical  phenomena  besides.  Both 
the  adaptation  and  the  non-adaptation  which 
occur  in  organic  structures  have  been  explained, 
The  scientific  thought  of  Dr.  Darwin,  of  Mr. 
Herbert  Spencer,  and  of  Mr.  Wallace,  has 


166  LECTURES  AND  ESSAYS 

described  that  hitherto  unknown  process  of 
adaptation  as  consisting  of  perfectly  well-known 
and  familiar  processes.  There  are  two  kinds  of 
these  :  the  direct  processes,  in  which  the  physical 
changes  required  to  produce  a  structure  are 
worked  out  by  the  very  actions  for  which  that 
structure  becomes  adapted — as  the  backbone  or 
notochord  has  been  modified  from  generation 
to  generation  by  the  bendings  which  it  has 
undergone  ;  and  the  indirect  processes  included 
under  the  head  of  Natural  Selection — the 
reproduction  of  children  slightly  different  from 
their  parents,  and  the  survival  of  those  which  are 
best  fitted  to  hold  their  own  in  the  struggle  for 
existence.  Naturalists  might  give  you  some 
idea  of  the  rate  at  which  we  are  getting  ex- 
planations of  the  evolution  of  all  parts  of  animals 
and  plants — the  growth  of  the  skeleton,  of  the 
nervous  system  and  its  mind,  of  leaf  and  flower. 
But  what  then  do  we  mean  by  explanation  ? 

We  were  considering  just  now  an  explana- 
tion of  a  law  of  gases — the  law  according  to 
which  pressure  increases  in  the  same  proportion 
in  which  volume  diminishes.  The  explanation 
consisted  in  supposing  that  a  gas  is  made  up  of 
a  vast  number  of  minute  particles  always  flying 
about  and  striking  against  one  another,  and 
then  showing  that  the  rate  of  impact  of  such  a 
crowd  of  particles  on  the  sides  of  the  vessel 
containing  them  would  vary  exactly  as  the 


AIMS  OF  SCIENTIFIC  THOUGHT  167 

pressure  is  found  to  vary.  Suppose  the  vessel 
to  have  parallel  sides,  and  that  there  is  only  one 
particle  rushing  backwards  and  forwards  between 
them  ;  then  it  is  clear  that  if  we  bring  the  sides 
together  to  half  the  distance,  the  particle  will 
hit  each  of  them  twice  as  often,  or  the  pressure 
will  be  doubled.  Now  it  turns  out  that  this 
would  be  just  as  true  for  millions  of  particles  as 
for  one,  and  when  they  are  flying  in  all  directions 
instead  of  only  in  one  direction  and  its  opposite. 
Observe  now  ;  it  is  a  perfectly  well-known  and 
familiar  thing  that  a  body  should  strike  against 
an  opposing  surface  and  bound  off  again  ;  and 
it  is  a  mere  everyday  occurrence  that  what  has 
only  half  so  far  to  go  should  be  back  in  half  the 
time  ;  but  that  pressure  should  be  strictly  pro- 
portional to  density  is  a  comparatively  strange, 
unfamiliar  phenomenon.  The  explanation  de- 
scribes the  unknown  and  unfamiliar  as  being 
made  up  of  the  known  and  the  familiar  ;  and 
this,  it  seems  to  me,  is  the  true  meaning  of 
explanation.1 

Here  is  another  instance.  If  small  pieces 
of  camphor  are  dropped  into  water,  they  will 
begin  to  spin  round  and  swim  about  in  a  most 
marvellous  way.  Mr.  Tomlinson  gave,  I  believe, 

1  This  view  differs  from  those  of  Mr.  J.  S.  Mill  and  Mr.  Herbert 
Spencer  in  requiring  every  explanation  to  contain  an  addition 
to  our  knowledge  about  the  thing  explained.  Both  these  writers 
regard  subsumption  under  a  general  law  as  a  species  of  explana- 
tion. See  also  Ferrier's  Remains,  vol.  ii.  p.  436. 


168  LECTURES  AND  ESSAYS 

the  explanation  of  this.  We  must  observe,  to 
begin  with,  that  every  liquid  has  a  skin  which 
holds  it ;  you  can  see  that  to  be  true  in  the  case 
of  a  drop,  which  looks  as  if  it  were  held  in  a 
bag.  But  the  tension  of  this  skin  is  greater  in 
some  liquids  than  in  others  ;  and  it  is  greater 
in  camphor  and  water  than  in  pure  water. 
When  the  camphor  is  dropped  into  water  it 
begins  to  dissolve  and  get  surrounded  with 
camphor  and  water  instead  of  water.  If  the 
fragment  of  camphor  were  exactly  symmetrical, 
nothing  more  would  happen  ;  the  tension  would 
be  greater  in  its  immediate  neighbourhood, 
but  no  motion  would  follow.  The  camphor, 
however,  is  irregular  in  shape  ;  it  dissolves 
more  on  one  side  than  the  other ;  and  con- 
sequently gets  pulled  about,  because  the  tension 
of  the  skin  is  greater  where  the  camphor  is  most 
dissolved.  Now  it  is  probable  that  this  is  not 
nearly  so  satisfactory  an  explanation  to  you  as 
it  was  to  me  when  I  was  first  told  of  it  ;  and 
for  this  reason.  By  that  time  I  was  already 
perfectly  familiar  with  the  notion  of  a  skin  upon 
the  surface  of  liquids,  and  I  had  been  taught  by 
means  of  it  to  work  out  problems  in  capillarity. 
The  explanation  was  therefore  a  description  of 
the  unknown  phenomenon  which  I  did  not 
know  how  to  deal  with  as  made  up  of  known 
phenomena  which  I  did  know  how  to  deal  with. 
But  to  many  of  you  possibly  the  liquid  skin 


AIMS  OF  SCIENTIFIC  THOUGHT  169 

may  seem  quite  as  strange  and  unaccountable 
as  the  motion  of  camphor  on  water. 

And  this  brings  me  to  consider  the  source 
of  the  pleasure  we  derive  from  an  explanation. 
By  known  and  familiar  I  mean  that  which  we 
know  how  to  deal  with,  either  by  action  in  the 
ordinary  sense,  or  by  active  thought.  When 
therefore  that  which  we  do  not  know  how  to 
deal  with  is  described  as  made  up  of  things  that 
we  do  know  how  to  deal  with,  we  have  that 
sense  of  increased  power  which  is  the  basis  of 
all  higher  pleasures.  Of  course  we  may  after- 
wards by  association  come  to  take  pleasure  in 
explanation  for  its  own  sake.  Are  we  then  to 
say  that  the  observed  order  of  events  is  reason- 
able, in  the  sense  that  all  of  it  admits  of 
explanation  ?  That  a  process  may  be  capable 
of  explanation,  it  must  break  up  into  simpler 
constituents  which  are  already  familiar  to  us. 
Now,  first,  the  process  may  itself  be  simple,  and 
not  break  up  ;  secondly,  it  may  break  up  into 
elements  which  are  as  unfamiliar  and  impractic- 
able as  the  original  process. 

It  is  an  explanation  of  the  moon's  motion 
to  say  that  she  is  a  falling  body,  only  she  is 
going  so  fast  and  is  so  far  off  that  she  falls 
quite  round  to  the  other  side  of  the  earth, 
instead  of  hitting  it ;  and  so  goes  on  for  ever. 
But  it  is  no  explanation  to  say  that  a  body 
falls  because  of  gravitation.  That  means  that 


170  LECTURES  AND  ESSAYS 

the  motion  of  the  body  may  be  resolved  into 
a  motion  of  every  one  of  its  particles  towards 
every  one  of  the  particles  of  the  earth,  with  an 
acceleration  inversely  as  the  square  of  the 
distance  between  them.  But  this  attraction  of 
two  particles  must  always,  I  think,  be  less 
familiar  than  the  original  falling  body,  however 
early  the  children  of  the  future  begin  to  read 
their  Newton.  Can  the  attraction  itself  be 
explained  ?  Le  Sage  said  that  there  is  an 
everlasting  hail  of  innumerable  small  ether- 
particles  from  all  sides,  and  that  the  two 
material  particles  shield  each  other  from  this 
and  so  get  pushed  together.  This  is  an  explana- 
tion ;  it  may  or  may  not  be  a  true  one.  The 
attraction  may  be  an  ultimate  simple  fact ;  or  it 
may  be  made  up  of  simpler  facts  utterly  unlike 
anything  that  we  know  at  present  ;  and  in  either 
of  these  cases  there  is  no  explanation.  We  have 
no  right  to  conclude,  then,  that  the  order  of 
events  is  always  capable  of  being  explained. 

There  is  yet  another  way  in  which  it  is  said 
that  Nature  is  reasonable  ;  namely,  inasmuch 
as  every  effect  has  a  cause.  What  do  we  mean 
by  this  ? 

In  asking  this  question,  we  have  entered 
upon  an  appalling  task.  The  word  represented 
by  "  cause "  has  sixty-four  meanings  in  Plato 
and  forty-eight  in  Aristotle.  These  were  men 
who  liked  to  know  as  near  as  might  be  what 


AIMS  OF  SCIENTIFIC  THOUGHT  171 

they  meant ;  but  how  many  meanings  it  has  had 
in  the  writings  of  the  myriads  of  people  who 
have  not  tried  to  know  what  they  meant  by  it 
will,  I  hope,  never  be  counted.  It  would  not 
only  be  the  height  of  presumption  in  me  to 
attempt  to  fix  the  meaning  of  a  word  which 
has  been  used  by  so  grave  authority  in  so  many 
and  various  senses  ;  but  it  would  seem  a  thank- 
less task  to  do  that  once  more  which  has  been 
done  so  often  at  sundry  times  and  in  divers 
manners  before.  And  yet  without  this  we 
cannot  determine  what  we  mean  by  saying 
that  the  order  of  nature  is  reasonable.  I  shall 
evade  the  difficulty  by  telling  you  Mr.  Grote's 
opinion.1  You  come  to  a  scarecrow  and  ask, 
what  is  the  cause  of  this  ?  You  find  that  a 
man  made  it  to  frighten  the  birds.  You  go 
away  and  say  to  yourself,  "Everything  resembles 
this  scarecrow.  Everything  has  a  purpose." 
And  from  that  day  the  word  "  cause "  means 
for  you  what  Aristotle  meant  by  "  final  cause." 
Or  you  go  into  a  hairdresser's  shop,  and  wonder 
what  turns  the  wheel  to  which  the  rotatory 
brush  is  attached.  On  investigating  other 
parts  of  the  premises,  you  find  a  man  working 
away  at  a  handle.  Then  you  go  away  and 
say,  "  Everything  is  like  that  wheel.  If  I  in- 
vestigated enough,  I  should  always  find  a  man 
at  a  handle."  And  the  man  at  the  handle,  or 
1  Plato,  vol.  ii.  (Phaedo). 


172  LECTURES  AND  ESSAYS 

whatever  corresponds  to  him,  is  from  henceforth 
known  to  you  as  "  cause." 

And  so  generally.  When  you  have  made 
out  any  sequence  of  events  to  your  entire 
satisfaction,  so  that  you  know  all  about  it,  the 
laws  involved  being  so  familiar  that  you  seem 
to  see  how  the  beginning  must  have  been 
followed  by  the  end,  then  you  apply  that  as  a 
simile  to  all  other  events  whatever,  and  your 
idea  of  cause  is  determined  by  it.  Only  when 
a  case  arises,  as  it  always  must,  to  which  the 
simile  will  not  apply,  you  do  not  confess  to 
yourself  that  it  was  only  a  simile  and  need  not 
apply  to  everything,  but  you  say,  "  The  cause 
of  that  event  is  a  mystery  which  must  remain 
for  ever  unknown  to  me."  On  equally  just 
grounds  the  nervous  system  of  my  umbrella  is 
a  mystery  which  must  remain  for  ever  unknown 
to  me.  My  umbrella  has  no  nervous  system  ; 
and  the  event  to  which  your  simile  did  not 
apply  has  no  cause  in  your  sense  of  the  word. 
When  we  say  then  that  every  effect  has  a 
cause,  we  mean  that  every  event  is  connected 
with  something  in  a  way  that  might  make 
somebody  call  that  the  cause  of  it.  But  I,  at 
least,  have  never  yet  seen  any  single  meaning 
of  the  word  that  could  be  fairly  applied  to  the 
whole  order  of  nature. 

From  this  remark  I  cannot  even  except  an 
attempt  recently  made  by  Mr.  Bain  to  give  the 


AIMS  OF  SCIENTIFIC  THOUGHT  173 

word  a  universal  meaning,  though  I  desire  to 
speak  of  that  attempt  with  the  greatest  respect. 
Mr.  Bain l  wishes  to  make  the  word  "  cause " 
hang  on  in  some  way  to  what  we  call  the  law 
of  energy ;  but  though  I  speak  with  great 
diffidence  I  do  think  a  careful  consideration 
will  show  that  the  introduction  of  this  word 
"  cause  "  can  only  bring  confusion  into  a  matter 
which  is  distinct  and  clear  enough  to  those  who 
have  taken  the  trouble  to  understand  what 
energy  means.  It  would  be  impossible  to 
explain  that  this  evening ;  but  I  may  mention 
that  "  energy "  is  a  technical  term  out  of 
mathematical  physics,  which  requires  of  most 
men  a  good  deal  of  careful  study  to  understand 
it  accurately. 

Let  us  pass  on  to  consider,  with  all  the 
reverence  which  it  demands,  another  opinion 
held  by  great  numbers  of  the  philosophers  who 
have  lived  in  the  Brightening  Ages  of  Europe  ; 
the  opinion  that  at  the  basis  of  the  natural 
order  there  is  something  which  we  can  know 
to  be  unreasonable,  to  evade  the  processes  of 
human  thought.  The  opinion  is  set  forth  first 
by  Kant,  so  far  as  I  know,  in  the  form  of  his 
famous  doctrine  of  the  antinomies  or  contra- 
dictions, a  later  form  2  of  which  I  will  endeavour 


1  Inductive  Logic,  chap.  iv. 

2  That  of  Mr.    Herbert  Spencer,   First  Principles.      I  believe 
Kant  himself  would  have  admitted  that  the  antinomies  do  not 


174  LECTURES  AND  ESSAYS 

to  explain  to  you.  It  is  said,  then,  that  space 
must  either  be  infinite  or  have  a  boundary. 
Now  you  cannot  conceive  infinite  space  ;  and 
you  cannot  conceive  that  there  should  be  any 
end  to  it  Here,  then,  are  two  things,  one  of 
which  must  be  true,  while  each  of  them  is  in- 
conceivable ;  so  that  our  thoughts  about  space 
are  hedged  in,  as  it  were,  by  a  contradiction. 
Again,  it  is  said  that  matter  must  either  be 
infinitely  divisible,  or  must  consist  of  small 
particles  incapable  of  further  division.  Now 
you  cannot  conceive  a  piece  of  matter  divided 
into  an  infinite  number  of  parts,  while,  on  the 
other  hand,  you  cannot  conceive  a  piece  of 
matter,  however  small,  which  absolutely  cannot 
be  divided  into  two  pieces  ;  for,  however  great 
the  forces  are  which  join  the  parts  of  it  together, 
you  can  imagine  stronger  forces  able  to  tear  it 
in  pieces.  Here,  again,  there  are  two  state- 
ments, one  of  which  must  be  true,  while  each 
of  them  is  separately  inconceivable  ;  so  that 
our  thoughts  about  matter  also  are  hedged  in 
by  a  contradiction.  There  are  several  other 
cases  of  the  same  thing,  but  I  have  selected 
these  two  as  instructive  examples.  And  the 
conclusion  to  which  philosophers  were  led  by 
the  contemplation  of  them  was  that  on  every 


exist  for  the  empiricist.  [Much  less  does  he  say  that  either  of  a 
pair  of  antinomies  must  be  true.  The  real  Kantian  position  is 
that  both  assertions  are  illegitimate.] 


AIMS  OF  SCIENTIFIC  THOUGHT  175 

side,  when  we  approach  the  limits  of  existence, 
a  contradiction  must  stare  us  in  the  face.  The 
doctrine  has  been  developed  and  extended  by 
the  great  successors  of  Kant ;  and  this  un- 
reasonable, or  unknowable,  which  is  also  called 
the  absolute  and  the  unconditioned,  has  been 
set  forth  in  various  ways  as  that  which  we 
know  to  be  the  true  basis  of  all  things.  As  I 
said  before,  I  approach  this  doctrine  with  all 
the  reverence  which  should  be  felt  for  that 
which  has  guided  the  thoughts  of  so  many  of 
the  wisest  of  mankind.  Nevertheless  I  shall 
endeavour  to  show  that  in  these  cases  of 
supposed  contradiction  there  is  always  some- 
thing which  we  do  not  know  now,  but  of  which 
we  cannot  be  sure  that  we  shall  be  ignorant 
next  year.  The  doctrine  is  an  attempt  to 
found  a  positive  statement  upon  this  ignorance, 
which  can  hardly  be  regarded  as  justifiable. 
Spinoza  said,  "A  free  man  thinks  of  nothing 
so  little  as  of  death  ; "  it  seems  to  me  we  may 
parallel  this  maxim  in  the  case  of  thought,  and 
say,  "  A  wise  man  only  remembers  his  ignorance 
in  order  to  destroy  it."  A  boundary  is  that 
which  divides  two  adjacent  portions  of  space. 
The  question,  then,  "  Has  space  (in  general)  a 
boundary  ?  "  involves  a  contradiction  in  terms, 
and  is,  therefore,  unmeaning.  But  the  question, 
"  Does  space  contain  a  finite  number  of  cubic 
miles,  or  an  infinite  number  ? "  is  a  perfectly 


176  LECTURES  AND  ESSAYS 

intelligible  and  reasonable  question  which  re- 
mains to  be  answered  by  experiment.1  The 
surface  of  the  sea  would  still  contain  a  finite 
number  of  square  miles,  if  there  were  no  land 
to  bound  it  Whether  or  fno  the  space  in 
which  we  live  is  of  this  nature  remains  to  be 
seen.  If  its  extent  is  finite,  we  may  quite 
possibly  be  able  to  assign  that  extent  next 
year  ;  if,  on  the  other  hand,  it  has  no  end,  it  is 
true  that  the  knowledge  of  that  fact  would  be 
quite  different  from  any  knowledge  we  at 
present  possess,  but  we  have  no  right  to  say 
that  such  knowledge  is  impossible.  Either  the 
question  will  be  settled  once  for  all,  or  the 
extent  of  space  will  be  shown  to  be  greater 
than  a  quantity  which  will  increase  from  year 
to  year  with  the  improvement  of  our  sources 
of  knowledge.  Either  alternative  is  perfectly 
conceivable,  and  there  is  no  contradiction. 
Observe  especially  that  the  supposed  contra- 
diction arises  from  the  assumption  of  theoretical 
exactness  in  the  laws  of  geometry.  The  other 
case  that  I  mentioned  has  a  very  similar  origin. 
The  idea  of  a  piece  of  matter  the  parts  of  which 
are  held  together  by  forces,  and  are  capable  of 
being  torn  asunder  by  greater  forces,  is  entirely 
derived  from  the  large  pieces  of  matter  which 
we  have  to  deal  with.  We  do  not  know 

'  1  The  very  important  distinction  between  unboundedntss  and 
infinite  extent  is  made  by  Riemann,  loc.  cit. 


AIMS  OF  SCIENTIFIC  THOUGHT  177 

whether  this  idea  applies  in  any  sense  even  to 
the  molecules  of  gases  ;  still  less  can  we  apply 
it  to  the  atoms  of  which  they  are  composed. 
The  word  force  is  used  of  two  phenomena  :  the 
pressure,  which  when  two  bodies  are  in  contact 
connects  the  motion  of  each  with  the  position 
of  the  other  ;  and  attraction  or  repulsion, — 
that  is  to  say,  a  change  of  velocity  in  one  body 
depending  on  the  position  of  some  other  body 
which  is  not  in  contact  with  it.  We  do  not 
know  that  there  is  anything  corresponding  to 
either  of  these  phenomena  in  the  case  of  a 
molecule.  A  meaning  can,  however,  be  given 
to  the  question  of  the  divisibility  of  matter  in 
this  way.  We  may  ask  if  there  is  any  piece  of 
matter  so  small  that  its  properties  as  matter 
depend  upon  its  remaining  all  in  one  piece. 
This  question  is  reasonable ;  but  we  cannot 
answer  it  at  present,  though  we  are  not  at  all 
sure  that  we  shall  be  equally  ignorant  next 
year.  If  there  is  no  such  piece  of  matter,  no 
such  limit  to  the  division  which  shall  leave  it 
matter,  the  knowledge  of  that  fact  would  be 
different  from  any  of  our  present  knowledge  ; 
but  we  have  no  right  to  say  that  it  is  impossible. 
If,  on  the  other  hand,  there  is  a  limit,  it  is  quite 
possible  that  we  may  have  measured  it  by  the 
time  the  Association  meets  at  Bradford.  Again, 
when  we  are  told  that  the  infinite  extent  of 
space,  for  example,  is  something  that  we  cannot 
VOL.  I  N 


i;8  LECTURES  AND  ESSAYS 

conceive  at  present,  we  may  reply  that  this  is 
only  natural,  since  our  experience  has  never 
yet  supplied  us  with  the  means  of  conceiving 
such  things.  But  then  we  cannot  be  sure  that 
the  facts  will  not  make  us  learn  to  conceive 
them  ;  in  which  case  they  will  cease  to  be 
inconceivable.  In  fact,  the  putting  of  limits  to 
human  conception  must  always  involve  the 
assumption  that  our  previous  experience  is 
universally  valid  in  a  theoretical  sense ;  an 
assumption  which  we  have  already  seen  reason 
to  reject.  Now  you  will  see  that  our  considera- 
tion of  this  opinion  has  led  us  to  the  true  sense 
of  the  assertion  that  the  Order  of  Nature  is 
reasonable.  If  you  will  allow  me  to  define  a 
reasonable  question  as  one  which  is  asked  in 
terms  of  ideas  justified  by  previous  experience, 
without  itself  contradicting  that  experience, 
then  we  may  say,  as  the  result  of  our  investi- 
gation, that  to  every  reasonable  question  there 
is  an  intelligible  answer  which  either  we  or 
posterity  may  know. 

We  have,  then,  come  somehow  to  the  follow- 
ing conclusions.  By  scientific  thought  we  mean 
the  application  of  past  experience  to  new  circum- 
stances by  means  of  an  observed  order  of  events. 
By  saying  that  this  order  of  events  is  exact  we 
mean  that  it  is  exact  enough  to  correct  experi- 
ments by,  but  we  do  not  mean  that  it  is  theo- 
retically or  absolutely  exact,  because  we  do 


AIMS  OF  SCIENTIFIC  THOUGHT  179 

not  know.  The  process  of  inference  we  found 
to  be  in  itself  an  assumption  of  uniformity,  and 
we  found  that,  as  the  known  exactness  of  the 
uniformity  became  greater,  the  stringency  of 
the  inference  increased.  By  saying  that  the 
order  of  events  is  reasonable  we  do  not  mean 
that  everything  has  a  purpose,  or  that  every- 
thing can  be  explained,  or  that  everything  has 
a  cause  ;  for  neither  of  these  is  true.  But  we 
mean  that  to  every  reasonable  question  there 
is  an  intelligible  answer,  which  either  we  or 
posterity  may  know  by  the  exercise  of  scientific 
thought. 

For  I  specially  wish  you  not  to  go  away 
with  the  idea  that  the  exercise  of  scientific 
thought  is  properly  confined  to  the  subjects 
from  which  my  illustrations  have  been  chiefly 
drawn  to-night.  When  the  Roman  jurists 
applied  their  experience  of  Roman  citizens  to 
dealings  between  citizens  and  aliens,  showing 
by  the  difference  of  their  actions  that  they  re- 
garded the  circumstances  as  essentially  different, 
they  laid  the  foundations  of  that  great  structure 
which  has  guided  the  social  progress  of  Europe. 
That  procedure  was  an  instance  of  strictly 
scientific  thought.  When  a  poet  finds  that  he 
has  to  move  a  strange  new  world  which  his 
predecessors  have  not  moved  ;  when,  neverthe- 
less, he  catches  fire  from  their  flashes,  arms 
from  their  armoury,  sustentation  from  their 


i8o  LECTURES  AND  ESSAYS 

footprints,  the  procedure  by  which  he  applies 
old  experience  to  new  circumstances  is  nothing 
greater  or  less  than  scientific  thought.  When 
the  moralist,  studying  the  conditions  of  society 
and  the  ideas  of  right  and  wrong  which  have 
come  down  to  us  from  a  time  when  war  was 
the  normal  condition  of  man  and  success  in  war 
the  only  chance  of  survival,  evolves  from  them 
the  conditions  and  ideas  which  must  accompany 
a  time  of  peace,  when  the  comradeship  of  equals 
is  the  condition  of  national  success  ;  the  process 
by  which  he  does  this  is  scientific  thought  and 
nothing  else.  Remember,  then,  that  it  is  the 
guide  of  action  ;  that  the  truth  which  it  arrives 
at  is  not  that  which  we  can  ideally  contemplate 
without  error,  but  that  which  we  may  act  upon 
without  fear  ;  and  you  cannot  fail  to  see  that 
scientific  thought  is  not  an  accompaniment  or 
condition  of  human  progress,  but  human  pro- 
gress itself.  And  for  this  reason  the  question 
what  its  characters  are,  of  which  I  have  so 
inadequately  endeavoured  to  give  you  some 
glimpse,  is  the  question  of  all  questions  for  the 
human  race. 


ATOMS l 

IF  I  were  to  wet  my  finger  and  then  rub  it 
along  the  edge  of  this  glass,  I  should  no  doubt 
persuade  the  glass  to  give  out  a  certain  musical 
note.  So  also  if  I  were  to  sing  to  that  glass 
the  same  note  loud  enough,  I  should  get  the 
glass  to  answer  me  back  with  a  note. 

I  want  you  to  remember  that  fact,  because 
it  is  of  capital  importance  for  the  arguments 
we  shall  have  to  consider  to-night.  The  very 
same  note  which  I  can  get  the  tumbler  to  give 
out  by  agitating  it,  by  rubbing  the  edge,  that 
same  note  I  can  also  get  the  tumbler  to  answer 
back  to  me  when  I  sing  to  it.  Now,  remember- 
ing that,  please  to  conceive  a  rather  complicated 
thing  that  I  am  now  going  to  try  to  describe 
to  you.  The  same  property  that  belongs  to 
the  glass  belongs  also  to  a  bell  which  is  made 
out  of  metal.  If  that  bell  is  agitated  by  being 
struck,  or  in  any  other  way,  it  will  give  out  the 
same  sound  that  it  will  answer  back  if  you  sing 

1  Sunday  Lecture   Society,   January  7,   1872  ;    Hulme  Town 
Hall,  Manchester,  November  20,  1872. 


i8z  LECTURES  AND  ESSAYS 

that  sound  to  it ;  but  if  you  sing  a  different 
sound  to  it  then  it  will  not  answer. 

Now  suppose  that  I  have  several  of  these 
metal  bells  which  answer  to  quite  different 
notes,  and  that  they  are  all  fastened  to  a  set 
of  elastic  stalks  which  spring  out  of  a  certain 
centre  to  which  they  are  fastened.  All  these 
bell,  then,  are  not  only  fastened  to  these  stalks, 
but  they  are  held  there  in  such  a  way  that  they 
can  spin  round  upon  the  points  to  which  they 
are  fastened. 

And  then  the  centre  to  which  these  elastic 
stalks  are  fastened  or  suspended,  you  may 
imagine  as  able  to  move  in  all  manner  of 
directions,  and  that  the  whole  structure  made 
up  of  these  bells  and  stalks  and  centre  is  able 
to  spin  round  any  axis  whatever.  We  must 
also  suppose  that  there  is  surrounding  this 
structure  a  certain  framework.  We  will  sup- 
pose the  framework  to  be  made  of  some  elastic 
material,  so  that  it  is  able  to  be  pressed  in  to 
a  certain  extent  Suppose  that  framework  is 
made  of  whalebone,  if  you  like.  This  structure 
I  am  going  for  the  present  to  call  an  "  atom." 
I  do  not  mean  to  say  that  atoms  are  made  of  a 
structure  like  that.  I  do  not  mean  to  say  that 
there  is  anything  in  an  atom  which  is  in  the 
shape  of  a  bell ;  and  I  do  not  mean  to  say 
that  there  is  anything  analogous  to  an  elastic 
stalk  in  it.  But  what  I  mean  is  this — that  an 


ATOMS  183 

atom  is  something  that  is  capable  of  vibrating 
at  certain  definite  rates  ;  also  that  it  is  capable 
of  other  motions  of  its  parts  besides  those 
vibrations  at  certain  definite  rates  ;  and  also 
that  it  is  capable  of  spinning  round  about  any 
axis.  Now  by  the  framework  which  I  suppose 
to  be  put  round  that  structure  made  out  of 
bells  and  elastic  stalks,  I  mean  this — that  sup- 
posing you  had  two  such  structures,  then  you 
cannot  put  them  closer  together  than  a  certain 
distance,  but  they  will  begin  to  resist  being  put 
close  together  after  you  have  put  them  as  near 
as  that,  and  they  will  push  each  other  away  if 
you  attempt  to  put  them  closer.  That  is  all  I 
mean  then.  You  must  only  suppose  that  that 
structure  is  described,  and  that  set  of  ideas  is 
put  together,  just  for  the  sake  of  giving  us  some 
definite  notion  of  a  thing  which  has  similar 
properties  to  that  structure.  But  you  must 
not  suppose  that  there  is  any  special  part  of  an 
atom  which  has  got  a  bell-like  form,  or  any 
part  like  an  elastic  stalk  made  out  of  whale- 
bone. 

Now  having  got  the  idea  of  such  a  compli- 
cated structure,  which  is  capable,  as  we  said,  of 
vibratory  motion,  and  of  other  sorts  of  motion, 
I  am  going  on  to  explain  what  is  the  belief  of 
those  people  who  have  studied  the  subject 
about  the  composition  of  the  air  which  fills  this 
room.  The  air  which  fills  this  room  is  what  is 


i34  LECTURES  AND  ESSAYS 

called  a  gas  ;  but  it  is  not  a  simple  gas  ;  it  is 
a  mixture  of  two  different  gases,  oxygen  and 
nitrogen.  What  is  believed  about  this  air  is 
that  it  consists  of  quite  distinct  portions  or  little 
masses  of  air — that  is,  of  little  masses  each  of 
which  is  either  oxygen  or  nitrogen  ;  and  that 
these  little  masses  are  perpetually  flying  about 
in  all  directions.  The  number  of  them  in  this 
room  is  so  great  that  it  strains  the  powers  of 
our  numerical  system  to  count  them.  They 
are  flying  about  in  all  directions  and  mostly  in 
straight  lines,  except  where  they  get  quite  near 
to  one  another,  and  then  they  rebound  and  fly 
off  in  other  directions.  Part  of  these  little 
masses  which  compose  the  air  are  of  one  sort — 
they  are  called  oxygen.  All  those  little  masses 
which  are  called  oxygen  are  alike  ;  they  are  of 
the  same  weight ;  they  have  the  same  rates  of 
vibration  ;  and  they  go  about  on  the  average 
at  a  certain  rate.  The  other  part  of  these 
little  masses  is  called  nitrogen,  and  they  have  a 
different  weight ;  but  the  weight  of  all  the 
nitrogen  masses  is  the  same,  as  nearly  as  we 
can  make  out  They  have  again  the  same 
rates  of  vibration  ;  but  the  rates  of  vibration 
that  belong  to  them  are  different  from  the  rates 
of  vibration  that  belong  to  the  oxygen  masses  ; 
and  the  nitrogen  masses  go  about  on  the  aver- 
age at  a  certain  rate,  but  this  rate  is  different 
from  the  average  rate  at  which  the  oxygen 


ATOMS  185 

masses  go  about.  So  then,  taking  up  that 
structure  which  I  endeavoured  to  describe  to 
you  at  first,  we  should  represent  the  state  of 
the  air  in  this  room  as  being  made  up  of  such 
a  lot  of  compound  atoms  of  those  structures  of 
bells  and  stalks,  with  frameworks  round  them, 
that  I  described  to  you,  being  thrown  about  in 
all  directions  with  great  rapidity,  and  continu- 
ally impinging  against  one  another,  each  fly- 
ing off  in  a  different  direction,  so  that  they 
would  go  mostly  in  straight  lines  (you  must 
suppose  them  for  a  moment  not  to  fall  down 
towards  the  earth),  excepting  where  they  come 
near  enough  for  their  two  frameworks  to  be  in 
contact,  and  then  their  frameworks  throw  them 
off  in  different  directions  :  that  is  a  conception 
of  the  state  of  things  which  actually  takes  place 
inside  of  gas. 

Now,  the  conception  which  scientific  men 
have  of  the  state  of  things  which  takes  place 
inside  of  a  liquid  is  different  from  that.  We 
should  conceive  it  in  this  way  :  We  should 
suppose  that  a  number  of  these  structures  are 
put  so  close  together  that  their  frameworks  are 
always  in  contact  ;  and  yet  they  are  moving 
about  and  rolling  among  one  another,  so  that 
no  one  of  them  keeps  the  same  place  for  two 
instants  together,  and  any  one  of  them  is 
travelling  all  over  the  whole  space.  Inside  of 
this  glass,  where  there  is  a  liquid,  all  the  small 


186  LECTURES  AND  ESSAYS 

particles  or  molecules  are  running  about  among 
one  another,  and  yet  none  of  them  goes  for  any 
appreciable  portion  of  its  path  in  a  straight 
line,  because  there  is  no  distance  however  small 
that  it  goes  without  being  in  contact  with  others 
all  around  it ;  and  the  effect  of  this  contact  of 
the  others  all  around  it  is  that  they  press 
against  it  and  force  it  out  of  a  straight  path. 
So  that  the  path  of  a  particle  in  a  liquid  is  a 
sort  of  wavy  path  ;  it  goes  in  and  out  in  all 
directions,  and  a  particle  at  one  part  of  the 
liquid  will,  at  a  certain  time,  have  traversed  all 
the  different  parts  one  after  another. 

The  conception  of  what  happens  inside  of 
a  solid  body,  say  a  crystal  of  salt,  is  different 
again  from  this.  It  is  supposed  that  the  very 
small  particles  which  constitute  that  crystal  of 
salt  do  not  travel  about  from  one  part  of  the 
crystal  to  another,  but  that  each  one  of  them 
remains  pretty  much  in  the  same  place.  I  say 
"  pretty  much,"  but  not  exactly,  and  the  motion 
of  it  is  like  this  :  Suppose  one  of  my  structures, 
with  its  framework  round  it,  to  be  fastened  up 
by  elastic  strings,  so  that  one  string  goes  to  the 
ceiling,  and  another  to  the  floor,  and  another  to 
each  wall,  so  that  it  is  fastened  by  all  these 
strings.  Then  if  these  strings  are  stretched, 
and  a  particle  is  displaced  in  any  way,  it  will 
just  oscillate  about  its  mean  position,  and  will 
not  go  far  away  from  it ;  and  if  forced  away 


%\  ATOMS  187 

from  that  position  it  will  come  back  again. 
That  is  the  sort  of  motion  that  belongs  to  a 
particle  in  the  inside  of  a  solid  body.  A  solid 
body,  such  as  a  crystal  of  salt,  is  made  up,  just 
as  a  liquid  or  a  gas  is  made  up,  of  innumerable 
small  particles,  but  they  are  so  attached  to  one 
another  that  each  of  them  can  only  oscillate 
about  its  mean  position.  It  is  very  probable 
that  it  is  also  able  to  spin  about  any  axis  in 
that  position  or  near  it ;  but  it  is  not  able  to 
leave  that  position  finally,  and  to  go  and  take 
up  another  position  in  the  crystal  ;  it  must  stop 
in  or  near  about  the  same  position. 

These,  then,  are  the  views  which  are  held 
by  scientific  men  at  present  about  what  actually 
goes  on  inside  of  a  gaseous  body,  or  a  liquid 
body,  or  a  solid  body.  In  each  case  the  body 
is  supposed  to  be  made  up  of  a  very  large 
number  of  very  small  particles  ;  but  in  one  case 
these  particles  are  very  seldom  in  contact  with 
one  another,  that  is,  very  seldom  within  range 
of  each  other's  action  ;  in  this  case  they  are 
during  the  greater  part  of  the  time  moving 
separately  along  straight  lines.  In  the  case  of 
a  liquid  they  are  constantly  within  the  range 
of  each  other's  action  ;  but  they  do  not  move 
along  straight  lines  for  any  appreciable  part  of 
the  time ;  they  are  always  changing  their 
position  relatively  to  the  other  particles,  and 
one  of  them  gets  about  from  one  part  of  the 


i88  LECTURES  AND  ESSAYS 

liquid  to  another.  In  the  case  of  a  solid  they 
are  always  also  within  the  range  of  each  other's 
action,  and  they  are  so  much  within  that  range 
that  they  are  not  able  to  change  their  relative 
positions  ;  and  each  one  of  them  is  obliged  to 
remain  in  very  nearly  the  same  position. 

Now  what  I  want  to  do  this  evening  is  to 
explain  to  you,  as  far  as  I  can,  the  reasons 
which  have  led  scientific  men  to  adopt  these 
views  ;  and  what  I  wish  especially  to  impress 
upon  you  is  this,  that  what  is  called  the  "  atomic 
theory " — that  is,  what  I  have  just  been  ex- 
plaining— is  no  longer  in  the  position  of  a 
theory,  but  that  such  of  the  facts  as  I  have  just 
explained  to  you  are  really  things  which  are 
definitely  known  and  which  are  no  longer 
suppositions ;  that  the  arguments  by  which 
scientific  men  have  been  led  to  adopt  these 
views  are  such  as,  to  anybody  who  fairly 
considers  them,  justify  that  person  in  believing 
that  the  statements  are  true. 

Now  first  of  all  I  want  to  explain  what  the 
reasons  are  why  we  believe  that  the  air  consists 
of  separate  portions,  and  that  these  portions  are 
repetitions  of  the  same  structures.  That  is  to 
say  that  in  the  air  we  have  two  structures  really, 
each  of  them  a  great  number  of  times  repeated. 
Take  a  simple  illustration,  which  is  a  rather 
easier  one  to  consider.  Suppose  we  take  a 
vessel  which  is  filled  with  oxygen.  I  want  to 


&i  ATOMS  189 

show  what  the  reasons  are  which  lead  us  to 
believe  that  that  gas  consists  of  a  certain 
structure  which  is  a  great  number  of  times 
repeated,  and  that  between  two  examples  of 
that  structure  which  exist  inside  of  the  vessel 
there  is  a  certain  empty  space  which  does  not 
contain  any  oxygen.  That  oxygen  gas  con- 
tained in  the  vessel  is  made  up  of  small 
particles  which  are  not  close  together,  and 
each  of  these  particles  has  a  certain  structure, 
which  structure  also  belongs  to  the  rest  of  the 
particles.  This  argument  is  rather  a  difficult 
one,  and  I  shall  ask  you  therefore  to  follow  it 
as  closely  as  possible,  because  it  is  an  extremely 
complicated  argument  to  follow  out  the  first 
time  that  it  is  presented  to  you. 

I  want  to  consider  again  the  case  of  this 
finger-glass.  You  must  often  have  tried  that 
experiment — that  a  glass  will  give  out  when  it 
is  agitated  the  same  note  which  it  will  return 
when  it  is  sung  to.  Well,  now,  suppose  that  I 
have  got  this  room  filled  with  a  certain  number 
of  such  atomic  structures  as  I  have  endeavoured 
to  describe — that  is  to  say,  of  sets  of  bells,  the 
bells  answering  to  certain  given  notes.  Each 
of  these  little  structures  is  exactly  alike,  that  is 
to  say,  it  contains  just  the  bells  corresponding 
to  the  same  notes.  Well,  now,  suppose  that 
you  sing  to  a  glass  or  to  a  bell,  there  are  three 
things  that  may  happen.  First,  you  may  sing 


190  LECTURES  AND  ESSAYS 

a  note  which  does  not  belong  to  the  bell  at  all. 
In  that  case  the  bell  will  not  answer ;  it  will 
not  be  affected  or  agitated  by  your  singing  that 
note,  but  it  will  remain  quite  still.  Next,  if 
you  sing  a  note  that  belongs  to  the  bell,  but  if 
you  sing  it  rather  low,  then  the  effect  of  that 
note  will  be  to  make  the  bell  move  a  little,  but 
the  bell  will  not  move  so  much  as  to  give  back 
the  note  in  an  audible  form.  Thirdly,  if  you 
sing  the  note  which  belongs  to  the  bell  loud 
enough,  then  you  will  so  far  agitate  the  bell  that 
it  will  give  back  the  note  to  you  again.  Now 
exactly  that  same  property  belongs  to  a 
stretched  string  or  the  string  of  a  piano.  You 
know  that  if  you  sing  a  certain  note  in  a  room 
where  there  is  a  piano,  the  string  belonging  to 
that  note  will  answer  you  if  you  sing  loud 
enough.  The  other  strings  won't  answer  at  all. 
If  you  don't  sing  loud  enough  the  string  will  be 
affected,  but  not  enough  to  answer  you.  Now 
let  us  imagine  a  screen  of  piano  strings,  all  of 
exactly  the  same  length,  of  the  same  material, 
and  stretched  equally,  and  that  this  screen  of 
strings  is  put  across  the  room  ;  that  I  am  at 
one  end  and  that  you  are  at  another,  and  that 
I  proceed  to  sing  notes  straight  up  the  scale. 
While  I  sing  notes  which  are  different  from  that 
note  which  belongs  to  the  screen  of  strings,  they 
will  pass  through  the  screen  without  being 
altered,  because  the  agitation  of  the  air  which 


8V  ATOMS  191 

I  produce  will  not  affect  the  strings.  But  that 
note  will  be  heard  quite  well  at  the  other  side 
of  the  screen.  You  must  remember  that  when 
the  air  carries  a  sound  it  vibrates  at  a  certain 
rate  belonging  to  the  sound.  I  make  the  air 
vibrate  by  singing  a  particular  note,  and  if  that 
rate  of  vibration  corresponds  to  the  strings  the 
air  will  pass  on  part  of  its  vibration  to  the 
strings,  and  so  make  the  strings  move.  But  if 
the  rate  of  vibration  is  not  the  one  that  corre- 
sponds to  the  strings,  then  the  air  will  not  pass 
on  any  of  its  vibrations  to  the  strings,  and 
consequently  the  sound  will  be  heard  equally 
loud  after  it  has  passed  through  the  strings. 
Having  put  the  strings  of  the  piano  across  the 
room,  if  I  sing  up  the  scale,  when  I  come  to  the 
note  which  belongs  to  each  of  the  strings  my 
voice  will  suddenly  appear  to  be  deadened,  be- 
cause at  the  moment  that  the  rate  of  vibration 
which  I  impress  upon  the  air  coincides  with  that 
belonging  to  the  strings,  part  of  it  will  be  taken 
up  in  setting  the  strings  in  motion.  As  I  pass 
the  note,  then,  which  belongs  to  the  strings, 
that  note  will  be  deadened. 

Instead  of  a  screen  of  piano  strings  let  us  put 
in  a  series  of  sets  of  bells,  three  or  four  belonging 
to  each  set,  so  that  each  set  of  bells  answers  to 
three  or  four  notes,  and  so  that  all  the  sets  are 
exactly  alike.  Now  suppose  that  these  sets  of 
bells  are  distributed  all  over  the  middle  part  of 


192  LECTURES  AND  ESSAYS 

the  room,  and  that  I  sing  straight  up  the  scale 
from  one  note  to  another  until  I  come  to  the 
note  that  corresponds  to  one  of  the  bells  in  these 
sets,  then  that  note  will  appear  to  be  deadened 
at  the  other  end,  because  part  of  the  vibration 
communicated  to  the  air  will  be  taken  up  in 
setting  those  bells  in  motion.  When  I  come  to 
another  note  which  belongs  to  them,  that  note 
will  also  be  deadened  ;  so  that  a  person  listening 
at  the  other  end  of  the  room  would  observe  that 
certain  notes  were  deadened,  or  even  had  dis- 
appeared altogether.  If,  however,  I  sing  loud 
enough,  I  then  should  set  all  these  bells  vibrating. 
What  would  be  heard  at  the  other  end  of  the 
room  ?  Why,  just  the  chord  compounded  out 
of  those  sounds  that  belonged  to  the  bells, 
because  the  bells  having  been  set  vibrating 
would  give  out  the  corresponding  notes.  So 
you  see  there  are  here  three  facts.  When  I 
sing  a  note  which  does  not  belong  to  the  bells, 
my  voice  passes  to  the  end  of  the  room  without 
diminution.  When  I  sing  a  note  that  does 
belong  to  the  bells,  then  if  it  is  not  loud 
enough  it  is  deadened  by  passing  through  the 
screen  ;  but  if  it  is  loud  enough  it  sets  the  bells 
vibrating,  and  is  heard  afterwards.  Now  just 
notice  this  consequence.  We  have  supposed  a 
screen  made  out  of  these  structures  that  I  have 
imagined  to  represent  atoms,  and  when  I  sing 
through  the  scale  at  one  end  of  the  room  certain 


£  ATOMS  193 

notes  appear  to  be  deadened.  If  I  take  away 
half  of  those  structures,  what  will  be  the  effect  ? 
Exactly  the  same  notes  will  be  deadened,  but 
they  will  not  be  deadened  so  much  ;  the  notes 
which  are  picked  out  of  the  thinner  screen  to  be 
deadened  will  be  exactly  the  same  notes,  but 
the  amount  of  the  deadening  will  not  be  the  same. 
So  far  we  have  only  been  talking  about  the 
transmission  of  sound.  You  know  that  sound 
consists  of  certain  waves  which  are  passed  along 
in  the  air  ;  they  are  called  "  aerial  vibrations." 
We  also  know  that  light  consists  of  certain  waves 
which  are  passed  along,  not  in  the  air,  but 
along  another  medium.  I  cannot  stop  at 
present  to  explain  to  you  what  the  sort  of  evi- 
dence is  upon  which  that  assertion  rests,  but  it 
is  the  same  sort  of  evidence  as  that  which  I 
shall  try  to  show  you  belongs  to  the  statement 
about  atoms ;  that  is  to  say,  the  "  undulatory 
theory,"  as  it  is  called,  of  light,  the  theory  that 
light  consists  of  waves  transmitted  along  a 
certain  medium,  has  passed  out  of  the  stage  of 
being  a  theory,  and  has  passed  into  the  stage 
of  being  a  demonstrated  fact.  The  difference 
between  a  theory  and  a  demonstrated  fact  is 
something  like  this  :  If  you  supposed  a  man  to 
have  walked  from  Chorlton  Town  Hall  down 
here  say  in  ten  minutes,  the  natural  conclusion 
would  be  that  he  had  walked  along  the  Stret- 
ford  Road.  Now  that  theory  would  entirely 
VOL.  I  O 


194  LECTURES  AND  ESSAYS 

account  for  all  the  facts,  but  at  the  same  time 
the  facts  would  not  be  proved  by  it.  But  sup- 
pose it  happened  to  be  winter  time,  with  snow 
on  the  road,  and  that  you  could  trace  the  man's 
footsteps  all  along  the  road,  then  you  would 
know  that  he  had  walked  along  that  way. 
The  sort  of  evidence  we  have  to  show  that 
light  does  consist  of  waves  transmitted  through 
a  medium  is  the  sort  of  evidence  that  footsteps 
upon  the  snow  make  ;  it  is  not  a  theory  merely 
which  simply  accounts  for  the  facts,  but  it  is  a 
theory  which  can  be  reasoned  back  to  from  the 
facts  without  any  other  theory  being  possible. 
So  that  you  must  just  for  the  present  take  it 
for  granted  that  the  arguments  in  favour  of  the 
hypothesis  that  light  consists  of  waves  are  such 
as  to  take  it  out  of  the  region  of  hypothesis, 
and  make  it  into  demonstrated  fact. 

Very  well,  then,  light  consists  of  waves 
transmitted  along  this  medium  in  the  same 
way  that  sound  is  transmitted  along  the  air. 
The  waves  are  not  of  the  same  kind  ;  but  still 
they  are  waves,  and  they  are  transmitted  as  such; 
and  the  different  colours  of  light  correspond  to 
the  different  lengths  of  these  waves,  or  to  the 
different  rates  of  the  vibration  of  the  medium, 
just  as  the  different  pitches  of  sound  corre- 
spond to  the  different  lengths  of  the  air-waves 
or  to  the  different  rates  of  the  vibration  of  the 
air.  Now,  if  we  take  any  gas,  such  as  oxygen, 


8  ATOMS  195 

and  we  pass  light  through  it,  we  find  that  that 
gas  intercepts,  or  weakens,  certain  particular 
colours.  If  we  take  any  other  gas,  such  as 
hydrogen,  and  pass  light  through  it,  we  find 
that  that  gas  intercepts,  or  weakens,  certain 
other  particular  colours  of  the  light.  There 
are  two  ways  in  which  it  can  do  that:  it  is 
clear  that  the  undulations,  or  waves,  are  made 
weaker,  because  they  happen  to  coincide  with 
the  rate  of  vibration  of  the  gas  they  are  passing 
through.  But  the  gas  may  vibrate  as  a  whole 
in  the  same  way  that  the  air  does  when  you 
transmit  sound.  Or  the  waves  may  be  stopped, 
because  the  gas  consists  of  a  number  of  small 
structures  ;  just  as  my  screen,  which  I  imagine 
to  consist  of  structures  ;  or  just  as  the  screen 
of  piano  strings  is  made  up  of  the  same  struc- 
ture many  times  repeated.  Either  of  these 
suppositions  would  apparently  at  first  account 
for  the  fact  that  certain  waves  of  light  are  inter- 
cepted by  the  gas,  while  others  are  let  through. 
But  how  is  it  that  we  can  show  one  of  these 
suppositions  is  wrong  and  the  other  is  right  ? 
Instead  of  taking  so  small  a  structure  as  piano 
strings,  let  us  suppose  we  had  got  a  series  of 
fiddles,  the  strings  of  all  of  them  being  stretched 
exactly  in  tune.  I  suppose  this  case  because 
it  makes  a  more  complicated  structure,  for  there 
would  be  two  or  three  notes  corresponding  in 
each  'fiddle.  If  you  suppose  this  screen  of 


196  LECTURES  AND  ESSAYS 

fiddles  to  be  hung  up  and  then  compressed, 
what  will  be  the  effect?  The  effect  of  the 
compression  will  be,  if  they  are  all  in  contact, 
that  each  fiddle  itself  will  be  altered.  If  the 
fiddles  are  compressed  longways,  the  strings 
will  give  lower  notes  than  before,  and  con- 
sequently the  series  of  notes  which  will  be 
intercepted  by  that  screen  will  be  different  from 
the  series  of  notes  which  were  intercepted 
before.  But  if  you  have  a  screen  made  out  of 
fiddles  which  are  at  a  distance  from  one  another, 
and  then  if  you  compress  them  into  a  smallerspace 
by  merely  bringing  them  nearer  together,  without 
making  them  touch,  then  it  is  clear  that  exactly 
the  same  notes  will  be  intercepted  as  before  ; 
only,  as  there  will  be  more  fiddles  in  the  same 
space,  the  deadening  of  the  sound  will  be  greater. 
Now  when  you  compress  any  gas  you  find 
that  it  intercepts  exactly  the  same  colours  of 
light  which  it  intercepted  before  it  was  com- 
pressed. It  follows,  therefore,  that  the  rates  of 
vibration  which  it  intercepts  depend  not  upon 
the  mass  of  the  gas  whose  properties  are  altered 
by  the  compression,  but  upon  some  individual 
parts  of  it  which  were  at  a  distance  from  one 
another  before,  and  which  are  only  brought 
nearer  together  without  being  absolutely 
brought  into  contact  so  as  to  squeeze  them. 
That  is  the  sort  of  reasoning  by  which  it  is 
made  clear  that  the  interception  of  light,  or 


1  ATOMS  197 

particular  waves  of  light,  by  means  of  a  gas, 
must  depend  on  certain  individual  structures 
in  the  gas  which  are  at  a  distance  from  one 
another,  and  which  by  compression  are  not 
themselves  compressed,  but  only  brought  nearer 
to  one  another. 

There  is  an  extremely  interesting  con- 
sequence which  follows  from  this  reasoning, 
and  which  was  deduced  from  it  by  Professor 
Stokes  in  the  year  1851,  and  which  was  after- 
wards presented  in  a  more  developed  form  in 
the  magnificent  researches  of  Kirchhoff — 
namely,  the  reasoning  about  the  presence  of 
certain  matter  in  the  sun.  If  you  analyse  the 
solar  light  by  passing  it  through  a  prism,  the 
effect  of  the  prism  is  to  divide  it  off  so  as  to 
separate  the  light  into  the  different  colours 
which  it  contains.  That  line  of  variously 
coloured  light  which  is  produced  by  the  prism 
is,  as  you  know,  called  the  Spectrum.  When 
that  spectrum  is  made  in  a  very  accurate  way, 
so  that  the  parts  of  it  are  well  defined,  it  is 
observed  to  contain  certain  dark  lines.  That 
is,  there  is  a  certain  kind  of  light  which  is 
missing  in  the  sunlight ;  certain  kinds  of  light, 
as  we  travel  along  the  scale  of  lights,  are  miss- 
ing. Why  are  they  missing?  Because  there 
is  something  that  the  light  has  passed  through 
which  intercepts  or  weakens  those  kinds  of 
light.  Now  that  something  which  the  light 


198  LECTURES  AND  ESSAYS 

has  passed  through,  how  shall  we  find  out 
what  it  is  ?  It  ought  to  be  the  same  sort  of 
substance  which  if  it  were  heated  would  give 
out  exactly  that  kind  of  light.  Now  there  is 
a  certain  kind  of  light  which  is  intercepted 
which  makes  a  group  of  dark  lines  in  the  solar 
spectrum.  There  are  two  principal  lines  which 
together  are  called  the  line  D  ;  and  it  is  found 
that  exactly  that  sort  of  light  is  emitted  by 
sodium  when  heated  hot  enough.  The  con- 
clusion therefore  is  that  that  matter  which 
intercepts  that  particular  part  of  the  solar  light 
is  sodium,  or  that  there  is  sodium  somewhere 
between  us  and  the  hot  portion  of  the  sun 
which  sends  us  the  light  And  other  reasons 
lead  us  to  conclude  that  this  sodium  is  not  in 
the  atmosphere  of  the  earth,  but  in  the  neigh- 
bourhood of  the  sun — that  it  exists  in  a  gaseous 
state  in  the  sun's  atmosphere.  And  nearly  all 
the  lines  in  the  solar  spectrum  have  been 
explained  in  that  way,  and  shown  to  belong 
to  certain  substances  which  we  are  able  to  heat 
here,  and  to  show  that  when  they  are  heated 
they  give  out  exactly  the  same  kind  of  light 
which  they  intercepted  when  the  light  was  first 
given  out  by  the  sun  and  they  stood  in  the 
way.  So  you  see  that  is  a  phenomenon 
exactly  like  the  phenomenon  presented  by  the 
finger-glass  that  we  began  with. 

Precisely  the  same  light  which  any  gas  will 


I  ATOMS  199 

give  out  when  it  is  heated,  that  same  kind  of 
light  it  will  stop  or  much  weaken  if  the  light  is 
attempted  to  be  passed  through  it.  That  means 
that  this  medium  which  transmits  light,  and 
which  we  call  the  "  luminiferous  ether,"  has  a 
certain  rate  of  vibration  for  every  particular 
colour  of  the  spectrum.  When  that  rate  of 
vibration  coincides  with  one  of  the  rates  of 
vibration  of  an  atom,  then  it  will  be  stopped 
by  that  atom,  because  it  will  set  the  atom 
vibrating  itself.  If  therefore  you  pass  light  of 
any  particular  colour  through  a  gas  whose 
atoms  are  capable  of  the  corresponding  rate  of 
vibration,  the  light  will  be  cut  off  by  the  gas. 
If  on  the  other  hand  you  so  far  heat  the  gas 
that  the  atoms  are  vibrating  strongly  enough  to 
give  out  light,  it  will  give  out  a  light  of  a  kind 
which  it  previously  stopped. 

We  have  reason  then  for  believing  that  a 
simple  gas  consists  of  a  great  number  of  atoms  ; 
that  it  consists  of  very  small  portions,  each 
of  which  has  a  complicated  structure,  but  that 
structure  is  the  same  for  each  of  them,  and 
that  these  portions  are  separate,  or  that  there 
is  space  between  them. 

In  the  next  place  I  want  to  show  you  what 
is  the  evidence  upon  which  we  believe  that 
these  portions  of  the  gas  are  in  motion — that 
they  are  constantly  moving. 

If  this  were  a  political  instead  of  a  scientific 


300  LECTURES  AND  ESSAYS 

meeting,  there  would  probably  be  some  people 
who  would  be  inclined  to  disagree  with  us, 
instead  of  all  being  inclined  to  agree  with  one 
another ;  and  these  people  might  have  taken 
it  into  their  heads,  as  has  been  done  in  certain 
cases,  to  stop  the  meeting  by  putting  a  bottle 
of  sulphuretted  hydrogen  in  one  corner  of  the 
room  and  taking  the  cork  out  You  know  that 
after  a  certain  time  the  whole  room  would 
contain  sulphuretted  hydrogen,  which  is  a  very 
unpleasant  thing  to  come  in  contact  with. 
Now  how  is  it  that  that  gas  which  was  con- 
tained in  a  small  bottle  could  get  in  a  short 
time  over  the  whole  room  unless  it  was  in 
motion  ?  What  we  mean  by  motion  is  change 
of  place.  The  gas  was  in  one  corner,  and  it  is 
afterwards  all  over  the  room.  There  has  there- 
fore been  motion  somewhere,  and  this  motion 
must  have  been  of  considerable  rapidity,  because 
we  know  that  there  was  the  air  which  filled  the 
room  beforehand  to  oppose  resistance  to  that 
motion.  We  cannot  suppose  that  the  sul- 
phuretted hydrogen  gas  was  the  only  thing 
that  was  in  motion,  and  that  the  air  was 
not  in  motion  itself,  because  if  we  had  used 
any  other  gas  we  should  find  that  it  would 
diffuse  itself  in  exactly  the  same  way.  An 
argument  just  like  that  applies  also  to  the  case 
of  a  liquid.  Suppose  this  room  were  a  large 
tank  entirely  filled  with  water  and  anybody 


ATOMS  201 

were  to  drop  a  little  iodine  into  it,  after  a 
certain  time  the  whole  of  the  water  would  be 
found  to  be  tinged  of  a  blue  colour.  Now  that 
drop  may  be  introduced  into  any  part  of  the 
tank  you  like,  either  at  the  top  or  bottom,  and 
it  will  always  diffuse  itself  over  the  whole  water. 
There  has  here  again  been  motion.  We  cannot 
suppose  that  the  drop  which  was  introduced 
was  the  only  thing  that  moved  about,  because 
any  other  substance  would  equally  have  moved 
about.  And  the  water  has  moved  into  the 
place  where  the  drop  was,  because  in  the  place 
where  you  put  the  drop  there  is  not  so  much 
iodine  as  there  was  to  begin  with.  Well  then 
it  is  clear  that  in  the  case  of  a  gas,  these 
particles  of  which  we  have  shown  it  to  consist 
must  be  constantly  in  motion  ;  and  we  have 
shown  also  that  a  liquid  must  consist  of  parts 
that  are  in  motion,  because  it  is  able  to  admit 
the  particles  of  another  body  among  them. 

When  we  have  decided  that  the  particles  of 
a  gas  are  in  motion,  there  are  two  things  that 
they  may  do — they  may  either  hit  against  one 
another,  or  they  may  not.  Now  it  is  established 
that  they  do  hit  against  one  another,  and  that 
they  do  not  proceed  along  straight  lines  inde- 
pendent of  one  another.  But  I  cannot  at 
present  explain  to  you  the  whole  of  the  reason- 
ing upon  which  that  conclusion  is  grounded. 
It  is  grounded  upon  some  rather  hard  mathe- 


302  LECTURES  AND  ESSAYS 

matics.  It  was  shown  by  Professor  Clerk 
Maxwell  that  a  gas  cannot  be  a  medium  con- 
sisting of  small  particles  moved  about  in  all 
directions  in  straight  lines,  which  do  not  inter- 
fere with  one  another,  but  which  bound  off 
from  the  surfaces  which  contain  this  medium. 
Supposing  we  had  a  box  containing  a  gas  of 
this  sort.  Well,  these  particles  do  not  interfere 
with  one  another,  but  only  rebound  when  they 
come  against  the  sides  of  the  box  ;  then  that 
portion  of  the  gas  will  behave  not  like  a  gas 
but  like  a  solid  body.  The  peculiarity  of 
liquids  and  gases  is  that  they  do  not  mind 
being  bent  and  having  their  shape  altered.  It 
has  been  shown  by  Clerk  Maxwell  that  a 
medium  whose  particles  do  not  interfere  with 
one  another  would  behave  like  a  solid  body 
and  object  to  be  bent.  It  was  a  most  extra- 
ordinary conclusion  to  come  to,  but  it  is  entirely 
borne  out  by  the  mathematical  formulae.  It  is 
certain  that  if  there  were  a  medium  composed 
of  small  particles  flying  about  in  all  directions 
and  not  interfering  with  one  another,  then  that 
medium  would  be  to  a  certain  extent  solid,  that 
is,  would  resist  any  bending  or  change  of  shape. 
By  that  means  then  it  is  known  that  these 
particles  do  run  against  one  another.  And 
they  come  apart  again.  There  were  two  things 
of  course  they  might  do  :  they  might  either  go 
on  in  contact,  or  they  might  come  apart  Now 


ATOMS  203 

we  know  that  they  come  apart  for  this  reason 
— we  have  already  considered  how  two  gases 
in  contact  will  diffuse  into  one  another.  If 
you  were  to  put  a  bucket  containing  carbonic 
acid  (which  is  very  heavy)  upon  the  floor  of 
this  room,  it  would  after  a  certain  time  diffuse 
itself  over  all  the  room  ;  you  would  find  carbonic 
acid  gas  in  every  part  of  the  room.  Graham 
found  that  if  you  were  to  cover  over  the  top  of 
that  bucket  with  a  very  thin  cover  made  out 
of  graphite,  or  blacklead,  then  the  gas  would 
diffuse  itself  over  the  room  pretty  nearly  as 
fast  as  before.  The  graphite  acts  like  a  porous 
body,  as  a  sponge  does  to  water,  and  lets  the 
gas  get  through.  The  remarkable  thing  is 
that  if  the  graphite  is  thin  the  gas  will  get 
through  nearly  as  fast  as  it  will  if  nothing  is 
put  between  to  stop  it.  Graham  found  out 
another  fact.  Suppose  that  bucket  to  contain 
two  very  different  gases,  say  a  mixture  of 
hydrogen  and  carbonic  acid  gas.  Then  the 
hydrogen  would  come  out  through  the  black- 
lead  very  much  faster  than  the  carbonic  acid 
gas.  It  is  found  by  mathematical  calculation 
that  if  you  have  two  gases,  which  are  supposed 
to  consist  of  small  particles  which  are  all  bang- 
ing about,  the  gas  whose  particles  are  lightest 
will  come  out  quickest ;  that  a  gas  which  is 
four  times  as  light  will  come  out  twice  as  fast ; 
and  a  gas  nine  times  as  light  will  come  out 


104  LECTURES  AND  ESSAYS 

three  times  as  fast,  and  so  on.  Consequently, 
when  you  mix  two  gases  together  and  then 
pass  them  through  a  thin  piece  of  blacklead, 
the  lightest  gas  comes  out  quickest,  and  is  as  it 
were  sifted  from  the  other.  Now  suppose  we 
put  pure  hydrogen  into  a  bucket  and  put 
blacklead  on  the  top,  and  then  see  how  fast 
the  hydrogen  comes  out.  If  the  particles  of 
the  hydrogen  are  different  from  one  another, 
if  some  are  heavier,  the  lighter  ones  will  come 
out  first.  Now  let  us  suppose  we  have  got  a 
vessel  which  is  divided  into  two  parts  by  a  thin 
wall  of  blacklead.  We  will  put  hydrogen  into 
one  of  these  parts  and  allow  it  to  come  through 
this  blacklead  into  the  other  part ;  then  if  the 
hydrogen  contains  any  molecules  or  atoms  which 
are  lighter  than  the  others,  those  will  come 
through  first.  If  we  test  the  hydrogen  that 
has  come  through,  we  shall  find  that  the  atoms, 
as  a  rule,  on  one  side  of  this  wall  are  lighter 
than  the  atoms  on  the  other  side.  How  should 
we  find  that  out?  Why  we  should  take  these 
two  portions  of  gas,  and  we  should  try  whether 
one  of  them  would  pass  through  another  piece 
of  blacklead  quicker  than  the  other  ;  because  if 
it  did,  it  would  consist  of  lighter  particles. 
Graham  found  that  it  did  not  pass  any  quicker. 
Supposing  you  put  hydrogen  into  one  half  of 
such  a  vessel,  and  then  allow  the  gas  to  diffuse 
itself  through  the  blacklead,  the  gas  on  the  two 


ATOMS  205 

sides  would  be  found  to  be  of  precisely  the 
same  qualities.  Consequently,  there  has  not 
been  in  this  case  any  sifting  of  the  lighter 
particles  from  the  heavier  ones  ;  and  con- 
sequently there  could  not  have  been  any  lighter 
particles  to  sift,  because  we  know  that  if  there 
were  any  they  would  have  come  through  quicker 
than  the  others.  Therefore  we  are  led  to  the 
conclusion  that  in  any  simple  gas,  such  as 
hydrogen  or  oxygen,  all  the  atoms  are,  as  nearly 
as  possible,  of  the  same  weight.  We  have  no 
right  to  conclude  that  they  are  exactly  of  the 
same  weight,  because  there  is  no  experiment 
in  the  world  that  enables  us  to  come  to  an 
exact  conclusion  of  that  sort.  But  we  are 
enabled  to  conclude  that,  within  the  limits  of 
experiment,  all  the  atoms  of  a  simple  gas  are 
of  the  same  weight.  What  follows  from  that  ? 
It  follows  that  when  they  bang  against  one 
another,  they  must  come  apart  again  ;  for  if 
two  of  them  were  to  go  on  as  one,  that  one 
would  be  twice  as  heavy  as  the  others,  and 
would  consequently  be  sifted  back.  It  follows 
therefore  that  two  particles  of  a  gas  which  bang 
against  one  another  must  come  apart  again, 
because  if  they  were  to  cling  together  they 
would  form  a  particle  twice  as  heavy,  and  so 
this  clinging  would  show  itself  when  the  gas 
was  passed  through  the  screen  of  blacklead. 
Now  there  are  certain  particles  or  small 


206  LECTURES  AND  ESSAYS 

masses  of  matter  which  we  know  to  bang  against 
one  another  according  to  certain  laws  ;  such, 
for  example,  as  billiard  balls.  The  way  in 
which  different  bodies,  after  hitting  together, 
come  apart  again,  depends  on  the  constitution 
of  those  bodies.  The  earlier  hypothesis  about 
the  constitution  of  a  gas  supposed  that  the 
particles  of  them  came  apart  according  to  the 
same  law  that  billiard  balls  do  ;  but  that  hypo- 
thesis, although  it  was  found  to  explain  a  great 
number  of  phenomena,  did  not  explain  them 
all.  And  it  was  Professor  Clerk  Maxwell  again 
who  found  the  hypothesis  which  does  explain 
all  the  rest  of  the  phenomena.  He  found  that 
particles  when  they  come  together  separate  as 
if  they  repelled  one  another,  or  pushed  one 
another  away  ;  and  as  if  they  did  that  much 
more  strongly  when  close  together  than  when 
further  apart  You  know  that  what  is  called 
the  great  law  of  gravitation  asserts  that  all 
bodies  pull  one  another  together  according  to  a 
certain  rule,  and  that  they  pull  one  another 
more  when  close  than  when  further  apart.  Now 
that  law  differs  from  the  law  which  Clerk 
Maxwell  found  out  as  affecting  the  repulsion  of 
gaseous  particles.  The  law  of  attraction  of 
gravitation  is  this  ;  that  when  you  halve  the 
distance,  you  have  to  multiply  the  attraction 
four  times — twice  two  make  four.  If  you  divide 
the  distance  into  three,  you  must  multiply  the 


ATOMS  207 

attraction  nine  times — three  times  three  are 
nine.  Now  in  the  case  of  atomic  repulsion 
you  have  got  to  multiply  not  twice  two,  or  three 
times  three,  but  five  twos  together — which 
multiplied  make  32.  If  you  halve  the  distance 
between  two  particles  you  increase  the  repulsion 
32  times.  So  also  five  threes  multiplied  to- 
gether make  243  ;  and  if  you  divide  the  dis- 
tance between  two  particles  by  three,  then  you 
increase  the  repulsion  by  243.  So  you  see  the 
repulsion  increases  with  enormous  rapidity  as 
the  distance  diminishes.  That  law  is  expressed 
by  saying  that  the  repulsion  of  two  gases  is 
inversely  as  the  fifth  power  of  the  distance. 
But  I  must  warn  you  against  supposing  that 
that  law  is  established  in  the  same  sense  that 
these  other  statements  that  we  have  been  mak- 
ing are  established.  That  law  is  true  provided 
that  there  is  a  repulsion  between  two  gaseous 
particles,  and  that  it  varies  as  a  power  of  the 
distance  ;  it  is  proved  that  if  there  is  any  law 
of  repulsion,  and  if  the  law  is  that  it  varies  as 
some  power  of  the  distance,  then  that  power 
cannot  be  any  other  than  the  fifth.  It  has 
not  been  shown  that  the  action  between  the 
two  particles  is  not  something  perhaps  more 
complicated  than  this,  but  which  on  the  average 
produces  the  same  results.  But  still  the  state- 
ment that  the  action  of  gaseous  molecules  upon 
one  another  can  be  entirely  explained  by  the 


208  LECTURES  AND  ESSAYS 

assumption  of  a  law  like  that,  is  the  newest 
statement  in  physics  since  the  law  of  gravitation 
was  discovered.  You  know  that  there  are 
other  actions  of  matter  which  apparently  take 
place  through  intervening  spaces  and  which 
always  follow  the  same  law  as  gravitation,  such 
as  the  attraction  or  repulsion  of  magnetical  or 
electrical  particles  :  those  follow  the  same  law 
as  gravitation.  But  here  is  a  law  of  repulsion 
which  follows  a  different  law  from  that  of 
gravitation,  and  in  that  lies  the  extreme  interest 
of  Professor  Clerk  Maxwell's  investigation. 

The  next  thing  that  I  want  to  give  you 
reasoning  for  is  again  rather  a  hard  thing  in 
respect  of  the  reasoning,  but  the  fact  is  an 
extremely  simple  and  beautiful  one.  It  is  this. 
Suppose  I  have  two  vessels,  say  cylinders,  with 
stoppers  which  do  not  fit  upon  the  top  of  the 
vessel,  but  slide  up  and  down  inside  and  yet  fit 
exactly.  These  two  vessels  are  of  exactly  the 
same  size  ;  one  of  them  contains  hydrogen  and 
the  other  contains  oxygen.  They  are  to  be  of 
the  same  temperature  and  pressure,  that  is  to 
say  they  will  bear  exactly  the  same  weight  on 
the  top.  Very  well,  these  two  vessels  having 
equal  volumes  of  gas  of  the  same  pressure  and 
temperature  will  contain  just  the  same  number 
of  atoms  in  each,  only  the  atoms  of  oxygen 
will  be  heavier  than  the  atoms  of  hydrogen. 
Now  how  is  it  that  we  arrive  at  that  result  ?  I 


ATOMS  209 

shall  endeavour  to  explain  the  process  of 
reasoning.  Boyle  discovered  a  law  about  the 
dependence  of  the  pressure  of  a  gas  upon  its 
volume  which  showed  that  if  you  squeezed  a 
gas  into  a  smaller  space  it  will  press  so  much 
the  more  as  the  space  has  been  diminished.  If 
the  space  has  been  diminished  one-half,  then 
the  pressure  is  doubled  ;  if  the  space  is  dimin- 
ished to  one-third,  then  the  pressure  is  increased 
to  three  times  what  it  was  before.  This  holds 
for  a  varying  volume  of  the  same  gas.  That 
same  law  would  tell  us  that  if  we  put  twice  the 
quantity  of  gas  into  the  same  space,  we  should 
get  twice  the  amount  of  pressure.  Dalton 
made  a  new  statement  of  that  law,  which  ex- 
presses it  in  this  form,  that  when  you  put  more 
gas  into  a  vessel  which  already  contains  gas, 
the  pressure  that  you  get  is  the  sum  of  the  two 
pressures  which  would  be  got  from  the  two 
gases  separately.  You  will  see  directly  that 
that  is  equivalent  to  the  other  law.  But  the 
importance  of  Dalton's  statement  of  the  law  is 
this,  that  it  enabled  the  law  to  be  extended 
from  the  case  of  the  same  gas  to  the  case  of 
two  different  gases.  If  instead  of  putting  a 
pint  of  oxygen  into  a  vessel  already  containing 
a  pint,  I  were  to  put  in  a  pint  of  nitrogen,  I 
should  equally  get  a  double  pressure.  The 
oxygen  and  nitrogen,  when  mixed  together, 
would  exert  the  sum  of  the  pressures  upon  the 
VOL.  I  P 


2io  LECTURES  AND  ESSAYS 

vessel  that  the  oxygen  and  nitrogen  would 
exert  separately.  Now  the  explanation  of  that 
pressure  is  this.  The  pressure  of  the  gas  upon 
the  sides  of  the  vessel  is  due  to  the  impact  of 
these  small  particles  which  are  constantly  flying 
about  and  impinging  upon  the  sides  of  the 
vessel.  It  is  first  of  all  shown  mathematically 
that  the  effect  of  that  impinging  would  be  the 
same  as  the  pressure  of  the  gas.  But  the 
amount  of  the  pressure  could  be  found  if  we 
knew  how  many  particles  there  were  in  a  given 
space,  and  what  was  the  effect  of  each  one 
when  it  impinged  on  the  sides  of  the  vessel. 
You  see  directly  why  it  is  that  putting  twice  as 
many  particles,  which  are  going  at  the  same 
rate,  into  the  same  vessel,  we  should  get  twice 
the  effect.  Although  there  are  just  twice  as 
many  particles  to  hit  the  sides  of  the  vessel, 
they  are  apparently  stopped  by  each  other 
when  they  bound  off.  But  the  effect  of  there 
being  more  particles  is  to  make  them  come 
back  quicker  ;  so  that  altogether  the  number  of 
impacts  upon  the  sides  of  the  vessel  is  just 
doubled  when  you  double  the  number  of  par- 
ticles. Supposing  we  have  got  a  cubic  inch 
of  space,  then  the  amount  of  pressure  upon  the 
side  of  that  cubic  inch  depends  upon  the  num- 
ber of  particles  inside  the  cube,  and  upon  the 
energy  with  which  each  one  of  them  strikes 
against  the  sides  of  the  vessel. 


ATOMS  2ii 

Again  there  is  a  law  which  connects  together 
the  pressure  of  a  gas  and  its  temperature.  It 
is  found  that  there  is  a  certain  absolute  zero  of 
temperature,  and  that  if  you  reckon  your 
temperature  from  that,  then  the  pressure  of  the 
gas  is  directly  proportional  to  the  temperature, 
that  twice  the  temperature  will  give  twice  the 
pressure  of  the  same  gas,  and  three  times  the 
temperature  will  give  three  times  the  pressure 
of  the  same  gas. 

Well  now  we  have  just  got  to  remember 
these  two  rules — the  law  of  Boyle,  as  expressed 
by  Dalton,  connecting  together  the  pressure  of 
a  gas  and  its  volume,  and  this  law  which  con- 
nects together  the  pressure  with  the  absolute 
temperature.  You  must  remember  that  it 
has  been  calculated  by  mathematics  that  the 
pressure  upon  one  side  of  a  vessel  of  a  cubic 
inch  has  been  got  by  multiplying  together  the 
number  of  particles  into  the  energy  with  which 
each  of  them  strikes  against  the  side  of  the 
vessel.  If  we  keep  that  same  gas  in  a  vessel 
and  alter  its  temperature,  then  we  find  that  the 
pressure  is  proportional  to  the  temperature  ;  but 
since  the  number  of  molecules  remains  the  same 
when  we  double  the  pressure,  we  must  alter  that 
other  factor  in  the  pressure,  we  must  double 
the  energy  with  which  each  of  the  particles 
attacks  the  side  of  the  vessel.  That  is  to  say, 
when  we  double  the  temperature  of  the  gas  we 


212  LECTURES  AND  ESSAYS 

double  the  energy  of  each  particle  ;  consequently 
the  temperature  of  the  gas  is  proportional  always 
to  the  energy  of  its  particles.  That  is  the  case 
with  a  single  gas.  If  we  mix  two  gases,  what 
happens?  They  come  to  exactly  the  same 
temperature.  It  is  calculated  also  by  mathe- 
matics that  the  particles  of  one  gas  have  the 
same  effect  as  those  of  the  other  ;  that  is,  the 
light  particles  go  faster  to  make  up  for  their 
want  of  weight.  If  you  mix  oxygen  and 
hydrogen,  you  find  that  the  particles  of  hydrogen 
go  four  times  as  fast  as  the  particles  of  oxygen. 
Now  we  have  here  a  mathematical  statement 
— that  when  two  gases  are  mixed  together,  the 
energy  of  the  two  particles  is  the  same  ;  and 
with  any  one  gas  considered  by  itself  that  energy 
is  proportional  to  the  temperature.  Also  when 
two  gases  are  mixed  together  the  two  tempera- 
tures become  equal.  If  you  think  over  that  a 
little,  you  will  see  that  it  proves  that  whether 
we  take  the  same  gas  or  different  gases,  the 
energy  of  the  single  particles  is  always  pro- 
portional to  the  temperature  of  the  gas. 

What  follows  ?  If  I  have  two  vessels  con- 
taining gas  at  the  same  pressure  and  the  same 
temperature  (suppose  that  hydrogen  is  in  one 
and  oxygen  in  the  other),  then  I  know  that 
the  temperature  of  the  hydrogen  is  the  same  as 
the  temperature  of  the  oxygen,  and  that  the 
pressure  of  the  hydrogen  is  the  same  as  the 


ATOMS  213 

pressure  of  the  oxygen.  I  also  know  (because 
the  temperatures  are  equal)  that  the  average 
energy  of  a  particle  of  the  hydrogen  is  the  same 
as  that  of  a  particle  of  the  oxygen.  Now  the 
pressure  is  made  up  by  multiplying  the  energy 
by  the  number  of  particles  in  both  gases  ;  and 
as  the  pressure  in  both  cases  is  the  same, 
therefore  the  number  of  particles  is  the  same. 
That  is  the  reasoning ;  I  am  afraid  it  will  seem 
rather  complicated  at  first  hearing,  but  it  is  this 
sort  of  reasoning  which  establishes  the  fact  that 
in  two  equal  volumes  of  different  gases  at  the 
same  temperature  and  pressure,  the  number  of 
particles  is  the  same. 

Now  there  is  an  exceedingly  interesting 
conclusion  which  was  arrived  at  very  early  in 
the  theory  of  gases,  and  calculated  by  Mr. 
Joule.  It  is  found  that  the  pressure  of  a  gas 
upon  the  sides  of  a  vessel  may  be  represented 
quite  fairly  in  this  way.  Let  us  divide  the 
particles  of  gas  into  three  companies  or  bands. 
Suppose  I  have  a  cubical  vessel  in  which  one 
of  these  companies  is  to  go  forward  and  back- 
ward, another  right  and  left,  and  the  other  to 
go  up  and  down.  If  we  make  those  three 
companies  of  particles  to  go  in  their  several 
directions,  then  the  effect  upon  the  sides  of  the 
vessel  will  not  be  altered  ;  there  will  be  the 
same  impact  and  pressure.  It  was  also  found 
out  that  the  effect  of  this  pressure  would  not  be 


214  LECTURES  AND  ESSAYS 

altered  if  we  combined  together  all  the  particles 
forming  one  company  into  one  mass,  and  made 
them  impinge  with  the  same  velocity  upon  the 
sides  of  the  vessel.  The  effect  of  the  pressure 
would  be  just  the  same.  Now  we  know  what 
the  weight  of  a  gas  is,  and  we  know  what  the 
pressure  is  that  it  produces,  and  we  want  to 
find  the  velocity  it  is  moving  at  on  the  average. 
We  can  find  out  at  what  velocity  a  certain 
weight  has  to  move  in  order  to  produce  a 
certain  definite  impact  Therefore  we  have 
merely  to  take  the  weight  of  the  gas,  divide  it 
by  three,  and  to  find  how  fast  that  has  to  move 
in  order  to  produce  the  pressure,  and  that  will 
give  us  the  average  rate  at  which  the  gas  is 
moving.  By  that  means  Mr.  Joule  calculated 
that  in  air  of  ordinary  temperature  and  pressure 
the  velocity  is  about  500  metres  per  second, 
nearly  five  miles  in  sixteen  seconds,  or  nearly 
twenty  miles  a  minute — about  sixty  times  the 
rate  of  an  ordinary  train. 

The  average  velocity  of  the  particles  of  gas 
is  about  i^  times  as  great  as  the  velocity  of 
sound.  You  can  easily  remember  the  velocity  of 
sound  in  air  at  freezing  point — it  is  3  3  3  metres 
per  second  ;  so  that  about  i \  times,  really 
1.432  of  that  would  be  the  average  velocity  of 
a  particle  of  air.  At  the  ordinary  temperature 
— 60  degrees  Fahrenheit — the  velocity  would, 
of  course,  be  greater. 


ATOMS  215 

Let  us  consider  how  much  we  have  estab- 
lished so  far  about  these  small  particles  of  which 
we  find  that  the  gas  consists.  We  have  so  far 
been  treating  mainly  of  gases.  We  find  that  a 
gas,  such  as  the  air  in  this  room,  consists  of 
small  particles,  which  are  separate  with  spaces 
between  them.  They  are  as  a  matter  of  fact 
of  two  different  types,  oxygen  and  nitrogen. 
All  the  particles  of  oxygen  contain  the  same 
structure,  and  the  rates  of  internal  vibration  are 
the  same  for  all  these  particles.  It  is  also 
compounded  of  particles  of  nitrogen  which  have 
different  rates  of  internal  vibration.  We  have 
shown  that  these  particles  are  moving  about 
constantly.  We  have  shown  that  they  impinge 
against  and  interfere  with  one  another's  motion  ; 
and  we  have  shown  that  they  come  apart  again. 
We  have  shown  that  in  vessels  of  the  same  size 
containing  two  different  gases  of  the  same 
pressure  and  temperature  there  is  the  same 
number  of  those  two  different  sorts  of  particles. 
We  have  shown  also  that  the  average  velocity 
of  these  particles  in  the  air  of  this  room  is  about 
twenty  miles  a  minute. 

There  is  one  other  point  of  very  great 
interest  to  which  I  want  to  call  your  attention. 
The  word  "  atom,"  as  you  know,  has  a  Greek 
origin  ;  it  means  that  which  is  not  divided. 
Various  people  have  given  it  the  meaning  of 
that  which  cannot  be  divided  ;  but  if  there  is 


2i6  LECTURES  AND  ESSAYS 

anything  which  cannot  be  divided  we  do  not 
know  it,  because  we  know  nothing  about 
possibilities  or  impossibilities,  only  about  what 
has  or  has  not  taken  place.  Let  us  then  take 
the  word  in  the  sense  in  which  it  can  be  applied 
to  a  scientific  investigation.  An  atom  means 
something  which  is  not  divided  in  certain  cases 
that  we  are  considering.  Now  these  atoms  I 
have  been  talking  about  may  be  called  physical 
atoms,  because  they  are  not  divided  under  those 
circumstances  that  are  considered  in  physics. 
These  atoms  are  not  divided  under  the  ordinary 
alteration  of  temperature  and  pressure  of  gas, 
and  variation  of  heat ;  they  are  not  in  general 
divided  by  the  application  of  electricity  to  the 
gas,  unless  the  stream  is  very  strong.  But 
there  is  a  science  which  deals  with  operations 
by  which  these  atoms  which  we  have  been 
considering  can  be  divided  into  two  parts,  and 
in  which  therefore  they  are  no  longer  atoms. 
That  science  is  chemistry.  The  chemist  there- 
fore will  not  consent  to  call  these  little  particles 
that  we  are  speaking  of  by  the  name  of  atoms, 
because  he  knows  that  there  are  certain  processes 
to  which  he  can  subject  them  which  will  divide 
them  into  parts,  and  then  they  cease  to  be 
things  which  have  not  been  divided.  I  will  give 
you  an  instance  of  that.  The  atoms  of  oxygen 
which  exist  in  enormous  numbers  in  this  room 
consist  of  two  portions,  which  are  of  exactly 


ATOMS  217 

the  same  structure.  Every  molecule,  as  the 
chemist  would  call  it,  travelling  in  this  room, 
is  made  up  of  two  portions  which  are  exactly 
alike  in  their  structure.  It  is  a  complicated 
structure  ;  but  that  structure  is  double.  It  is 
like  the  human  body — one  side  is  like  the  other 
side.  How  do  we  know  that  ?  We  know  it  in 
this  way.  Suppose  that  I  take  a  vessel  which 
is  divided  into  two  parts  by  a  division  which  I 
can  take  away.  One  of  these  parts  is  twice  as 
large  as  the  other  part,  and  will  contain  twice 
as  much  gas.  Into  that  part  which  is  twice  as 
big  as  the  other  I  put  hydrogen  ;  into  the  other 
I  put  oxygen.  Suppose  that  one  contains  a 
quart  and  the  other  a  pint ;  then  I  have  a  quart 
of  hydrogen  and  a  pint  of  oxygen  in  this  vessel. 
Now  I  will  take  away  the  division  so  that  they 
can  permeate  one  another,  and  then  if  the  vessel 
is  strong  enough  I  pass  an  electric  spark 
through  them.  The  result  will  be  an  explosion 
inside  the  vessel ;  it  will  not  break  if  it  is 
strong  enough  ;  but  the  quart  of  hydrogen  and 
the  pint  of  oxygen  will  be  converted  into  steam  ; 
they  will  combine  together  to  form  steam.  If 
I  choose  to  cool  down  that  steam  until  it  is 
just  as  hot  as  the  two  gases  were  before  I  passed 
the  electric  spark  through  them,  then  I  shall 
find  that  at  the  same  pressure  there  will  only 
be  a  quart  of  steam.  Now  let  us  remember 
what  it  was  that  we  established  about  two  equal 


2i8  LECTURES  AND  ESSAYS 

volumes  of  different  gases  at  the  same  tempera- 
ture and  pressure.  First  of  all,  we  had  a  quart 
of  hydrogen  with  a  pint  of  oxygen.  We  know 
that  that  quart  of  hydrogen  contains  twice  as 
many  hydrogen  molecules  as  the  pint  of  oxygen 
contains  of  oxygen  molecules.  Let  us  take 
particular  numbers.  Suppose  instead  of  a 
quart  or  a  pint  we  take  a  smaller  quantity, 
and  say  that  there  are  100  hydrogen  and  50 
oxygen  molecules.  Well,  after  the  cooling  has 
taken  place,  I  should  find  a  volume  of  steam 
which  was  equal  to  the  volume  of  hydrogen, 
that  is,  I  should  find  100  steam  molecules. 
Now  these  steam  molecules  are  made  up  of 
hydrogen  and  oxygen  molecules.  I  have  got 
therefore  100  things  which  are  all  exactly  alike, 
made  up  of  100  things  and  50  things — 100 
hydrogen  and  50  oxygen,  making  100  steam 
molecules.  Now  since  the  I  oo  steam  molecules 
are  exactly  alike,  we  have  those  50  oxygen 
molecules  distributed  over  the  whole  of  these 
steam  molecules.  Therefore,  unless  the  oxygen 
contains  something  which  is  common  to  the 
hydrogen  also,  it  is  clear  that  each  of  those  50 
molecules  of  oxygen  must  have  been  divided 
into  two,  because  you  cannot  put  50  horses  into 
100  stables,  so  that  there  shall  be  exactly  the 
same  amount  of  horse  in  each  stable  ;  but  you 
can  divide  50  pairs  of  horses  among  100  stables. 
There  we  have  the  supposition  that  there  is 


ATOMS  219 

nothing  common  to  the  oxygen  and  hydrogen, 
that  there  is  no  structure  that  belongs  to  each 
of  them.  Now  that  supposition  is  made  by 
a  great  majority  of  chemists.  Sir  Benjamin 
Brodie,  however,  has  made  a  supposition  that 
there  is  a  structure  in  hydrogen  which  is  also 
common  to  certain  other  elements.  He  has 
himself,  for  particular  reasons,  restricted  that 
supposition  to  the  belief  that  hydrogen  is 
contained  as  a  whole  in  many  of  the  other 
elements.  Let  us  make  that  further  supposition 
and  it  will  not  alter  our  case  at  all.  We  have 
then  100  hydrogen  and  50  oxygen  molecules, 
but  there  is  something  common  to  the  two. 
Well,  this  something  we  will  call  X.  Of  this 
we  have  to  make  100  equal  portions.  Now 
that  cannot  be  the  case  unless  that  structure 
occurred  twice  as  often  in  each  molecule  of 
oxygen  as  in  each  molecule  of  hydrogen. 
Consequently,  whether  the  oxygen  molecule 
contains  something  common  to  hydrogen  or 
not,  it  is  equally  true  that  the  oxygen  molecule 
must  contain  the  same  thing  repeated  twice 
over ;  it  must  be  divisible  into  two  parts  which 
are  exactly  alike. 

Similar  reasoning  applies  to  a  great  number 
of  other  elements  ;  to  all  those  which  are  said 
to  have  an  even  number  of  atomicities.  But 
with  regard  to  those  which  are  said  to  have  an 
odd  number,  although  many  of  these  also  are 


aao  LECTURES  AND  ESSAYS 

supposed  to  be  double,  yet  the  evidence  in 
favour  of  that  supposition  is  of  a  different 
kind  ;  and  we  must  regard  the  supposition  as 
still  a  theory  and  not  yet  a  demonstrated  fact. 

Now  I  have  spoken  so  far  only  of  gases. 
I  must  for  one  or  two  moments  refer  to  some 
calculations  of  Sir  William  Thomson,  which 
are  of  exceeding  interest  as  showing  us  what 
is  the  proximity  of  the  molecules  in  liquids 
and  in  solids.  By  four  different  modes  of 
argument  derived  from  different  parts  of  science, 
and  pointing  mainly  to  the  same  conclusion, 
he  has  shown  that  the  distance  between  two 
molecules  in  a  drop  of  water  is  such  that  there 
are  between  five  hundred  millions  and  five 
thousand  millions  of  them  in  an  inch.  He 
expresses  that  result  in  this  way — that  if  you 
were  to  magnify  a  drop  of  water  to  the  size  of 
the  earth,  then  the  coarseness  of  the  graining 
of  it  would  be  something  between  that  of 
cricket -balls  and  small  shot.  Or  we  may 
express  it  in  this  rather  striking  way.  You 
know  that  the  best  microscopes  can  be  made 
to  magnify  from  6000  to  8000  times.  A 
microscope  which  would  magnify  that  result 
as  much  again  would  show  the  molecular 
structure  of  water. 

There  is  another  scientific  theory  analogous 
to  this  one  which  leads  us  to  hope  that  some 
time  we  shall  know  more  about  these  molecules. 


ATOMS  221 

You  know  that  since  the  time  that  we  have 
known  all  about  the  motions  of  the  solar 
system,  people  have  speculated  about  the  origin 
of  it ;  and  a  theory  started  by  Laplace  and 
worked  out  by  other  people  has,  like  the  theory 
of  luminiferous  ether,  been  taken  out  of  the 
rank  of  hypothesis  into  that  of  fact.  We  know 
the  rough  outlines  of  the  history  of  the  solar 
system,  and  there  are  hopes  that  when  we 
know  the  structure  and  properties  of  a  molecule, 
what  its  internal  motions  are  and  what  are  the 
parts  and  shape  of  it,  somebody  may  be  able 
to  form  a  theory  as  to  how  that  was  built  up 
and  what  it  was  built  out  of.  It  is  obvious 
that  until  we  know  the  shape  and  structure  of 
it,  nobody  will  be  able  to  form  such  a  theory. 
But  we  can  look  forward  to  the  time  when  the 
structure  and  motions  in  the  inside  of  a  molecule 
will  be  so  well  known  that  some  future  Kant 
or  Laplace  will  be  able  to  make  an  hypothesis 
about  the  history  and  formation  of  matter.1 

1  The  mathematical  development  of  this  subject  is  due  to 
Clausius  and  Maxwell.  Reference  to  the  chief  papers  will  be 
found  at  the  beginning  of  Maxwell's  Memoir,  ' '  On  the  Dynamical 
Theory  of  Gases,"  Phil.  Trans.  1867. 


THE  FIRST  AND  THE  LAST 
CATASTROPHE1 

A  CRITICISM  ON  SOME  RECENT  SPECULATIONS 
ABOUT  THE  DURATION   OF  THE  UNIVERSE 

I  PROPOSE  in  this  lecture  to  consider  specula- 
tions of  quite  recent  days  about  the  beginning 
and  the  end  of  the  world.  The  world  is  a 
very  interesting  thing,  and  I  suppose  that  from 
the  earliest  times  that  men  began  to  form  any 
coherent  idea  of  it  at  all,  they  began  to  guess 
in  some  way  or  other  how  it  was  that  it  all 
began,  and  how  it  was  all  going  to  end.  But 
there  is  one  peculiarity  about  these  speculations 
which  I  wish  now  to  consider,  that  makes  them 
quite  different  from  the  early  guesses  of  which 
we  read  in  many  ancient  books.  These  modern 
speculations  are  attempts  to  find  out  how  things 
began,  and  how  they  are  to  end,  by  consider- 
ation of  the  way  in  which  they  are  going  on 
now.  And  it  is  just  that  character  of  these 

1  Sunday  Lecture  Society.  April  12,  1874  ;  afterwards  revised 
for  publication. 


FIRST  AND  LAST  CATASTROPHE  223 

speculations  that  gives  them  their  interest  for 
you  and  for  me  ;  for  we  have  only  to  consider 
these  questions  from  the  scientific  point  of  view. 
By  the  scientific  point  of  view  I  mean  one 
which  attempts  to  apply  past  experience  to 
new  circumstances  according  to  an  observed 
order  of  nature.  So  that  we  shall  only  con- 
sider the  way  in  which  things  began,  and  the 
way  in  which  they  are  to  end,  in  so  far  as  we 
seem  able  to  draw  inferences  about  the  questions 
from  facts  which  we  know  about  the  way  in 
which  things  are  going  on  now.  And,  in  fact, 
the  great  interest  of  the  subject  to  me  lies  in 
the  amount  of  illustration  which  it  offers  of  the 
degree  of  knowledge  which  we  have  now 
attained  of  the  way  in  which  the  universe  is 
going  on. 

The  first  of  these  speculations  is  one  set 
forth  by  Professor  Clerk  Maxwell,  in  a  lecture  on 
Molecules  delivered  before  the  British  Associa- 
tion at  Bradford.  Now,  this  argument  of  his 
which  he  put  before  the  British  Association  at 
Bradford  depends  entirely  upon  the  modern 
theory  of  the  molecular  constitution  of  matter. 
I  think  this  the  more  important,  because  a  great 
number  of  people  appear  to  have  been  led  to 
the  conclusion  that  this  theory  is  very  similar 
to  the  guesses  which  we  find  in  ancient  writers 
— Democritus  and  Lucretius.  It  so  happens 
that  these  ancient  writers  did  hold  a  view  of 


224  LECTURES  AND  ESSAYS 

the  constitution  of  things  which  in  many  strik- 
ing respects  agrees  with  the  view  which  we  hold 
in  modern  times.  This  parallelism  has  been 
brought  recently  before  the  public  by  Professor 
Tyndall  in  his  excellent  address  at  Belfast. 
And  it  is  perhaps  on  account  of  the  parallelism, 
which  he  pointed  out  at  that  place,  between 
the  theories  held  amongst  the  ancients  and  the 
theory  held  amongst  the  moderns,  that  many 
people  who  are  acquainted  with  classic  literature 
have  thought  that  a  knowledge  of  the  views  of 
Democritus  and  Lucretius  would  enable  them 
to  understand  and  criticise  the  modern  theory 
of  matter.  That,  however,  is  a  mistake.  The 
difference  between  the  two  is  mainly  this  :  the 
atomic  theory  of  Democritus  was  a  guess,  and 
no  more  than  a  guess.  Everybody  around 
him  was  guessing  about  the  origin  of  things, 
and  they  guessed  in  a  great  number  of  ways  ; 
but  he  happened  to  make  a  guess  which  was 
more  near  the  right  thing  than  any  of  the 
others.  This  view  was  right  in  its  main 
hypothesis — that  all  things  are  made  up  of 
elementary  parts,  and  that  the  different 
properties  of  different  things  depend  rather 
upon  difference  of  arrangement  than  upon 
ultimate  difference  in  the  substance  of  which 
they  are  composed.  Although  this  was  con- 
tained in  the  atomic  theory  of  Democritus,  as 
expounded  by  Lucretius,  yet  it  will  be  found 


FIRST  AND  LAST  CATASTROPHE  225 

by  any  one  who  examines  further  the  con- 
sequences which  are  drawn  from  it  that  it  very 
soon  diverges  from  the  truth  of  things,  as  we 
might  naturally  expect  it  would.  On  the 
contrary,  the  view  of  the  constitution  of  matter 
which  is  held  by  scientific  men  in  the  present 
day  is  not  a  guess  at  all. 

In  the  first  place  I  will  endeavour  to  explain 
what  are  the  main  points  in  this  theory.  First 
of  all  we  must  take  the  simplest  form  of  matter, 
which  turns  out  to  be  a  gas — such,  for  example, 
as  the  air  in  this  room.  The  belief  of  scientific 
men  in  the  present  day  is  that  this  air  is  not 
a  continuous  thing,  that  it  does  not  fill  the 
whole  of  the  space  in  the  room,  but  is  made  up 
of  an  enormous  number  of  exceedingly  small 
particles.  There  are  two  sorts  of  particles  : 
one  sort  of  particle  is  oxygen,  and  another  sort 
of  particle  nitrogen.  All  the  particles  of 
oxygen  are  as  near  as  possible  alike  in  these 
two  respects ;  first  in  weight,  and  secondly  in 
certain  peculiarities  of  mechanical  structure. 
These  small  molecules  are  not  at  rest  in  the 
room,  but  are  flying  about  in  all  directions  with 
a  mean  velocity  of  seventeen  miles  a  minute. 
They  do  not  fly  far  in  one  direction  ;  but  any 
particular  molecule,  after  going  over  an  in- 
credibly short  distance — the  measure  of  which 
has  been  made — meets  another,  not  exactly 
plump,  but  a  little  on  one  side,  so  that  they 
VOL.  I  Q 


226  LECTURES  AND  ESSAYS 

behave  to  one  another  somewhat  in  the  same 
way  as  two  people  do  who  are  dancing  Sir 
Roger  de  Coverley;  they  join  hands,  swing 
round,  and  then  fly  away  in  different  directions. 
All  these  molecules  are  constantly  changing 
the  direction  of  each  other's  motion  ;  they  are 
flying  about  with  very  different  velocities, 
although,  as  I  have  said,  their  mean  velocity 
is  about  seventeen  miles  a  minute.  If  the 
velocities  were  all  marked  off  on  a  scale,  they 
would  be  found  distributed  about  the  mean 
velocity  just  as  shots  are  distributed  about  a 
mark.  If  a  great  many  shots  are  fired  at  a 
target,  the  hits  will  be  found  thickest  at  the 
bull's-eye,  and  they  will  gradually  diminish  as 
we  go  away  from  that,  according  to  a  certain 
law  which  is  called  the  law  of  error.  It  was 
first  stated  clearly  by  Laplace ;  and  it  is  one 
of  the  most  remarkable  consequences  of  theory 
that  the  molecules  of  a  gas  have  their  velocities 
distributed  amongst  them  precisely  according 
to  this  law  of  error.  In  the  case  of  a  liquid, 
it  is  believed  that  the  state  of  things  is  quite 
different.  We  said  that  in  the  gas  the  mole- 
cules are  moved  in  straight  lines,  and  that  it  is 
only  during  a  small  portion  of  their  motion 
that  they  are  deflected  by  other  molecules ;  but  in 
a  liquid  we  may  say  that  the  molecules  go  about 
as  if  they  were  dancing  the  grand  chain  in  the 
Lancers.  Every  molecule  after  parting  com- 


FIRST  AND  LAST  CATASTROPHE  227 

pany  with  one  finds  another,  and  so  is  constantly 
going  about  in  a  curved  path,  and  never  sent 
quite  clear  away  from  the  sphere  of  action  of 
the  surrounding  molecules.  But,  notwithstand- 
ing that,  all  molecules  in  a  liquid  are  constantly 
changing  their  places,  and  it  is  for  that  reason 
that  diffusion  takes  place  in  the  liquid.  Take 
a  large  tank  of  water  and  drop  a  little  iodine 
into  it,  and  you  will  find  after  a  certain  time 
all  the  water  turned  slightly  blue.  That  is 
because  all  the  iodine  molecules  have  changed 
like  the  others  and  spread  themselves  over  the 
whole  of  the  tank.  Because,  however,  you 
cannot  see  this,  except  where  you  use  different 
colours,  you  must  not  suppose  that  it  does  not 
take  place  where  the  colours  are  the  same. 
In  every  liquid  all  the  molecules  are  running 
about  and  continually  changing  and  mixing 
themselves  up  in  fresh  forms.  In  the  case  of 
a  solid  quite  a  different  thing  takes  place.  In 
a  solid  every  molecule  has  a  place  which  it 
keeps  ;  that  is  to  say,  it  is  not  at  rest  any  more 
than  a  molecule  of  a  liquid  or  a  gas,  but  it  has 
a  certain  mean  position  which  it  is  always 
vibrating  about  and  keeping  fairly  near  to,  and 
it  is  kept  from  losing  that  position  by  the 
action  of  the  surrounding  molecules.  These 
are  the  main  points  of  the  theory  of  the  con- 
stitution of  matter  as  at  present  believed.  v^I*;? 
It  differs  from  the  theory  of  Democritus  in 


228  LECTURES  AND  ESSAYS 

this  way.  There  is  no  doubt  that  in  the  first 
origin  of  it,  when  it  was  suggested  to  the 
mind  of  Daniel  Bernouilli  as  an  explanation 
of  the  pressure  of  gases,  and  to  the  mind 
of  Dalton  as  an  explanation  of  chemical 
reactions,  it  was  a  guess  ;  that  is  to  say,  it 
was  a  supposition  which  would  explain  these 
facts  of  physics  and  chemistry,  but  which 
was  not  known  to  be  true.  Some  theories 
are  still  in  that  position  ;  other  theories 
are  known  to  be  true,  because  they  can  be 
argued  back  to  from  the  facts.  In  order  to 
make  out  that  your  supposition  is  true,  it  is 
necessary  to  show,  not  merely  that  that  parti- 
cular supposition  will  explain  the  facts,  but  also 
that  no  other  one  will.  Now,  by  the  efforts  of 
Clausius  and  Clerk  Maxwell,  the  molecular 
theory  of  matter  has  been  put  in  this  other 
position.  Namely,  instead  of  saying,  Let  us 
suppose  such  and  such  things  are  true, — and 
then  deducing  from  that  supposition  what  the 
consequences  ought  to  be,  and  showing  that 
these  consequences  are  just  the  facts  which  we 
observe — instead  of  doing  that,  I  say,  we  make 
certain  experiments  ;  we  show  that  certain  facts 
are  undoubtedly  true,  and  from  these  facts  we 
go  back  by  a  direct  chain  of  logical  reasoning, 
which  there  is  no  way  of  getting  out  of,  to  the 
statement  that  all  matter  is  made  up  of  separate 
pieces  or  molecules,  and  that  in  matter  of  a 


FIRST  AND  LAST  CATASTROPHE  229 

given  kind,  in  oxygen,  or  in  hydrogen,  or  in 
nitrogen,  these  molecules  are  of  very  nearly  the 
same  weight,  and  have  certain  mechanical  pro- 
perties which  are  common  to  all  of  them.  In 
order  to  show  you  something  of  the  kind  of 
evidence  for  that  statement,  I  must  mention 
another  theory  which,  as  it  seems  to  me,  is  in 
the  same  position  ;  namely,  the  doctrine  of  the 
luminiferous  ether,  or  that  wonderful  substance 
which  is  distributed  all  over  space,  and  which 
carries  light  and  radiant  heat.  By  means  of 
certain  experiments  upon  interference  of  light 
we  can  show,  not  by  any  hypothesis,  not  by 
any  guess  at  all,  but  by  a  pure  interpretation  of 
the  experiment — that  in  every  ray  of  light  there 
is  some  change  or  other,  whatever  it  is,  which 
is  periodic  in  time  and  in  space.  By  saying  it 
is  periodic  in  time,  I  mean  that,  at  a  given 
point  of  the  ray  of  light,  this  change  increases 
up  to  a  certain  instant,  then  decreases,  then 
increases  in  the  opposite  direction,  and  then 
decreases  again,  and  so  on  alternately.  That 
is  shown  by  experiments  of  interference  ;  it  is 
not  a  theory  which  will  explain  the  facts,  but 
it  is  a  fact  which  is  got  out  of  observation. 
By  saying  that  this  phenomenon  is  periodic  in 
space,  I  mean  that,  if  at  any  given  instant  you 
could  examine  the  ray  of  light,  you  would  find 
that  some  change  or  disturbance,  whatever  it 
is,  has  taken  place  all  along  it  in  different 


230  LECTURES  AND  ESSAYS 

degrees.  It  vanishes  at  certain  points,  and 
between  these  it  increases  gradually  to  a 
maximum  on  one  side  and  the  other  alternately. 
That  is  to  say,  in  travelling  along  a  ray  of  light 
there  is  a  certain  change  (which  can  be  observed 
by  experiments,  by  operating  upon  a  ray  of 
light  with  other  rays  of  light)  which  goes 
through  a  periodic  variation  in  amount.  The 
height  of  the  sea,  as  you  know  if  you  travel 
along  it,  goes  through  certain  periodic  changes  ; 
it  increases  and  decreases,  and  increases  and 
decreases  again  at  definite  intervals.  And  if 
you  take  the  case  of  waves  travelling  over  the 
sea,  and  place  yourself  at  a  given  point,  or 
mark  a  point  by  putting  a  cork  upon  the 
surface,  you  will  find  that  the  cork  will  rise  up 
and  down  ;  that  is  to  say,  there  will  be  a  change 
or  displacement  of  the  cork's  position,  which  is 
periodic  in  time,  which  increases  and  decreases, 
then  increases  in  the  opposite  direction,  and 
decreases  again.  Now  this  fact,  which  is 
established  by  experiment,  and  which  is  not  a 
guess  at  all — the  fact  that  light  is  a  phenomenon 
periodic  in  time  and  space — is  what  we  call  the 
wave  theory  of  light.  The  word  "  theory  "  here 
does  not  mean  a  guess  ;  it  means  an  organised 
account  of  the  facts,  such  that  from  it  you  may 
deduce  results  which  are  applicable  to  future 
experiments,  the  like  of  which  have  not  yet 
been  made.  But  we  can  see  more  than  this. 


FIRST  AND  LAST  CATASTROPHE  231 

So  far  we  say  that  light  consists  of  waves, 
merely  in  the  sense  that  it  consists  of  some 
phenomenon  or  other  which  is  periodic  in  time 
and  in  place  ;  but  we  know  that  a  ray  of  light 
or  heat  is  capable  of  doing  work.  Radiant 
heat,  for  example,  striking  on  a  body,  will 
warm  it  and  enable  it  to  do  work  by  expansion  ; 
therefore  this  periodic  phenomenon  which  takes 
place  in  the  ray  of  light  is  something  or  other 
which  possesses  mechanical  energy,  which  is 
capable  of  doing  work.  We  may  make  it,  if 
you  like,  a  mere  matter  of  definition,  and  say  : 
Any  change  which  possesses  energy  is  a  motion 
of  matter ;  and  this  is  perhaps  the  most  in- 
telligible definition  of  matter  that  we  can  frame. 
In  that  sense,  and  in  that  sense  only,  it  is  a 
matter  of  demonstration,  and  not  a  matter  of 
guess,  that  light  consists  of  the  periodic  motion 
of  matter,  of  something  which  is  between  the 
luminous  object  and  our  eyes. 

But  that  something  is  not  matter  in  the 
ordinary  sense  of  the  term  ;  it  is  not  made  up 
of  such  molecules  as  gases  and  liquids  and 
solids  are  made  up  of.  This  last  statement 
again  is  no  guess,  but  a  proved  fact.  There 
are  people  who  ask  :  Why  is  it  necessary  to 
suppose  a  luminiferous  ether  to  be  anything 
else  except  molecules  of  matter  in  space,  in 
order  to  carry  light  about  ?  The  answer  is  a 
very  simple  one.  In  order  that  separate  mole- 


232  LECTURES  AND  ESSAYS 

cules  may  carry  about  a  disturbance,  it  is 
necessary  that  they  should  travel  at  least  as 
fast  as  the  disturbance  travels.  Now  we  know, 
by  means  that  I  shall  afterwards  come  to,  that 
the  molecules  of  gas  travel  at  a  very  ordinary 
rate — about  twenty  times  as  fast  as  a  good 
train.  But,  on  the  contrary,  we  know  by  the 
most  certain  of  all  evidence,  by  five  or  six 
different  means,  that  the  velocity  of  light  is 
200,000  miles  a  second.  By  that  very  simple 
consideration  we  are  able  to  tell  that  it  is  quite 
impossible  for  light  to  be  carried  by  the  mole- 
cules of  ordinary  matter,  and  that  it  wants 
something  else  that  lies  between  those  mole- 
cules to  carry  the  light.  Now,  remembering 
the  evidence  which  we  have  for  the  existence 
of  this  ether,  let  us  consider  another  piece  of 
evidence  ;  let  us  now  consider  what  evidence 
we  have  that  the  molecules  of  a  gas  are 
separate  from  one  another  and  have  something 
between  them.  We  find  out,  by  experiment 
again,  that  the  different  colours  of  light  depend 
upon  the  various  rapidity  of  these  waves,  depend 
upon  the  size  and  upon  the  length  of  the  waves 
that  travel  through  the  ether,  and  that  when 
we  send  light  through  glass  or  any  transparent 
medium  except  a  vacuum,  the  waves  of  different 
lengths  travel  with  different  velocities.  That 
is  the  case  with  the  sea  ;  we  find  that  long 
waves  travel  faster  than  short  ones.  In  much 


FIRST  AND  LAST  CATASTROPHE  233 

the  same  way,  when  light  comes  out  of  a 
vacuum  and  impinges  upon  any  transparent 
medium,  say  upon  glass,  we  find  that  the  rate 
of  transmission  of  all  the  light  is  diminished  ; 
that  it  goes  slower  when  it  gets  inside  of  a 
material  body  ;  and  that  this  change  is  greater 
in  the  case  of  small  waves  than  of  large  ones. 
The  small  waves  correspond  to  blue  light,  and 
the  large  waves  correspond  to  red  light.  The 
waves  of  red  light  are  not  made  to  travel  so 
slowly  as  the  waves  of  blue  light ;  but,  as  in 
the  case  of  waves  travelling  over  the  sea,  when 
light  moves  in  the  interior  of  a  transparent 
body  the  largest  waves  travel  most  quickly. 
Well,  then,  by  using  such  a  body  as  will 
separate  out  the  different  colours — a  prism — 
we  are  able  to  affirm  what  are  the  constituents 
of  the  light  which  strikes  upon  it.  The  light 
that  comes  from  the  sun  is  made  up  of  waves 
of  various  lengths  ;  but,  making  it  pass  through 
a  prism,  we  can  separate  it  out  into  a  spectrum, 
and  in  that  way  we  find  a  band  of  light  instead 
of  a  spot  coming  from  the  sun,  and  to  every 
band  in  the  spectrum  corresponds  a  wave  of  a 
certain  definite  length  and  definite  time  in 
vibration.  Now  we  come  to  a  very  singular 
phenomenon.  If  you  take  a  gas  such  as 
chlorine  and  interpose  it  in  the  path  of  that 
light,  you  will  find  that  certain  particular  rays 
of  the  spectrum  are  absorbed,  while  others  are 


234  LECTURES  AND  ESSAYS 

not.  How  is  it  that  certain  particular  rates  of 
vibration  can  be  absorbed  by  this  chlorine  gas, 
while  others  are  not  ?  That  happens  in  this 
way — that  the  chlorine  gas  consists  of  a  great 
number  of  very  small  structures,  each  of  which 
is  capable  of  vibrating  internally.  Each  of 
these  structures  is  complicated,  and  is  capable 
of  a  change  of  relative  position  amongst  its 
parts  of  a  vibratory  character.  We  know  that 
molecules  are  capable  of  such  internal  vibrations 
— for  this  reason,  that  if  we  heat  any  solid  body 
sufficiently  it  will  in  time  give  out  light ;  that 
is  to  say,  the  molecules  are  got  into  such  a  state 
of  vibration  that  they  start  the  ether  vibrating, 
and  they  start  the  ether  vibrating  at  the  same 
rate  at  which  they  vibrate  themselves.  So  that 
what  we  learn  from  the  absorption  of  certain 
particular  rays  of  light  by  chlorine  gas  is  that 
the  molecules  of  that  gas  are  structures  which 
have  certain  natural  rates  of  vibration  which 
they  absorb,  precisely  those  rates  of  vibration 
which  belong  to  the  molecules  naturally.  If 
you  sing  a  certain  note  to  a  string  of  a  piano, 
that  string  if  in  tune  will  vibrate.  If,  therefore, 
a  screen  of  such  strings  were  put  across  a  room, 
and  you  sang  a  note  on  one  side,  a  person  on  the 
other  side  would  hear  the  note  very  weakly  or 
not  at  all,  because  it  would  be  absorbed  by  the 
strings  ;  but  if  you  sang  another  note,  not  one 
to  which  the  strings  naturally  vibrated,  then  it 


FIRST  AND  LAST  CATASTROPHE  235 

would  pass  through,  and  would  not  be  eaten 
up  by  setting  the  strings  vibrating.  Now  this 
question  arises.  Let  us  put  the  molecules  aside 
for  a  moment.  Suppose  we  do  not  know  of 
their  existence,  and  say :  Is  this  rate  of  vibra- 
tion which  naturally  belongs  to  the  gas  a  thing 
which  belongs  to  it  as  a  whole,  or  does  it  belong 
to  the  separate  parts  of  it  ?  You  might  suppose 
that  it  belongs  to  the  gas  as  a  whole.  A  jar 
of  water,  if  you  shake  it,  has  a  perfectly  definite 
time  in  which  it  oscillates,  and  that  is  very 
easily  measured.  That  time  of  oscillation 
belongs  to  the  jar  of  water  as  a  whole.  It 
depends  upon  the  weight  of  the  water  and  the 
shape  of  the  jar.  But  now,  by  a  very  certain 
method,  we  know  that  the  time  of  vibration 
which  corresponds  to  a  certain  definite  gas  does 
not  belong  to  it  as  a  whole,  but  belongs  to  the 
separate  parts  of  it — for  this  reason,  that  if  you 
squeeze  the  gas  you  do  not  alter  the  time  of 
vibration.  Let  us  suppose  that  we  have  a  great 
number  of  fiddles  in  a  room  which  are  all  in 
contact,  and  have  strings  accurately  tuned  to 
vibrate  to  certain  notes.  If  you  sang  one  of 
those  notes  all  the  fiddles  would  answer ;  but 
if  you  compress  them  you  clearly  put  them  all 
out  of  tune.  They  are  all  in  contact,  and  they 
will  not  answer  to  the  note  with  the  same 
precision  as  before.  But  if  you  have  a  room 
which  is  full  of  fiddles,  placed  at  a  certain 


236  LECTURES  AND  ESSAYS 

distance  from  one  another,  then  if  you  bring 
them  within  shorter  distances  of  o:  :  another, 
so  that  they  still  do  not  touch,  they  will  not  be 
put  out  of  tune — they  will  answer  exactly  to 
the  same  note  as  before.  We  see,  therefore, 
that  since  compression  of  a  gas  within  certain 
limits  does  not  alter  the  rate  of  vibration  which 
belongs  to  it,  that  rate  of  vibration  cannot 
belong  to  the  body  of  gas  as  a  whole,  but  it 
must  belong  to  the  individual  parts  of  it.  Now, 
by  such  reasoning  as  this  it  seems  to  me  that 
the  modern  theory  of  the  constitution  of  matter 
is  put  upon  a  basis  which  is  absolutely  in- 
dependent of  hypothesis.  The  theory  is  simply 
an  organised  statement  of  the  facts  ;  a  state- 
ment, that  is,  which  is  rather  different  from  the 
experiments,  being  made  out  from  them  in  just 
such  a  way  as  to  be  most  convenient  for  rinding 
out  from  them  what  will  be  the  results  of  other 
experiments.  That  is  all  we  mean  at  present 
by  scientific  theory. 

Upon  this  theory  Professor  Clerk  Maxwell 
founded  a  certain  argument  in  his  lecture  before 
the  British  Association  at  Bradford.  It  is  a 
consequence  of  the  molecular  theory,  as  I  said 
before,  that  all  the  molecules  of  a  certain  given 
substance,  say  oxygen,  are  as  near  as  possible 
alike  in  two  respects — first  in  weight,  and 
secondly  in  their  times  of  vibration.  Professor 
Clerk  Maxwell's  argument  was  this.  He  first 


FIRST  AND  LAST  CATASTROPHE  237 

of  all  said  that  the  theory  required  us  to  believe, 
not  that  these  molecules  were  as  near  as  may 
be  alike,  but  that  they  were  exactly  alike  in 
these  two  respects  —  at  least  the  argument 
appeared  to  me  to  require  that  Then  he  said 
all  the  oxygen  we  know  of,  whatever  processes 
it  has  gone  through — whether  it  is  got  out  of 
the  atmosphere,  or  out  of  some  oxide  of  iron 
of  carbon,  or  whether  it  belongs  to  the  sun  or 
the  fixed  stars,  or  the  planets  or  the  nebulae — 
all  this  oxygen  is  alike.  And  all  these  mole- 
cules of  oxygen  we  find  upon  the  earth  must 
have  existed  unaltered,  or  appreciably  unaltered, 
during  the  whole  of  the  time  the  earth  has 
been  evolved.  Whatever  vicissitudes  they 
have  gone  through,  however  many  times  they 
have  entered  into  combination  with  iron  or 
carbon  and  been  carried  down  beneath  the 
crust  of  the  earth,  or  set  free  and  sent  up  again 
through  the  atmosphere,  they  have  remained 
steadfast  to  their  original  form  unaltered,  the 
monuments  of  what  they  were  when  the  world 
began.  Professor  Clerk  Maxwell  argues  that 
things  which  are  unalterable,  and  are  exactly 
alike,  cannot  have  been  formed  by  any  natural 
process.  Moreover,  being  exactly  alike,  they 
cannot  have  existed  for  ever,  and  therefore 
they  must  have  been  made.  As  Sir  John 
Herschel  said,  "  They  bear  the  stamp  of  the 
manufactured  article." 


238  LECTURES  AND  ESSAYS 

Into  these  further  deductions  I  do  not  pro- 
pose to  enter  at  all.  I  confine  myself  strictly 
to  the  first  of  the  deductions  which  Professor 
Clerk  Maxwell  made  from  the  molecular  theory. 
He  said  that  because  these  molecules  are  ex- 
actly alike,  and  because  they  have  not  been  in 
the  least  altered  since  the  beginning  of  time, 
therefore  they  cannot  have  been  produced  by 
any  process  of  evolution.  It  is  just  that  ques- 
tion which  I  want  to  discuss.  I  want  to  con- 
sider whether  the  evidence  we  have  to  prove 
that  these  molecules  are  exactly  alike  is 
sufficient  to  make  it  impossible  that  they  can 
have  been  produced  by  any  process  of  evolution. 

The  position  that  this  evidence  is  not 
sufficient  is  evidently  by  far  the  easier  to 
defend  ;  because  the  negative  is  proverbially 
hard  to  prove  ;  and  if  any  one  should  prove 
that  a  process  of  evolution  was  impossible,  it 
would  be  an  entirely  unique  thing  in  science 
and  philosophy.  In  fact,  we  may  see  from 
this  example  precisely  how  great  is  the  influence 
of  authority  in  matters  of  science.  If  there  is 
any  name  among  contemporary  natural  philo- 
sophers to  whom  is  due  the  reverence  of  all 
true  students  of  science,  it  is  that  of  Professor 
Clerk  Maxwell.  But  if  any  one  not  possessing 
his  great  authority  had  put  forward  an  argu- 
ment, founded  apparently  upon  a  scientific 
basis,  in  which  there  occurred  assumptions 


FIRST  AND  LAST  CATASTROPHE  239 

about  what  things  can  and  what  things  cannot 
have  existed  from  eternity,  and  about  the  exact 
similarity  of  two  or  more  things  established  by 
experiment,  we  should  say :  "  Past  eternity  ; 
absolute  exactness  ;  this  won't  do  ; "  and  we 
should  pass  on  to  another  book.  The  experi- 
ence of  all  scientific  culture  for  all  ages  during 
which  it  has  been  a  light  to  men  has  shown  us 
that  we  never  do  get  at  any  conclusions  of  that 
sort.  We  do  not  get  at  conclusions  about 
infinite  time  or  infinite  exactness.  We  get  at 
conclusions  which  are  as  nearly  true  as  experi- 
ment can  show,  and  sometimes  which  are  a 
great  deal  more  correct  than  direct  experiment 
can  be,  so  that  we  are  able  actually  to  correct 
one  experiment  by  deductions  from  another  ; 
but  we  never  get  at  conclusions  which  we  have 
a  right  to  say  are  absolutely  exact ;  so  that 
even  if  we  find  a  man  of  the  highest  powers 
saying  that  he  had  reason  to  believe  a  certain 
statement  to  be  exactly  true,  or  that  he  believed 
a  certain  thing  to  have  existed  from  the  begin- 
ning exactly  as  it  is  now,  we  must  say :  "  It 
is  quite  possible  that  a  man  of  so  great  eminence 
may  have  found  out  something  which  is  entirely 
different  from  the  whole  of  our  previous  know- 
ledge, and  the  thing  must  be  inquired  into.  But, 
notwithstanding  that,  it  remains  a  fact  that  this 
piece  of  knowledge  will  be  absolutely  of  a  differ- 
ent kind  from  anything  that  we  knew  before." 


240  LECTURES  AND  ESSAYS 

Now  let  us  examine  the  evidence  by  which 
we  know  that  the  molecules  of  the  same  gas 
are  as  near  as  may  be  alike  in  weight  and  in 
rates  of  vibration.  There  were  experiments 
made  by  Dr.  Graham,  late  Master  of  the  Mint, 
upon  the  rate  at  which  different  gases  were 
mixed  together.  He  found  that  if  he  divided 
a  vessel  by  a  thin  partition  made  of  blacklead 
or  graphite,  and  put  different  gases  on  the  two 
opposite  sides,  they  would  mix  together  nearly 
as  fast  as  though  there  was  nothing  between 
them.  The  difference  was  that  the  plate  of 
graphite  made  it  more  easy  to  measure  the 
rate  of  mixture ;  and  Dr.  Graham  made 
measurements  and  came  to  conclusions  which 
are  exactly  such  as  are  required  by  the  mole- 
cular theory.  It  is  found  by  a  process  of 
mathematical  calculation  that  the  rate  of 
diffusion  of  different  gases  depends  upon  the 
weight  of  the  molecules.  A  molecule  of  oxygen 
is  sixteen  times  as  heavy  as  a  molecule  of 
hydrogen,  and  it  is  found  upon  experiment 
that  hydrogen  goes  through  a  septum  or  wall 
of  graphite  four  times  as  fast  as  oxygen  does. 
Four  times  four  are  sixteen.  We  express  that 
rule  in  mathematics  by  saying  that  the  rate  of 
diffusion  of  gas  is  inversely  as  the  square  root 
of  the  mass  of  its  molecules.  If  one  molecule 
is  thirty-six  times  as  heavy  as  another — the 
molecule  of  chlorine  is  nearly  that  multiple  of 


FIRST  AND  LAST  CATASTROPHE  241 

hydrogen — it  will  diffuse  itself  at  one-sixth  of 
the  rate. 

This  rule  is  a  deduction  from  the  molecular 
theory,  and  it  is  found,  like  innumerable  other 
such  deductions,  to  come  right  in  practice.  But 
now  observe  what  is  the  consequence  of  this. 
Suppose  that,  instead  of  taking  one  gas  and 
making  it  diffuse  itself  through  a  wall,  we  take 
a  mixture  of  two  gases.  Suppose  we  put 
oxygen  and  hydrogen  into  one  side  of  a  vessel 
which  is  divided  into  two  parts  by  a  wall  of 
graphite,  and  we  exhaust  the  air  from  the  other 
side,  then  the  hydrogen  will  go  through  this 
wall  four  times  as  fast  as  the  oxygen  will. 
Consequently,  as  soon  as  the  other  side  is  full 
there  will  be  a  great  deal  more  hydrogen  in  it 
than  oxygen — that  is  to  say,  we  shall  have 
sifted  the  oxygen  from  the  hydrogen,  not  com- 
pletely, but  in  a  great  measure,  precisely  as  by 
means  of  a  screen  we  can  sift  large  coals  from 
small  ones.  Now  let  us  suppose  that  when  we 
have  oxygen  gas  unmixed  with  any  other  the 
molecules  are  of  two  sorts  and  of  two  different 
weights.  Then  you  see  that  if  we  make  that 
gas  pass  through  a  porous  wall,  the  lighter 
particles  would  pass  through  first,  and  we 
should  get  two  different  specimens  of  oxygen 
gas,  in  one  of  which  the  molecules  would  be 
lighter  than  in  the  other.  The  properties  of 
one  of  these  specimens  of  oxygen  gas  would 
VOL.  I  R 


242  LECTURES  AND  ESSAYS 

necessarily  be  different  from  those  of  the  other, 
and  that  difference  might  be  found  by  very 
easy  processes.  If  there  were  any  perceptible 
difference  between  the  average  weight  of  the 
molecules  on  the  two  sides  of  the  septum, 
there  would  be  no  difficulty  in  finding  that  out. 
No  such  difference  has  ever  been  observed. 
If  we  put  any  single  gas  into  a  vessel,  and  we 
filter  it  through  a  septum  of  blacklead  into 
another  vessel,  we  find  no  difference  between 
the  gas  on  one  side  of  the  wall  and  the  gas  on 
the  other  side.  That  is  to  say,  if  there  is  any 
difference  it  is  too  small  to  be  perceived  by  our 
present  means  of  observation.  It  is  upon  that 
sort  of  evidence  that  the  statement  rests  that 
the  molecules  of  a  given  gas  are  all  very  nearly 
of  the  same  weight.  Why  do  I  say  very 
nearly?  Because  evidence  of  that  sort  can 
never  prove  that  they  are  exactly  of  the  same 
weight.  The  means  of  measurement  we  have 
may  be  exceedingly  correct,  but  a  certain  limit 
must  always  be  allowed  for  deviation  ;  and  if 
the  deviation  of  molecules  of  oxygen  from  a 
certain  standard  of  weight  were  very  small, 
and  restricted  within  small  limits,  it  would  be 
quite  possible  for  our  experiments  to  give  us 
the  results  which  they  do  now.  Suppose,  for 
example,  the  variation  in  the  size  of  the  oxygen 
atoms  were  as  great  as  that  in  the  weight  of 
different  men,  then  it  would  be  very  difficult 


FIRST  AND  LAST  CATASTROPHE  243 

indeed  to  tell  by  such  a  process  of  sifting  what 
that  difference  was,  or  in  fact  to  establish  that 
it  existed  at  all.  But,  on  the  other  hand,  if  we 
suppose  the  forces  which  originally  caused  all 
those  molecules  to  be  so  nearly  alike  as  they 
are  to  be  constantly  acting  and  setting  the  thing 
right  as  soon  as  by  any  sort  of  experiment  we 
set  it  wrong,  then  the  small  oxygen  atoms  on 
one  side  would  be  made  up  to  their  right  size,  and 
it  would  be  impossible  to  test  the  difference  by 
any  experiment  which  was  not  quicker  than  the 
processes  by  which  they  were  made  right  again. 
There  is  another  reason  why  we  are  obliged 
to  regard  that  experiment  as  only  approximate, 
and  as  not  giving  us  any  exact  results.  There 
is  very  strong  evidence,  although  it  is  not  con- 
clusive, that  in  a  given  gas — say  in  a  vessel 
full  of  carbonic  acid — the  molecules  are  not 
all  of  the  same  weight.  If  we  compress  the 
gas,  we  find  that  when  in  the  state  of  a 
perfect  gas,  or  nearly  so,  the  pressure  increases 
just  in  the  ratio  that  the  volume  diminishes. 
That  law  is  entirely  explained  by  means  of  the 
molecular  theory.  It  is  what  ought  to  exist 
if  the  molecular  theory  is  true.  If  we  compress 
the  gas  further,  we  find  that  the  pressure  is 
smaller  than  it  ought  to  be  according  to  this 
law.  This  can  be  explained  in  two  ways.  First 
of  all  we  may  suppose  that  the  molecules  are 
so  crowded  that  the  time  during  which  they 


244  LECTURES  AND  ESSAYS 

are  sufficiently  near  to  attract  each  other 
sensibly  becomes  too  large  a  proportion  of  the 
whole  time  to  be  neglected ;  and  this  will 
account  for  the  change  in  the  law.  There 
is,  however,  another  explanation.  We  may 
suppose,  for  illustration,  that  two  molecules 
approach  one  another,  and  that  the  speed  at 
which  one  is  going  relatively  to  the  other  is 
very  small,  and  then  that  they  so  direct  one 
another  that  they  get  caught  together,  and  go 
on  circling,  making  only  one  molecule.  This, 
on  scientific  principles,  will  account  for  our 
fact,  that  the  pressure  in  a  gas  which  is  near 
a  liquid  state  is  too  small — that  instead  of  the 
molecules  going  about  singly,  some  are  hung 
together  in  couples  and  some  in  larger  numbers, 
and  making  still  larger  molecules.  This  sup- 
position is  confirmed  very  strikingly  by  the 
spectroscope.  If  we  take  the  case  of  chlorine 
gas,  we  find  that  it  changes  colour — that  it 
gets  darker  as  it  approaches  the  liquid  condi- 
tion. This  change  of  colour  means  that  there 
is  a  change  in  the  rate  of  vibration  which  belongs 
to  its  component  parts  ;  and  it  is  a  very  simple 
mechanical  deduction  that  the  larger  molecules 
will,  as  a  rule,  have  a  slower  rate  of  vibration 
than  the  smaller  ones — very  much  in  the  same 
way  as  a  short  string  gives  a  higher  note  than 
a  long  one.  The  colour  of  chlorine  changes 
just  in  the  way  we  should  expect  if  the  mole- 


FIRST  AND  LAST  CATASTROPHE  245 

cules,  instead  of  going  about  separately,  were 
hanging  together  in  couples ;  and  the  same 
thing  is  true  of  a  great  number,,  of  the  metals. 
Mr.  Lockyer,  in  his  admirable  researches,  has 
shown  that  several  of  the  metals  and  metalloids 
have  various  spectra,  according  to  the  tempera- 
ture and  the  pressure  to  which  they  are  exposed  ; 
and  he  has  made  it  exceedingly  probable  that 
these  various  spectra — that  is,  the  rates  of 
vibration  of  the  molecules — depend  upon  the 
molecules  being  actually  of  different  sizes. 
Dr.  Roscoe  has  a  few  months  ago  shown  an 
entirely  new  spectrum  of  the  metal  sodium, 
whereby  it  appears  that  this  metal  exists  in  a 
gaseous  state  in  four  different  degrees  of 
aggregation — as  a  simple  molecule,  and  as 
three  or  four  or  eight  molecules  together. 
Every  increase  in  the  complication  of  the 
molecules — every  extra  molecule  you  hang  on 
to  the  aggregate  that  goes  about  together — 
will  make  a  difference  in  the  rate  of  the  vibra- 
tion of  that  system,  and  so  will  make  a  difference 
in  the  colour  of  the  substance. 

So  then  we  have  an  evidence  of  an  entirely 
extraneous  character  that  in  a  given  gas  the 
actual  molecules  that  exist  are  not  all  of  the 
same  weight.  Any  experiment  which  failed 
to  detect  this  would  fail  to  detect  any  smaller 
difference.  And  here  also  we  can  see  a  reason 
why,  although  a  difference  in  the  size  of  the 


246  LECTURES  AND  ESSAYS 

molecules  does  exist,  yet  we  do  not  find  that 
out  by  sifting.  Suppose  you  take  oxygen  gas 
consisting  of  single  molecules  and  double  mole- 
cules, and  you  sift  it  through  a  plate ;  the 
single  molecules  get  through  first,  but,  when 
they  get  through,  some  of  them  join  themselves 
together  as  double  molecules ;  and  although 
more  double  molecules  are  left  on  the  other 
side,  yet  some  of  them  break  up  and  make 
single  molecules ;  so  the  process  of  sifting, 
which  ought  to  give  you  single  molecules  on 
the  one  side  and  double  on  the  other,  merely 
gives  you  a  mixture  of  single  and  double  on 
both  sides  ;  because  the  reasons  which  origin- 
ally decided  that  there  should  be  just  those 
two  forms  are  always  at  work  and  continually 
setting  things  right. 

Now  let  us  take  the  other  point  in  which 
molecules  are  very  nearly  alike — namely,  that 
they  have  very  nearly  the  same  rate  of  vibra- 
tion. The  metal  sodium  in  the  common  salt 
upon  the  earth  has  two  rates  of  vibration  ;  it 
sounds  two  notes,  as  it  were,  which  are  very 
near  to  each  other.  They  form  the  well-known 
double  line  D  in  the  yellow  part  of  the  spectrum. 
These  two  bright  yellow  lines  are  very  easy  to 
observe.  They  occur  in  the  spectra  of  a  great 
number  of  stars.  They  occur  in  the  solar 
spectrum  as  dark  lines,  showing  that  there  is 
sodium  in  the  outer  rim  of  the  sun,  which  is 


FIRST  AND  LAST  CATASTROPHE  247 

stopping  and  shutting  off  the  light  of  the 
bright  parts  behind.  All  these  lines  of  sodium 
are  just  in  the  same  position  in  the  spectrum, 
showing  that  the  rates  of  vibration  of  all  these 
molecules  of  sodium  all  over  the  universe,  so 
far  as  we  know,  are  as  near  as  possible  alike. 
That  implies  a  similarity  of  molecular  structure, 
which  is  a  great  deal  more  delicate  than  any 
mere  test  of  weight.  You  may  weigh  two 
fiddles  until  you  are  tired,  and  you  will  never 
find  out  whether  they  are  in  tune ;  the  one 
test  is  a  great  deal  more  delicate  than  the 
other.  Let  us  see  how  delicate  this  test  is. 
Lord  Rayleigh  has  remarked  that  there  is  a 
natural  limit  for  the  precise  position  of  a  given 
line  in  the  spectrum,  and  for  this  reason.  If  a 
body  which  is  emitting  a  sound  comes  towards 
you,  you  will  find  that  the  pitch  of  the  sound 
is  altered.  Suppose  that  omnibuses  run  every 
ten  minutes  in  the  streets,  and  you  walk  in  a 
direction  opposite  to  that  in  which  they  are 
coming,  you  will  obviously  pass  more  omni- 
buses in  an  hour  than  if  you  walked  in  an 
opposite  direction.  If  a  body  emitting  light 
is  coming  towards  you,  you  will  find  more 
waves  in  a  certain  direction  than  if  it  were 
going  from  you  ;  consequently,  if  you  are 
approaching  a  body  emitting  light,  the  waves 
will  come  at  shorter  intervals,  the  vibration 
will  be  of  shorter  period,  and  the  light  will  be 


248  LECTURES  AND  ESSAYS 

higher  up  in  the  spectrum  —  it  will  be  more 
blue.  If  you  are  going  away  from  the  body, 
then  the  rate  is  slower,  the  light  is  lower  down 
on  the  spectrum,  and  consequently  more  red. 
By  means  of  such  variations  in  the  positions  of 
certain  known  lines,  the  actual  rate  of  approach 
of  certain  fixed  stars  to  the  earth  has  been 
measured,  and  the  rate  of  going  away  of  certain 
other  fixed  stars  has  also  been  measured. 
Suppose  we  have  a  gas  which  is  glowing  in 
a  state  of  incandescence,  all  the  molecules  are 
giving  out  light  at  a  certain  specified  rate  of 
vibration  ;  but  some  of  these  are  coming 
towards  us  at  a  rate  much  greater  than  seven- 
teen miles  a  minute,  because  the  temperature 
is  higher  when  the  gas  is  glowing,  and  others 
are  also  going  away  at  a  much  higher  rate  than 
that.  The  consequence  is,  that  instead  of  hav- 
ing one  sharply  defined  line  on  the  spectrum, 
instead  of  having  light  of  exactly  one  bright 
colour,  we  have  light  which  varies  between 
certain  limits.  If  the  actual  rate  of  the  vibra- 
tion of  the  molecules  of  the  gas  were  marked 
down  upon  the  spectrum,  we  should  not  get 
that  single  bright  line  there,  but  we  should  get 
a  bright  band  overlapping  it  on  each  side. 
Lord  Rayleigh  calculated  that,  in  the  most 
favourable  circumstances,  the  breadth  of  this 
band  would  not  be  less  than  one-hundredth  of 
the  distance  between  the  sodium  lines.  It  is 


FIRST  AND  LAST  CATASTROPHE  249 

precisely  upon  that  experiment  that  the  evidence 
of  the  exact  similarity  of  molecules  rests.  We 
see,  therefore,  from  the  nature  of  the  experi- 
ment, that  we  should  get  exactly  the  same 
results  if  the  rates  of  vibration  of  all  the 
molecules  were  not  exactly  equal,  but  varied 
within  certain  very  small  limits.  If,  for 
example,  the  rates  of  vibration  varied  in  the 
same  way  as  the  heads  of  different  men,  then 
we  should  get  very  much  what  we  get  now 
from  the  experiment. 

From  the  evidence  of  these  two  facts,  then — 
the  evidence  that  molecules  are  of  the  same 
weight  and  degree  of  vibration — all  that  we 
can  conclude  is  that  whatever  differences  there 
are  in  their  weights,  and  whatever  differences 
there  are  in  their  degrees  of  vibration,  these 
differences  are  too  small  to  be  found  out  by 
our  present  modes  of  measurement.  And  that 
is  precisely  all  that  we  can  conclude  in  every 
similar  question  of  science. 

Now,  how  does  this  apply  to  the  question 
whether  it  is  possible  for  molecules  to  have 
been  evolved  by  natural  processes  ?  I  do  not 
understand  myself  how,  even  supposing  we 
knew  that  they  were  exactly  alike,  we  could 
infer  for  certain  that  they  had  not  been 
evolved ;  because  there  is  only  one  case  of 
evolution  that  we  know  anything  at  all  about 
— and  that  we  know  very  little  about  yet — 


250  LECTURES  AND  ESSAYS 

namely,  the  evolution  of  organised  beings.  The 
processes  by  which  that  evolution  takes  place 
are  long,  cumbrous,  and  wasteful  processes  of 
natural  selection  and  hereditary  descent.  They 
are  processes  which  act  slowly,  which  take  a 
great  lapse  of  ages  to  produce  their  natural 
effects.  But  it  seems  to  me  quite  possible  to 
conceive,  in  our  entire  ignorance  of  the  subject, 
that  there  may  be  other  processes  of  evolution 
which  result  in  a  definite  number  of  forms — 
those  of  the  chemical  elements — just  as  these 
processes  of  the  evolution  of  organised  beings 
have  resulted  [in  a  greater  number  of  forms. 
All  that  we  know  of  the  ether  shows  that  its 
actions  are  of  a  rapidity  very  much  exceeding 
anything  we  know  of  the  motions  of  visible 
matter.  It  is  a  possible  thing,  for  example, 
that  mechanical  conditions  should  exist  accord- 
ing to  which  all  bodies  must  be  made  of  regular 
solids,  that  molecules  should  all  have  flat  sides, 
and  that  these  sides  should  all  be  of  the  same 
shape.  I  suppose  that  it  is  just  conceivable 
that  it  might  be  impossible  for  a  molecule  to 
exist  with  two  of  its  faces  different.  In  that 
case  we  know  there  would  be  just  five  shapes 
for  a  molecule  to  exist  in,  and  these  would 
be  produced  by  a  process  of  evolution.  The 
various  forms  of  matter  that  chemists  call 
elements  seem  to  be  related  one  to  another 
very  much  in  that  sort  of  way  ;  that  is,  as  if 


FIRST  AND  LAST  CATASTROPHE  251 

they  rose  out  of  mechanical  conditions  which 
only  rendered  it  possible  for  a  certain  definite 
number  of  forms  to  exist,  and  which,  whenever 
any  molecule  deviates  slightly  from  one  of 
these  forms,  would  immediately  operate  to  set 
it  right  again.  I  do  not  know  at  all — we  have 
nothing  definite  to  go  upon — what  the  shape 
of  a  molecule  is,  or  what  is  the  nature  of  the 
vibration  it  undergoes,  or  what  its  condition  is 
compared  with  the  ether  ;  and  in  our  absolute 
ignorance  it  would  be  impossible  to  make  any 
conception  of  the  mode  in  which  it  grew  up. 
When  we  know  as  much  about  the  shape  of  a 
molecule  as  we  do  about  the  solar  system,  for 
example,  we  may  be  as  sure  of  its  mode  of 
evolution  as  we  are  of  the  way  in  which  the 
solar  system  came  about ;  but  in  our  present 
ignorance  all  we  have  to  do  is  to  show  that 
such  experiments  as  we  can  make  do  not  give 
us  evidence  that  it  is  absolutely  impossible  for 
molecules  of  matter  to  have  been  evolved  out 
of  ether  by  natural  processes. 

The  evidence  which  tells  us  that  the  mole- 
cules of  a  given  substance  are  alike  is  only  ap- 
proximate. The  theory  leaves  room  for  certain 
small  deviations ;  and  consequently  if  there 
are  any  conditions  at  work  in  the  nature  of  the 
ether  which  render  it  impossible  for  other  forms 
of  matter  than  those  we  know  of  to  exist,  the 
great  probability  is  that  when  by  any  process 


252  LECTURES  AND  ESSAYS 

we  contrive  to  sift  molecules  of  one  kind  from 
molecules  of  another,  these  very  conditions  at 
once  bring  them  back  and  restore  to  us  a  mass 
of  gas  consisting  of  molecules  whose  average 
type  is  a  normal  one. 

Now  I  want  to  consider  a  speculation  of  an 
entirely  different  character.  A  remark  was 
made  about  thirty  years  ago  by  Sir  William 
Thomson  upon  the  nature  of  certain  problems 
in  the  conduction  of  heat.  These  problems  had 
been  solved  by  Fourier  many  years  before  in  a 
beautiful  treatise.  The  theory  was  that  if  you 
knew  the  degree  of  warmth  of  a  body,  then  you 
could  find  what  would  happen  to  it  afterwards  ; 
you  would  find  how  the  body  would  gradually 
cool.  Suppose  you  put  the  end  of  a  poker  in 
the  fire  and  make  it  red  hot,  that  end  is  very 
much  hotter  than  the  other  end  ;  but  if  you 
take  it  out  and  let  it  cool,  you  will  find  that 
heat  is  travelling  from  the  hot  end  to  the  cool 
end  ;  and  the  amount  of  this  travelling,  and  the 
temperature  at  either  end  of  the  poker,  can  be 
calculated  with  great  accuracy.  This  comes 
out  of  Fourier's  theory.  Now  suppose  you  try 
to  go  backwards  in  time,  and  take  the  poker  at 
any  instant  when  it  is  about  half  cool,  and  say : 
"  Does  this  equation  give  me  the  means  of 
finding  out  what  was  happening  before  this 
time,  in  so  far  as  the  present  state  of  things  has 
been  produced  by  cooling  ?  "  You  will  find  the 


FIRST  AND  LAST  CATASTROPHE  253 

equation  will  give  you  an  account  of  the  state  of 
the  poker  before  the  time  when  it  came  into 
your  hands,  with  great  accuracy  up  to  a  certain 
point ;  but  beyond  that  point  it  refuses  to  give 
you  any  more  information,  and  it  begins  to 
talk  nonsense.  It  is  in  the  nature  of  a  problem 
of  the  conduction  of  heat  that  it  allows  you  to 
trace  the  forward  history  of  it  to  any  extent 
you  like ;  but  it  will  not  allow  you  to  trace  the 
history  of  it  backward  beyond  a  certain  point. 
There  is  another  case  in  which  a  similar  thing 
happens.  There  is  an  experiment  in  that 
excellent  manual,  the  Boy's  Own  Book,  which 
tells  you  that  if  you  half  fill  a  glass  with  beer, 
and  put  some  paper  on  it,  and  then  pour  in 
water  carefully,  and  draw  the  paper  out  without 
disturbing  the  two  liquids,  the  water  will  rest 
on  the  beer.  The  problem  then  is  to  drink  the 
beer  without  drinking  the  water,  and  it  is 
accomplished  by  means  of  a  straw.  Let  us 
suppose  these  two  liquids  resting  in  contact ; 
we  shall  find  they  begin  to  mix ;  and  it  is 
possible  to  write  down  an  equation  exactly  of 
the  same  form  as  the  equation  for  the  conduction 
of  heat,  which  would  tell  you  how  much  water 
had  passed  into  the  beer  at  any  given  time  after 
the  mixture  began.  So  that,  given  the  water 
and  the  beer  half  mixed,  you  could  trace  forward 
the  process  of  mixing,  and  measure  it  with 
accuracy,  and  give  a  perfect  account  of  it  ;  but 


254  LECTURES  AND  ESSAYS 

if  you  attempt  to  trace  that  back  you  will  have 
a  point  where  the  equation  will  stop,  and  will 
begin  to  talk  nonsense.  That  is  the  point  where 
you  took  away  the  paper,  and  allowed  the 
mixing  to  begin.  If  we  apply  that  same  con- 
sideration to  the  case  of  the  poker,  and  try  to 
trace  back  its  history,  you  will  find  that  the 
point  where  the  equation  begins  to  talk  nonsense 
is  the  point  where  you  took  it  out  of  the  fire. 
The  mathematical  theory  supposes  that  the 
process  of  conduction  of  heat  has  gone  on  in  a 
quiet  manner,  according  to  certain  defined  laws, 
and  that  if  at  any  time  there  was  a  catastrophe, 
an  event  not  included  in  the  laws  of  the  con- 
duction of  heat,  then  the  equation  could  give 
you  no  account  of  it.  There  is  another  thing 
which  is  of  the  same  kind — namely,  the  trans- 
mission of  fluid  friction.  If  you  take  your  tea 
in  your  cup,  and  stir  it  round  with  a  spoon,  it  will 
not  go  on  circulating  round  for  ever,  but  will 
come  to  a  stop ;  and  the  reason  is  that  there  is 
a  certain  friction  of  the  liquid  against  the  sides 
of  the  cup,  and  of  the  different  parts  of  the 
liquid  with  one  another.  The  friction  of  the 
different  parts  of  a  liquid  or  a  gas  is  precisely 
a  matter  of  mixing.  The  particles  which  are 
going  fast,  and  are  in  the  middle,  not  having 
been  stopped  by  the  side,  get  mixed  ;  and  the 
particles  at  the  side  going  slow  get  mixed  with 
the  particles  in  the  middle.  This  process  of 


FIRST  AND  LAST  CATASTROPHE  255 

mixing  can  be  calculated,  and  it  leads  to  an 
equation  of  exactly  the  same  sort  as  that  which 
applies  to  the  conduction  of  heat.  We  have, 
therefore,  in  these  problems  a  natural  process 
which  consists  in  mixing  things  together,  and 
this  always  has  the  property  that  you  can  go 
on  mixing  them  for  ever  without  coming  to 
anything  impossible ;  but  if  you  attempt  to 
trace  the  history  of  the  thing  backward,  you 
must  always  come  to  a  state  which  could  not 
have  been  produced  by  mixing — namely,  a 
state  of  complete  separation. 

Upon  this  remark  of  Sir  William  Thomson's, 
the  true  consequences  of  which  you  will  find 
correctly  stated  in  Mr.  Balfour  Stewart's  book 
on  the  Conservation  of  Energy,  a  most  singular 
doctrine  has  been  founded.  These  writers  have 
been  speaking  of  a  particular  problem  on  which 
they  were  employed  at  the  moment.  Sir 
William  Thomson  was  speaking  of  the  conduc- 
tion of  heat,  and  he  said  this  heat  problem  leads 
you  back  to  a  state  which  could  not  have  been 
produced  by  the  conduction  of  heat.  And  so 
Professor  Clerk  Maxwell,  speaking  of  the  same 
problem,  and  also  of  the  diffusion  of  gases,  said 
there  was  evidence  of  a  limit  in  past  time  to  the 
existing  order  of  things,  when  something  else 
than  mixing  took  place.  But  a  most  eminent 
man,  who  has  done  a  great  deal  of  service  to 
mankind,  Professor  Stanley  Jevons,  in  his  very 


256  LECTURES  AND  ESSAYS 

admirable  book,  the  Principles  of  Science^  which 
is  simply  marvellous  for  the  number  of  examples 
illustrating  logical  principles  which  he  has  drawn 
from  all  kinds  of  regions  of  science,  and  for  the 
small  number  of  mistakes  that  occur  in  it, 
takes  this  remark  of  Sir  W.  Thomson's,  and 
takes  out  two  very  important  words,  and  puts 
in  two  other  very  important  words.  He  says  : 
"  We  have  here  evidence  of  a  limit  of  a  state  of 
things  which  could  not  have  been  produced  by 
the  previous  state  of  things  according  to  the 
known  laws  of  nature."  It  is  not  according  to 
the  known  laws  of  nature,  it  is  according  to  the 
known  laws  of  conduction  of  heat,  that  Sir 
William  Thomson  is  speaking  ;  and  from  this 
we  may  see  the  fallacy  of  concluding  that  if  we 
consider  the  case  of  the  whole  universe  we 
should  be  able,  supposing  we  had  paper  and 
ink  enough,  to  write  down  an  equation  which 
would  enable  us  to  make  out  the  history  of  the 
world  forward — as  far  forward  as  we  liked  to 
go  ;  but  if  we  attempted  to  calculate  the  history 
of  the  world  backward,  we  should  come  to  a 
point  where  the  equation  would  begin  to  talk 
nonsense — we  should  come  to  a  state  of  things 
which  could  not  have  been  produced  from  any 
previous  state  of  things  by  any  known  natural 
laws.  You  will  see  at  once  that  that  is  an 
entirely  different  statement.  The  same  doctrine 
has  been  used  by  Mr.  Murphy,  in  a  very  able 


FIRST  AND  LAST  CATASTROPHE  257 

book,  the  Scientific  Basis  of  Faith,  to  build  upon 
it  an  enormous  superstructure — I  think  the 
restoration  of  the  Irish  Church  was  one  of  the 
results  of  it.  But  this  doctrine  is  founded,  as 
I  think,  upon  a  pure  misconception.  It  is 
founded  entirely  upon  forgetfulness  of  the  con- 
dition under  which  the  remark  was  originally 
made.  All  these  physical  writers,  knowing 
what  they  were  writing  about,  simply  drew  such 
conclusions  from  the  facts  which  were  before 
them  as  could  be  reasonably  drawn.  They  say  : 
"  Here  is  a  state  of  things  which  could  not  have 
been  produced  by  the  circumstances  we  are  at 
present  investigating."  Then  your  speculator 
comes  ;  he  reads  a  sentence,  and  says  :  "  Here 
is  an  opportunity  for  me  to  have  my  fling." 
And  he  has  his  fling,  and  makes  a  purely  base- 
less theory  about  the  necessary  origin  of  the 
present  order  of  nature  at  some  definite  point 
of  time  which  might  be  calculated.  But,  if  we 
consider  the  matter,  we  shall  see  that  this  is 
not  in  any  way  a  consequence  of  the  theory  of 
the  conduction  of  heat  ;  because  the  conduction 
of  heat  is  not  the  only  process  that  goes  on  in 
the  universe. 

If  we  apply  that  theory  to  the  case  of  the 
earth,  we  find  that  at  present  there  is  evidence 
of  a  certain  distribution  of  temperature  in  the 
interior  of  it ;  there  is  a  certain  rate  at  which 
the  temperature  increases  as  we  go  down  ;  and 

VOL.  I  S 


258  LECTURES  AND  ESSAYS 

9 

no  doubt,  if  we  made  further  investigations,  we 
should  find  that  if  we  went  deeper  an  accurate 
law  would  be  found,  according  to  which  the 
temperature  increases  in  the  interior. 

Now,  assuming  this  to  be  so,  taking  this  as 
the  basis  of  our  problem,  we  might  endeavour 
to  find  out  what  was  the  history  of  the  earth  in 
past  times,  and  when  it  began  cooling  down. 
That  is  exactly  what  Sir  William  Thomson  has 
done.  When  we  attempt  it,  we  find  that  there 
is  a  definite  point  to  which  we  can  go,  and 
beyond  which  our  equation  talks  nonsense. 
But  we  do  not  conclude  that  at  that  point  the 
laws  of  nature  began  to  be  what  they  are ;  we 
only  conclude  that  the  earth  began  to  solidify. 
Now  solidification  is  not  a  process  of  the  con- 
duction of  heat,  and  so  the  thing  cannot  be 
given  by  our  equation.  That  point  is  given 
definitely  as  a  point  of  time,  not  with  great 
accuracy,  but  still  as  near  as  we  can  expect  to 
get  it  with  such  means  of  measuring  as  we 
have  ;  and  Sir  William  Thomson  has  calculated 
that  the  earth  must  have  solidified  at  some  time 
between  a  hundred  millions  and  two  hundred 
millions  of  years  ago  ;  and  there  we  arrive  at 
the  beginning  of  the  present  state  of  things — 
the  process  of  cooling  the  earth  which  is  going 
on  now.  Before  that  it  was  cooling  as  a  liquid, 
and  in  passing  from  the  liquid  to  the  solid  state 
there  was  a  catastrophe  which  introduced  a  new 


FIRST  AND  LAST  CATASTROPHE  259 

rate  of  cooling.  So  that  by  means  of  that  law 
we  do  come  to  a  time  when  the  earth  began  to 
assume  its  present  state.  We  do  not  find  the 
time  of  the  commencement  of  the  universe,  but 
simply  of  the  present  structure  of  the  earth. 
If  we  went  farther  back  we  might  make  more 
calculations  and  find  how  long  the  earth  had 
been  in  a  liquid  state.  We  should  come  to 
another  catastrophe,  and  say  not  that  at  that 
time  the  universe  began  to  exist,  but  that  the 
present  earth  passed  from  the  gaseous  to  the 
liquid  state.  And  if  we  went  farther  back  still 
we  should  probably  find  the  earth  falling 
together  out  of  a  great  ring  of  matter  surround- 
ing the  sun  and  distributed  over  its  orbit.  The 
same  thing  is  true  of  every  body  of  matter :  if 
we  trace  its  history  back,  we  come  to  a  certain 
time  at  which  a  catastrophe  took  place  ;  and 
if  we  were  to  trace  back  the  history  of  all  the 
bodies  of  the  universe  in  that  way,  we  should 
continually  see  them  separating  up  into  smaller 
parts.  What  they  have  actually  done  is  to  fall 
together  and  get  solid.  If  we  could  reverse  the 
process  we  should  see  them  separating  and 
getting  fluid ;  and,  as  a  limit  to  that,  at  an 
indefinite  distance  in  past  time,  we  should  find 
that  all  these  bodies  would  be  resolved  into 
molecules,  and  all  these  would  be  flying  away 
from  each  other.  There  would  be  no  limit  to 
that  process,  and  we  could  trace  it  as  far  back 


26o  LECTURES  AND  ESSAYS 

as  ever  we  liked  to  trace  it.  So  that  on  the 
assumption — a  very  large  assumption — that  the 
present  constitution  of  the  laws  of  geometry  and 
mechanics  has  held  good  during  the  whole  of 
past  time,  we  should  be  led  to  the  conclusion 
that  at  an  inconceivably  long  time  ago  the 
universe  did  consist  of  ultimate  molecules,  all 
separate  from  one  another,  and  approaching  one 
another.  Then  they  would  meet  together  and 
form  a  great  number  of  small,  hot  bodies. 
Then  you  would  have  the  process  of  cooling 
going  on  in  these  bodies,  exactly  as  we  find  it 
going  on  now.  But  you  will  observe  that  we 
have  no  evidence  of  such  a  catastrophe  as 
implies  a  beginning  of  the  laws  of  nature.  We 
do  not  come  to  something  of  which  we  cannot 
make  any  further  calculation ;  we  find  that 
however  far  we  like  to  go  back,  we  approximate 
to  a  certain  state  of  things,  but  never  actually 
get  to  it. 

Here,  then,  we  have  a  doctrine  about  the 
beginning  of  things.  First,  we  have  a  pro- 
bability, about  as  great  as  science  can  make  it, 
of  the  beginning  of  the  present  state  of  things 
on  the  earth,  and  of  the  fitness  of  the  earth  for 
habitation  ;  and  then  we  have  a  probability 
about  the  beginning  of  the  universe  as  a  whole 
which  is  so  small  that  it  is  better  put  in  this 
form,  that  we  do  not  know  anything  at  all 
about  it  The  reason  why  I  say  that  we  do 


FIRST  AND  LAST  CATASTROPHE  261 

not  know  anything  at  all  of  the  beginning  of 
the  universe  is  that  we  have  no  reason  whatever 
for  believing  that  the  known  laws  of  geometry 
and  mechanics  are  exactly  and  absolutely 
true  at  present,  or  that  they  have  been  even 
approximately  true  for  any  period  of  time 
further  than  we  have  direct  evidence  of.  The 
evidence  we  have  of  them  is  founded  on 
experience  ;  and  we  should  have  exactly  the 
same  experience  of  them  now,  if  those  laws 
were  not  exactly  and  absolutely  true,  but  were 
only  so  nearly  true  that  we  could  not  observe 
the  difference.  So  that  in  making  the  assump- 
tion that  we  may  argue  upon  the  absolute 
uniformity  of  nature,  and  suppose  these  laws  to 
have  remained  exactly  as  they  are,  we  are 
assuming  something  we  know  nothing  about. 
My  conclusion  then  is  that  we  do  know,  with 
great  probability,  of  the  beginning  of  the 
habitability  of  the  earth  about  one  hundred  or 
two  hundred  millions  of  years  back,  but  that  of 
a  beginning  of  the  universe  we  know  nothing 
at  all. 

Now  let  us  consider  what  we  can  find  out 
about  the  end  of  things.  The  life  which  exists 
upon  the  earth  is  made  by  the  sun's  action, 
and  it  depends  upon  the  sun  for  its  continuance. 
We  know  that  the  sun  is  wearing  out,  that  it  is 
cooling ;  and  although  this  heat  which  it  loses 
day  by  day  is  made  up  in  some  measure, 


262  LECTURES  AND  ESSAYS 

perhaps  completely  at  present,  by  the  contrac- 
tion of  its  mass,  yet  that  process  cannot  go  on 
for  ever.  There  is  only  a  certain  amount  of 
energy  in  the  present  constitution  of  the  sun  ; 
and  when  that  has  been  used  up,  the  sun 
cannot  go  on  giving  out  any  more  heat.  Sup- 
posing, therefore,  the  earth  remains  in  her 
present  orbit  about  the  sun,  seeing  that  the 
sun  must  be  cooled  down  at  some  time,  we 
shall  all  be  frozen  out.  On  the  other  hand,  we 
have  no  reason  to  believe  that  the  orbit  of  the 
earth  about  the  sun  is  an  absolutely  stable 
thing.  It  has  been  maintained  for  a  long  time 
that  there  is  a  certain  resisting  medium  which 
the  planets  have  to  move  through  ;  and  it  may 
be  argued  that  in  time  all  the  planets  must  be 
gradually  made  to  move  in  smaller  orbits,  and 
so  to  fall  in  towards  the  sun.  But,  on  the 
other  hand,  the  evidence  upon  which  this 
assertion  was  based,  the  movement  of  Encke's 
comet  and  others,  has  been  recently  entirely 
overturned  by  Professor  Tait.  He  supposes 
that  these  comets  consist  of  bodies  of  meteors. 
Now  it  was  proved  a  long  time  ago  that  a  mass 
of  small  bodies  travelling  together  in  an  orbit 
about  a  central  body  will  always  tend  to  fall  in 
towards  it,  and  that  is  the  case  with  the  rings 
of  Saturn.  So  that,  in  fact,  the  movement  of 
Encke's  comet  is  entirely  accounted  for  on  the 
supposition  that  it  is  a  swarm  of  meteors,  with- 


FIRST  AND  LAST  CATASTROPHE  263 

out  regarding  the  assumption  of  a  resisting 
medium.  On  the  other  hand,  it  seems  exceed- 
ingly natural  to  suppose  that  some  matter  in  a 
very  thin  state  is  diffused  about  the  planetary 
spaces.  Then  we  have  another  consideration, 
— just  as  the  sun  and  moon  make  tides  upon 
the  sea,  so  the  planets  make  tides  upon 
the  sun.  Consider  the  tide  which  the  earth 
makes  upon  the  sun.  Instead  of  being  a  great 
wave  lifting  the  mass  of  the  sun  up  directly 
under  the  earth,  it  is  carried  forward  by  the 
sun's  rotation  ;  the  result  is  that  the  earth, 
instead  of  being  attracted  to  the  sun's  centre, 
is  attracted  to  a  point  before  the  centre.  The 
immediate  tendency  is  to  accelerate  the  earth's 
motion,  and  the  final  effect  of  this  upon  the 
planet  is  to  make  its  orbit  larger.  That  planet 
disturbing  all  the  other  planets,  the  consequence 
is  that  we  have  the  earth  gradually  going  away 
from  the  sun,  instead  of  falling  into  it.1 

In  any  case,  all  we  know  is  that  the  sun  is 
going  out  If  we  fall  into  the  sun  then  we 
shall  be  fried  ;  if  we  go  away  from  the  sun,  or 
the  sun  goes  out,  then  we  shall  be  frozen.  So 
that,  so  far  as  the  earth  is  concerned,  we  have 
no  means  of  determining  what  will  be  the 
character  of  the  end,  but  we  know  that  one  of 


1  I  learn  from  Sir  W.  Thomson  that  the  ultimate  effect  of  tidal 
deformation  on  a  number  of  bodies  is  to  reduce  them  to  two,  which 
move  as  if  they  were  rigidly  connected. 


264  LECTURES  AND  ESSAYS 

these  two  things  must  take  place  in  time.  But 
in  regard  to  the  whole  universe,  if  we  were  to 
travel  forward  as  we  have  travelled  backward 
in  time,  and  consider  things  as  falling  together, 
we  should  come  finally  to  a  great  central  mass, 
all  in  one  piece,  which  would  send  out  waves 
of  heat  through  a  perfectly  empty  ether,  and 
gradually  cool  itself  down.  As  this  mass  got 
cool  it  would  be  deprived  of  all  life  and  motion  ; 
it  would  be  just  a  mere  enormous  frozen  block 
in  the  middle  of  the  ether.  But  that  conclusion, 
which  is  like  the  one  that  we  discussed  about 
the  beginning  of  the  world,  is  one  which  we 
have  no  right  whatever  to  rest  upon.  It 
depends  upon  the  same  assumption  that  the 
laws  of  geometry  and  mechanics  are  exactly 
and  absolutely  true  ;  and  that  they  will  con- 
tinue exactly  and  absolutely  true  for  ever  and 
ever.  Such  an  assumption  we  have  no  right 
whatever  to  make.  We  may  therefore,  I  think, 
conclude  about  the  end  of  things  that,  so  far  as 
the  earth  is  concerned,  an  end  of  life  upon  it  is 
as  probable  as  science  can  make  anything ;  but 
that  in  regard  to  the  universe  we  have  no 
right  to  draw  any  conclusion  at  all. 

So  far,  we  have  considered  simply  the 
material  existence  of  the  earth  ;  but  of  course 
our  greatest  interest  lies  not  so  much  with  the 
material  life  upon  it,  the  organised  beings,  as 
with  another  fact  which  goes  along  with 


FIRST  AND  LAST  CATASTROPHE  265 

that,  and  which  is  an  entirely  different 
one — the  fact  of  the  consciousness  that  exists 
upon  the  earth.  We  find  very  good  reason 
indeed  to  believe  that  this  consciousness  in  the 
case  of  any  organism  is  itself  a  very  complex 
thing,  and  that  it  corresponds  part  for  part  to 
the  action  of  the  nervous  system,  and  more 
particularly  of  the  brain  of  that  organised  thing. 
There  are  some  whom  such  evidence  has  led  to 
the  conclusion  that  the  destruction  which  we 
have  seen  reason  to  think  probable  of  all 
organised  beings  upon  the  earth  will  lead  also 
to  the  final  destruction  of  the  consciousness 
that  goes  with  them.  Upon  this  point  I  know 
there  is  great  difference  of  opinion  amongst 
those  who  have  a  right  to  speak.  But  to  those 
who  do  see  the  cogency  of  the  evidences  of 
modern  physiology  and  modern  psychology  in 
this  direction  it  is  a  very  serious  thing  to  con- 
sider that  not  only  the  earth  itself  and  all  that 
beautiful  face  of  nature  we  see,  but  also  the 
living  things  upon  it,  and  all  the  consciousness 
of  men,  and  the  ideas  of  society,  which  have 
grown  up  upon  the  surface,  must  come  to  an 
end.  We  who  hold  that  belief  must  just  face 
the  fact  and  make  the  best  of  it ;  and  I  think  we 
are  helped  in  this  by  the  words  of  that  Jew 
philosopher,  who  was  himself  a  worthy  crown 
to  the  splendid  achievements  of  his  race  in  the 
cause  of  progress  during  the  Middle  Ages, 


266  LECTURES  AND  ESSAYS 

Benedict  Spinoza.  He  said  :  "  The  free  man 
thinks  of  nothing  so  little  as  of  death,  and  his 
wisdom  is  a  meditation  not  of  death  but  of 
life."  Our  interest  lies  with  so  much  of  the 
past  as  may  serve  to  guide  our  actions  in  the 
present,  and  to  intensify  our  pious  allegiance  to 
the  fathers  who  have  gone  before  us  and  the 
brethren  who  are  with  us  ;  and  our  interest  lies 
with  so  much  of  the  future  as  we  may  hope  will 
be  appreciably  affected  by  our  good  actions 
now.  Beyond  that,  as  it  seems  to  me,  we  do 
not  know,  and  we  ought  not  to  care.  Do  I 
seem  to  say  :  "  Let  us  eat  and  drink,  for  to- 
morrow we  die  ?  "  Far  from  it ;  on  the  contrary 
I  say  :  "  Let  us  take  hands  and  help,  for  this 
day  we  are  alive  together." 

The  following  note  was  afterwards  published 
by  the  author  (Fortnightly  Review,  vol.  xvii.  p. 
793)  :— 

The  passage  referred  to  from  the  Principles 
of  Science  is  as  follows  (vol.  ii.  p.  438)  : — 

"  For  a  certain  negative  value  of  the  time 
the  formulae  give  impossible  values,  indicating 
that  there  was  some  initial  distribution  of  heat 
which  could  not  have  resulted,  according  to 
known  laws  of  nature,  from  any  previous  dis- 
tribution." 

The  words  italicised  are  here  inserted  into  a 
sentence  from  Tait's  Thermo-dynamics,  p.  38. 


FIRST  AND  LAST  CATASTROPHE  267 

Had  the  words  conduction  of  heat  been  used 
instead  of  nature,  the  sentence  would  have 
remained  correct,  but  would  not  have  led  to 
the  alarming  inference  that 

"  The  theory  of  heat  places  us  in  the 
dilemma  either  of  believing  in  creation  at  some 
assignable  date  in  the  past,  or  else  of  suppos- 
ing that  some  inexplicable  change  in  the  work- 
ing of  natural  laws  then  took  place." 

It  has  been  pointed  out  by  Mr.  Higgins  that 
the  ultimate  effect  of  tides  in  the  sun  caused 
by  the  earth's  attraction  will  be  precisely  similar 
to  that  of  a  resisting  medium — that  is,  will 
diminish  the  orbit  of  the  earth  and  increase  its 
velocity  ;  and  that  I  was  wrong  in  supposing 
the  contrary  effect.  It  results  that  the  earth 
will  certainly  fall  into  the  sun  ;  but  whether 
before  or  after  the  sun  has  cooled  down  so 
much  as  not  to  be  able  to  support  life  on  this 
planet  remains  undetermined.  The  final  con- 
clusion remains  therefore  as  before — that  there 
must  be  an  end,  but  whether  by  heat  or  by 
cold  we  cannot  tell. 


THE  UNSEEN    UNIVERSE1 

THE  primary  motive  of  this  treatise  is  indicated 
by  its  second  title  :  "  Physical  Speculations  on 
a  Future  State."  A  sketch  of  the  beliefs  and 
yearnings  of  many  different  folk  in  regard  to  a 
life  after  death  leads  up  to  an  attempt  to  find 
room  for  it  within  the  limits  of  those  physical 
doctrines  of  continuity  and  the  conservation  of 
energy  which  are  regarded  as  the  established 
truths  of  science.  In  this  attempt  it  is  necessary 
to  discuss  the  ultimate  constitution  of  matter 
and  its  relation  to  the  ether.  When,  by  a 
singular  inconsequence  in  writers  possessing 
such  power  in  their  right  minds  of  sound 
scientific  reasoning,  room  has  been  found  for  a 
future  life  in  the  manner  indicated  above,  it  is 
discovered  that  there  is  room  for  a  great  deal 
more.  Accordingly  some  of  the  main  doctrines 
of  the  Christian  religion  are  interpreted  in  relation 
to  the  authors'  hypothesis,  and  placed  in  their 

1  "The  Unseen  Universe;  or,  Physical  Speculations  on  a 
Future  State."  London :  Macmillan  and  Co.  1875.  [Fort- 
nightly  Review,  June  1875.] 


THE  UNSEEN  UNIVERSE  269 

appropriate  niches.  It  will  perhaps  be  con- 
venient, therefore,  if  we  consider  these  three 
things  in  their  order :  first,  the  desire  for  a 
future  life  ;  secondly,  the  physical  speculations 
that  make  room  for  it  ;  and  lastly,  that  system, 
the  seemingly  innocent  dried  carcase  of  which 
is  to  be  smuggled  into  our  house  at  the  same 
time,  that  it  may  peradventure  find  means  of 
resurrection. 

I. 

It  is  often  said  that  the  universal  longing 
for  immortality  among  all  kinds  and  conditions 
of  men  is  a  presumption  that  there  is  some 
future  life  in  which  this  longing  shall  be  satis- 
fied. Let  us  endeavour,  therefore,  to  find  out 
in  what  this  longing  for  immortality  actually 
consists  ;  whether  the  existence  of  it,  when  its 
nature  is  understood,  can  be  explained  on 
grounds  which  do  not  require  it  to  have  any 
objective  fulfilment  other  than  the  life  and  the 
memory  of  those  who  come  after  us  ;  and  what 
relation  it  bears  to  the  equally  widespread 
dream  or  vision  of  a  spiritual  world  peopled  by 
supernatural  or  monstrous  beings,  ghosts  and 
gods  and  goblins. 

First,  let  us  notice  that  all  the  words  used 
to  describe  this  immortality  that  is  longed  for 
are  negative  words  :  m-mortality,  end-tess  life, 
z#-finite  existence.  Endless  life  is  an  incon- 


270  LECTURES  AND  ESSAYS 

ceivable  thing,  for  an  endless  time  would  be 
necessary  to  form  an  idea  of  it.  Now  it  is 
only  by  a  stretch  of  language  that  we  can  be 
said  to  desire  that  which  is  inconceivable.  No 
doubt  many  persons  say  that  they  are  smitten 
with  an  insatiable  longing  for  the  unattainable 
and  ineffable ;  but  this  means  that  they  feel 
generally  dissatisfied  and  do  not  at  all  know 
what  they  want.  Longing  for  deathlessness 
means  simply  shrinking  from  death.  However 
or  whenever  we  who  live  endeavour  to  realise 
an  end  to  this  healthy  life  of  action  in  ourselves 
or  in  our  brethren  the  effort  is  a  painful  one ; 
and  the  mind,  in  so  far  as  it  is  healthy,  tries  to 
put  it  off  and  avoid  it.  The  state  of  one  who 
really  wishes  for  death  is  firmly  linked  in  our 
thoughts  with  the  extreme  of  misery  and 
wretchedness  and  disease  ;  and,  in  so  far  as  it 
can  be  realised,  we  seem  to  feel  that  such  an 
one  is  fit  to  die.  In  those  cases  of  ripe  old 
age  not  hastened  by  disease,  where  the  physical 
structure  is  actually  worn  out,  having  finished 
its  work  right  honestly  and  well  ;  where  the 
love  of  life  is  worn  out  also,  and  the  grave 
appears  as  a  bed  of  rest  to  the  tired  limbs,  and 
death  as  a  mere  quiet  sleep  from  thought ; 
there  also,  in  so  far  as  we  are  able  to  realise 
the  state  of  the  aged  and  to  put  ourselves  in 
his  place,  death  seems  to  be  normal  and  natural, 
a  thing  to  be  neither  sought  nor  shunned.  But 


THE  UNSEEN  UNIVERSE  271 

such  putting  of  ourselves  in  the  place  of  one  to 
whom  death  is  no  evil  must  in  all  cases  be 
imperfect.  I  cannot,  in  my  present  life  and 
motion,  clearly  conceive  myself  in  so  parlous  a 
state  that  no  hope  of  better  things  should  make 
me  shrink  from  the  end  of  all.  However 
vividly  I  recall  the  feelings  of  pain  and  weak- 
ness, it  is  the  life  and  energy  of  my  present  self 
that  pictures  them  ;  and  this  life  and  energy 
cannot  help  raising  at  the  same  time  combative 
instincts  of  resistance  to  pain  and  weakness, 
whose  very  nature  it  is  to  demand  that  the  sun 
shall  not  go  down  upon  Gibeon  until  they  have 
slain  the  Amalekites.  Nor  can  I  really  and 
truly  put  myself  in  the  place  of  the  worn-out 
old  man  whose  consciousness  may  some  day 
have  a  memory  of  mine.  No  force  of  imagina- 
tion that  I  can  bring  to  bear  will  avail  to  cast 
out  the  youth  of  that  very  imagination  which 
endeavours  to  depict  its  latter  days  ;  no 
thoughts  of  final  and  supreme  fatigue  can  help 
suggesting  refreshment  and  new  rising  after  sleep. 
If,  then,  we  do  not  want  to  die  now,  nor 
next  year,  nor  the  year  after  that,  nor  at  any 
time  that  we  can  clearly  imagine  ;  what  is  this 
but  to  say  that  we  want  to  live  for  ever,  in  the 
only  meaning  of  the  words  that  we  can  at  all 
realise  ?  It  is  not  that  there  is  any  positive 
attraction  in  the  shadowy  vistas  of  eternity,  for 
the  effort  to  contemplate  even  any  very  long 


272  LECTURES  AND  ESSAYS 

time  is  weariness  and  vexation  of  spirit ;  it  is 
that  our  present  life,  in  so  far  as  it  is  healthy, 
rebels  once  for  all  against  its  own  final  and 
complete  destruction.  And  forasmuch  as  so 
many  and  so  mighty  generations  have  in  time 
past  ended  in  death  their  noble  and  brave 
battle  with  the  elements,  that  we  also  and  our 
brethren  can  in  nowise  hope  to  escape  their 
fate,  therefore  we  are  sorely  driven  to  find  some 
way  by  which  at  least  the  image  of  that  ending 
shall  be  avoided  and  set  aside.  As  the  fruit 
of  this  search  two  methods  have  been  found 
and  practised  among  men.  By  one  method 
we  detach  ourselves  from  the  individual  body 
and  its  actions  which  accompany  our  con- 
sciousness, to  identify  ourselves  with  something 
wider  and  greater  that  shall  live  when  we  as 
units  shall  have  done  with  living — that  shall 
work  on  with  new  hands  when  we,  its  worn-out 
limbs,  have  entered  into  rest.  The  soldier  who 
rushes  on  death  does  not  know  it  as  extinction ; 
in  thought  he  lives  and  marches  on  with  the 
army,  and  leaves  with  it  his  corpse  upon  the 
battlefield.  The  martyr  cannot  think  of  his 
own  end  because  he  lives  in  the  truth  he  has 
proclaimed  ;  with  it  and  with  mankind  he 
grows  into  greatness  through  ever  new  victories 
over  falsehood  and  wrong.  But  there  is  another 
way.  Since  when  men  have  died  such  orderly, 
natural,  and  healthy  activity  as  we  have  known 


THE  UNSEEN  UNIVERSE  273 

in  them  and  valued  their  lives  for  has  plainly 
ceased,  we  may  fashion  another  life  for  them, 
not  orderly,  not  natural,  not  healthy,  but 
monstrous  or  super-  natural ;  whose  cloudy 
semblance  shall  be  eked  out  with  the  dreams 
of  uneasy  sleep  or  the  crazes  of  a  mind  diseased. 
And  it  is  to  this  that  the  universal  shrinking 
of  men  from  death,  which  is  called  a  yearning 
for  immortality,  is  alleged  to  bear  witness. 

But  whence  now  does  it  really  come,  and 
what  is  the  true  lesson  of  it  ?  Surely  it  is  a 
necessary  condition  of  life  that  has  desires  at 
all  that  these  desires  should  be  towards  life  and 
not  away  from  it ;  seeing  how  cheap  and  easy 
a  thing  is  destruction  on  all  hands,  and  how 
hard  it  is  for  race  or  unit  to  hold  fast  in  the 
great  struggle  for  existence.  Surely  our  way 
is  paved  with  the  bones  of  those  who  have 
loved  life  and  movement  too  little,  and  lost  it 
before  their  time.  If  we  could  think  of  death 
without  shrinking  it  would  only  mean  that  this 
world  was  no  place  for  us,  and  that  we  should 
make  haste  to  be  gone  to  make  room  for  our 
betters.  And  therefore  that  love  of  action 
which  would  put  death  out  of  sight  is  to  be 
counted  good,  as  a  holy  and  healthy  thing  (one 
word  whose  meanings  have  become  unduly 
severed),  necessary  to  the  life  of  men,  serving 
to  knit  them  together  and  to  advance  them  in 
the  right.  Not  only  is  it  right  and  good  thus 
VOL.  I  T 


274  LECTURES  AND  ESSAYS 

to  cover  over  and  dismiss  the  thought  of  our 
own  personal  end,  to  keep  in  mind  and  heart 
always  the  good  things  that  shall  be  done, 
rather  than  ourselves  who  shall  or  shall  not 
have  the  doing  of  them  ;  but  also  to  our  friends 
and  loved  ones  we  shall  give  the  most  worthy 
honour  and  tribute  if  we  never  say  nor  remember 
that  they  are  dead,  but  contrariwise  that  they 
have  lived  ;  that  hereby  the  brotherly  force  and 
flow  of  their  action  and  work  may  be  carried 
over  the  gulfs  of  death  and  made  immortal  in 
the  true  and  healthy  life  which  they  worthily 
had  and  used.  It  is  only  when  the  bloody 
hands  of  one  who  has  fought  against  the  light 
and  the  right  are  folded  and  powerless  for 
further  crime,  that  it  is  most  kind  and  merciful 
to  bury  him  and  say,  "  The  dog  is  dead." 

But  for  you  noble  and  great  ones,  who  have 
loved  and  laboured  yourselves  not  for  your- 
selves but  for  the  universal  folk,  in  your  time 
not  for  your  time  only  but  for  the  coming 
generations,  for  you  there  shall  be  life  as  broad 
and  far-reaching  as  your  love,  for  you  life- 
giving  action  to  the  utmost  reach  of  the  great 
wave  whose  crest  you  sometimes  were. 

II. 

Believing  that  every  finite  intelligence  must 
be  "  conditioned  in  time  and  space,"  and  there- 


THE  UNSEEN  UNIVERSE  275 

fore  must  have  an  "  organ  of  memory "  and  a 
"power  of  varied  action,"  and  consequently 
must  be  associated  with  a  physical  organism, — 
recognising  also  that  the  world,  as  it  is  known 
at  present,  is  made  up  of  material  molecules 
and  of  ether, — our  authors  frankly  admit  that 
no  room  is  here  to  be  found  either  for  ghosts 
of  the  dead,  or  "superior  intelligences,"  or 
bogies  of  any  kind  whatever.  But  modifying 
a  hypothesis  of  Sir  W.  Thomson's  about  the 
ultimate  form  of  atoms  and  their  relation  to  the 
ether,  they  find  in  a  second  ether  the  material 
wherewith  to  refashion  all  these  marvels  which 
advancing  knowledge  had  banished  from  the 
realm  of  reality.  We  may  here,  then,  review 
with  advantage  for  a  short  time  the  state  of 
that  borderland  between  the  known  and  the 
unknown  in  physical  science  to  which  this  in- 
genious hypothesis  belongs  ;  with  the  view  of 
inquiring  what  measure  of  probability  is  to  be 
attached  to  the  modification  of  it  which  our 
authors  propose. 

Imagine  a  ring  of  indiarubber,  made  by 
joining  together  the  ends  of  a  cylindrical  piece 
(like  a  lead  pencil  before  it  is  cut),  to  be  put 
upon  a  round  stick  which  it  will  just  fit  with  a 
little  stretching.  Let  the  stick  be  now  pulled 
through  the  ring  while  the  latter  is  kept  in  its 
place  by  being  pulled  the  other  way  on  the 
outside.  The  indiarubber  has  then  what  is 


276  LECTURES  AND  ESSAYS 

called  vortex -motion.  Before  the  ends  were 
joined  together,  while  it  was  straight,  it  might 
have  been  made  to  turn  round  without  chang- 
ing position  by  rolling  it  between  the  hands. 
Just  the  same  motion  of  rotation  it  has  on  the 
stick,  only  that  the  ends  are  now  joined  together. 
All  the  inside  surface  of  the  ring  is  going  one 
way — namely,  the  way  the  stick  is  pulled  ;  and 
all  the  outside  is  going  the  other  way.  Such 
a  vortex -ring  is  made  by  the  smoker  who 
purses  his  lips  into  a  round  hole  and  sends  out 
a  puff  of  smoke.  The  outside  of  the  ring  is 
kept  back  by  the  friction  of  his  lips  while  the 
inside  is  going  forwards  ;  thus  a  rotation  is  set 
up  all  round  the  smoke-ring  as  it  travels  out 
into  the  air.  If  we  half  immerse  a  teaspoon 
in  our  tea  and  draw  it  across  the  surface,  we 
may  see  two  little  eddies  formed  at  the  edges 
of  the  spoon.  These  eddies  are  really  united 
by  a  sort  of  rope  of  fluid  underneath  the  surface, 
which  follows  the  shape  of  the  spoon,  and 
which  has  throughout  the  same  motion  of 
rotation  that  the  indiarubber  ring  had  when 
the  stick  was  drawn  through  it ;  except  that 
in  this  case  only  half  a  ring  is  formed,  being 
cut  off,  as  it  were,  by  the  surface  of  the  liquid. 
In  all  these  cases  vortex-motion  is  produced  by 
friction,  and  would  be  ultimately  destroyed  by 
friction.  But,  by  way  of  an  approximation  to 
the  study  of  water,  men  had  been  led  to  the 


THE  UNSEEN  UNIVERSE  277 

conception  of  a  perfect  liquid ;  that  is,  a  liquid 
absolutely  free  from  friction,  or  (which  is  the 
same  thing)  offering  no  resistance  to  change  of 
shape,  or  the  sliding  of  one  part  over  another. 
Water  at  rest  behaves  just  as  such  a  liquid 
would  behave  ;  but  water  in  motion  is  altogether 
a  different  thing.  Helmholtz  found,  by  a 
wonderfully  beautiful  calculation,  that  in  a  per- 
fect liquid  where  there  is  no  friction  it  is 
impossible  for  vortex-motion  to  be  generated 
or  destroyed  ;  in  any  part  of  the  liquid  where 
there  is  no  vortex-motion  no  mechanical  action 
can  possibly  start  it ;  but  where  it  once  exists 
there  it  is  for  ever,  and  no  mechanical  action 
can  possibly  stop  it.  A  vortex-ring  may  move 
from  place  to  place  ;  but  it  carries  with  it  the 
liquid  of  which  it  is  composed,  never  leaving 
any  particle  behind,  and  never  taking  up  any 
particle  from  the  surrounding  liquid.  If  we 
tried  to  cut  it  through  with  a  knife  it  would 
thin  out  like  a  stream  of  treacle,  and  the  thinner 
it  got  the  faster  it  would  go  round  ;  so  that  if 
we  multiplied  together  the  number  of  revolu- 
tions in  a  second,  and  the  number  of  square 
millimetres  in  the  cross-section  of  the  vortex- 
ring,  we  should  always  get  the  same  pro- 
duct, not  only  in  all  parts  of  the  ring,  but 
through  all  time.  Any  portion  of  liquid  which 
is  rotating  must  form  part  of  a  vortex -ring, 
either  returning  into  itself,  after  no  matter  how 


278  LECTURES  AND  ESSAYS 

many  knots  and  convolutions,  or  having  its  two 
ends  cut  off  at  the  surface  of  the  liquid.  That 
such  more  complex  forms  of  vortex-motion  may 
exist  is  easily  shown  by  making  knots  (to  be 
left  loose)  in  a  piece  of  string,  and  then  join- 
ing the  ends  :  motion  of  rotation  may  be  given 
to  any  part  of  it  by  rolling  it  between  two 
fingers,  and  will  be  carried  all  over  it.  Such  a 
knotted  vortex-ring  is  figured  on  the  cover  of 
the  "  Unseen  Universe  "  for  a  fitting  device. 

Thus  far  Helmholtz,  examining  into  the 
consequences  of  supposing  that  a  fiction,  serv- 
ing to  represent  the  actual  properties  of  liquids 
at  rest,  holds  good  also  in  the  case  of  motion. 
Here  steps  in  Sir  William  Thomson  with  a 
brilliant  conjecture.  The  ultimate  atom  of 
matter  is  required  to  be  indestructible,  to  have 
a  definite  mass,  and  definite  rates  of  vibration. 
A  vortex-ring  in  a  perfect  liquid  is  indestructible, 
has  a  definite  mass,  and  definite  rates  of  vibra- 
tion. Why  should  not  the  atom  be  a  vortex- 
ring  in  a  perfect  liquid  ?  If  the  whole  of  space 
were  filled  with  an  incompressible  frictionless 
fluid  in  which  vortex -rings  once  existed,  at 
least  some  of  the  known  phenomena  of  matter 
would  be  produced.  Why  should  it  not  be 
possible  in  this  way  to  explain  them  all  ? 

The  answer  to  this  question  is  only  to  be 
got  at  by  examining  further  into  the  con- 
sequences of  the  fundamental  supposition,  until 


THE  UNSEEN  UNIVERSE  279 

either  the  desired  explanation  of  all  phenomena 
is  reached  or  some  clear  discordance  with 
observed  results  shows  that  the  whole  hypo- 
thesis is  untenable.  To  this  task,  with  splendid 
energy  and  insight,  Sir  William  Thomson  has 
applied  himself;  arriving  at  results  which,  if 
they  are  not  the  foundation  of  the  final  theory 
of  matter,  are  at  least  imperishable  stones  in 
the  tower  of  dynamical  science. 

Independently,  however,  of  these  results  in 
the  theory  of  the  motion  of  perfect  liquids,  and 
independently  of  the  final  success  of  the  hypo- 
thesis itself,  it  has  led  to  two  very  important 
ideas  of  physical  explanation.  First,  there  is 
the  idea  that  matter  differs  from  ether  only  in 
being  another  state  or  mode  of  motion  of  the 
same  stuff;  which  suggests  the  hope  that  we 
may  by  and  by  get  to  know  something  about 
the  method  of  evolution  of  atoms,  and  the 
reason  why  there  are  so  many  kinds  of  them 
and  no  more.  It  must  not  be  supposed  that 
in  Sir  W.  Thomson's  hypothesis  the  part  of 
the  ether  is  played  simply  by  the  universal 
frictionless  fluid.  Such  a  fluid,  by  the  defini- 
tion of  it,  offers  no  resistance  to  a  change  of 
shape  of  any  part  of  it ;  but  the  actual  ether 
which  fills  space  is  so  elastic  that  the  slightest 
possible  distortion  produced  by  the  vibration  of 
a  single  atom  sends  a  shudder  through  it  with 
inconceivable  rapidity  for  billions  and  billions 


28o  LECTURES  AND  ESSAYS 

of  miles.  This  shudder  is  Light.  To  account 
for  such  elasticity  it  has  to  be  supposed  that 
even  where  there  are  no  material  molecules  the 
universal  fluid  is  full  of  vortex-motion,  but  that 
the  vortices  are  smaller  and  more  closely  packed 
than  those  of  matter,  forming  altogether  a  more 
finely  grained  structure.  So  that  the  difference 
between  matter  and  ether  is  reduced  to  a  mere 
difference  in  the  size  and  arrangement  of  the 
component  vortex-rings.  Now,  whatever  may 
turn  out  to  be  the  ultimate  nature  of  the  ether 
and  of  molecules,  we  know  that  to  some  extent 
at  least  they  obey  the  same  dynamic  laws,  and 
that  they  act  upon  one  another  in  accordance 
with  these  laws.  Until,  therefore,  it  is  absolutely 
disproved,  it  must  remain  the  simplest  and  most 
probable  assumption  that  they  are  finally  made 
of  the  same  stuff — that  the  material  molecule 
is  some  kind  of  knot  or  coagulation  of  ether. 

Secondly,  this  hypothesis  has  accustomed 
us  to  the  very  important  idea  that  the  hardness, 
resistance,  or  elasticity  of  solid  matter  may  be 
explained  by  the  very  rapid  motion  of  some- 
thing which  is  infinitely  soft  and  yielding. 
This  general  view  Sir  William  Thomson  has 
illustrated  by  exceedingly  beautiful  experiments. 
One  striking  form  is  the  complete  enclosure  of 
a  gyroscope  in  a  flat  cylindrical  box,  with  a 
sharp  projecting  edge,  so  that  the  motion  of 
the  contained  wheel  can  only  be  perceived  by 


THE  UNSEEN  UNIVERSE  281 

the  curious  resistance  to  rotation  of  the  box  ; 
which  will  balance  itself  on  its  edge  on  a  piece 
of  glass,  and  only  tremble  and  stand  firm  when 
it  is  struck  a  violent  blow  with  the  hand.  So 
also,  if  a  chain  hanging  straight  down  be  rapidly 
spun  round,  it  becomes  stiff  and  stark  like  a 
rigid  rod.  And,  lastly,  a  solid  suspended  in 
the  centre  of  a  globe  of  water  will,  when  the 
water  is  made  to  revolve  rapidly,  oscillate  about 
its  mean  position  as  if  it  were  fastened  by  a 
spring.  All  these  things  make  one  inclined  to 
look  to  the  rapid  motion  of  something  soft  for 
explanation  of  hardness  and  stiffness  ;  and  the 
value  of  this  explanation  does  not  depend  upon  the 
ultimate  success  of  the  hypothesis  of  vortex-atoms. 
But  these  things  being  admitted,  it  may 
perhaps  not  be  too  great  a  presumption  in  us 
to  make  some  criticisms  on  the  hypothesis  itself. 
A  true  explanation  describes  the  previous  un- 
known in  terms  of  the  known  ;  thus  light  is  de- 
scribed as  a  vibration,  and  such  properties  of  light 
as  are  also  properties  of  vibrations  are  thereby 
explained.  Now  a  perfect  liquid  is  not  a  known 
thing,  but  a  pure  fiction.  The  imperfect  liquids 
which  approximate  to  it,  and  from  which  the 
conception  is  derived,  consist  of  a  vast  number 
of  small  particles  perpetually  interfering  with  one 
another's  motion.  This  molecular  structure  not 
only  explains  the  fact  that  they  behave  like 
perfect  liquids  when  at  rest,  but  also  makes  it 


282  LECTURES  AND  ESSAYS 

necessary  that  they  should  not  behave  like 
perfect  liquids  when  in  motion.  Thus  a  liquid 
is  not  an  ultimate  conception,  but  is  explained 
— it  is  known  to  be  made  up  of  molecules  ; 
and  the  explanation  requires  that  it  should  not 
be  frictionless.  The  liquid  of  Sir  William 
Thomson's  hypothesis  is  continuous,  infinitely 
divisible,  not  made  of  molecules  at  all,  and  it  is 
absolutely  frictionless.  This  is  as  much  a  mere 
mathematical  fiction  as  the  attracting  and  repel- 
ling points  of  Boscovitch. 

The  authors  of  the  "  Unseen  Universe " 
modify  the  hypothesis  in  such  a  way  as  to  dis- 
pose of  this  objection.  They  regard  the  atoms 
as  not  absolutely  indestructible,  but  only  very 
long-lived.  Consequently  it  is  not  necessary 
for  them  that  the  universal  liquid  should  be 
quite  perfect,  but  only  that  its  viscosity  or 
friction  should  be  exceedingly  small — small 
enough  to  let  the  atoms  keep  going  for  billions 
of  years  when  they  are  once  started,  with  no 
appreciable  change  in  their  properties  during  the 
short  time  in  which  we  can  observe  them.  Thus, 
instead  of  a  fiction,  we  have  indeed  a  known 
thing,  an  imperfect  liquid,  by  which  to  explain 
the  molecules  that  are  wanted  to  explain  the 
properties  of  water.  Can  we,  then,  explain 
this  universal  imperfect  liquid  ?  Certainly  ;  it 
consists  of  molecules  inconceivably  smaller  than 
those  of  ordinary  matter.  But  how  to  explain 


THE  UNSEEN  UNIVERSE  283 

the  molecules  ?  Why,  clearly,  they  are  vortex- 
rings  in  a  liquid  of  still  finer  grain  and  less 
viscosity.  Molecules,  liquid,  molecules,  liquid, 
alternately  for  ever  ;  each  term  of  the  infinite 
series  being  fully  explained  by  the  next  follow- 
ing. Could  anything  be  more  satisfactory  ? 

It  is,  moreover,  to  be  observed  that  known 
facts  about  the  ether  and  about  atoms  do  lead 
us  a  very  great  way  towards  a  conception  of 
their  relative  structure.  The  experimental  dis- 
coveries and  the  geometric  insight  of  Faraday, 
and  the  application  to  these  of  mathematical 
analysis  by  Thomson,  Helmholtz,  and  above  all 
by  Clerk  Maxwell,  have  shown  that  the  ether 
which  was  required  for  the  theory  of  light  is 
capable  also  of  explaining  magnetic  and  electric 
phenomena.  Whatever  that  motion  is  which  is 
periodically  reversed  in  a  ray  of  light,  we  have 
very  strong  evidence  to  show  that  the  same 
motion  is  continuous  along  an  electric  current. 
This  stream  makes  vortex-motion  all  round  it, 
as  if  it  were  a  stick  drawn  through  indiarubber 
rings;  and  the  vortex -rings  are  Faraday's 
"  lines  of  magnetic  force."  The  direction  in 
which  a  small  magnet  will  point  indicates  at 
any  place  the  axis  of  rotation  of  the  ether : 
thus,  except  in  the  neighbourhood  of  magnets 
or  batteries,  the  ether  in  this  country  is  all 
rotating  in  a  plane  rather  tilted  up  on  the  north 
side.  According  to  Maxwell's  provisional  con- 


284  LECTURES  AND  ESSAYS 

ception,  we  may  suppose  that  this  rotation 
belongs  to  soft  balls,  all  spinning  the  same  way, 
and  separated  by  smaller  "  idle  wheels,"  which 
turn  in  the  opposite  direction.  It  is  a  con- 
tinuous stream  of  these  idle  wheels  that 
constitutes  an  electric  current.  Now  there  is 
great  reason  to  believe  that  every  material 
atom  carries  upon  it  a  small  electric  current, 
if  it  does  not  wholly  consist  of  this  current. 
For,  in  the  first  place,  every  particle  of  a 
magnet  is  itself  a  magnet.  Now,  when  a  piece 
of  iron  is  magnetised,  there  are  two  possible 
suppositions :  either  every  particle  is  made  into 
a  magnet  as  it  stands,  having  had  no  previous 
magnetism  ;  or  else  all  the  particles  were 
originally  magnets  which  neutralise  one  another 
because  they  were  turned  in  all  manner  of 
directions,  but  which  by  the  process  of  mag- 
netising have  been  made  to  approximate  to 
the  same  direction.  The  latter  supposition  is 
conclusively  picked  out  by  experiment  as  the 
true  one.  Thus  it  seems  that  the  molecule  of 
iron  is  a  magnet.  If,  however,  the  magnetism 
of  the  molecules  were  so  much  increased  that 
they  held  each  other  tight,  and  so  could  not  be 
turned  round  by  ordinary  magnetising  forces,  it 
is  shown  that  effects  would  be  produced  like  those 
of  diamagnetism.  Faraday  gave  reasons  for 
believing  that  all  bodies  are  either  ferromagnetic 
or  diamagnetic.  Next,  the  theory  of  Ampere, 


THE  UNSEEN  UNIVERSE  285 

confirmed  by  many  subsequent  experiments 
and  calculations,  makes  all  magnetism  to 
depend  upon  small  electric  currents.  But 
magnetism  is  an  affair  of  molecules  ;  if  the 
molecules  are  groups  of  atoms  we  find  in  this 
way  good  reason  to  suppose  that  all  atoms 
carry  upon  them  electric  currents. 

Three  important  sets  of  phenomena  are 
(among  many  others)  still  unexplained — the 
action  of  molecules  upon  one  another,  the 
action  of  transparent  bodies  on  light,  and 
gravitation.  The  precise  law  of  action  of  mole- 
cules on  one  another  is  in  fact  unknown,  the 
inverse  fifth  power  of  the  distance,  proposed  by 
Maxwell,  having  been  given  up  on  the  evidence 
of  later  experiments.  The  study  of  the  mutual 
action  of  free  small  magnets  in  space  offers 
mathematical  difficulties  which  at  present  pre- 
vent us  from  saying  whether  a  great  number 
of  these  magnets  would  have  such  known  pro- 
perties of  gases  as  depend  upon  the  law  of 
mutual  action  of  molecules.  Transparent 
bodies  act  upon  light  as  if  the  ether  in  their 
interior  were  somewhat  less  elastic  than  the 
ether  outside  them.  It  is  possible  that  this 
change  of  elasticity  may  be  explained  by  the 
electric  field  surrounding  their  molecules, 
although  the  most  powerful  fields  that  we  can 
produce  have  not  yet  been  observed  to  have 
any  such  effect.  There  is  something  left  for 


286  LECTURES  AND  ESSAYS 

gravitation.  In  the  theories  of  electric  and 
magnetic  action  the  motion  of  the  "  idle  wheels," 
except  in  actual  currents,  is  neglected  in  com- 
parison with  that  of  the  revolving  soft  spheres. 
It  is,  perhaps,  conceivable  that  in  some  way  or 
other  an  explanation  may  be  found  in  them  for 
the  relatively  weaker  force  of  gravitation.  If 
— and  what  an  if! — these  three  explanations 
were  made  out,  we  might  reasonably  suppose 
not  merely  that  an  atom  carries  an  electric 
current,  but  that  it  is  nothing  else.  We  should 
thus  be  led  to  find  an  atom,  not  in  the  rota- 
tional motion  of  a  vortex-ring,  but  in  irrotational 
motion  round  a  re-entering  channel.  It  might 
well  be  that  such  motion,  to  be  permanent, 
must  have  some  definite  relation  to  the  size  of 
the  rotating  spheres  and  their  interstices,  so 
that  only  certain  kinds  of  atoms  could  survive. 
In  this  way  we  may  get  an  explanation  of  the 
definite  number  of  chemical  elements,  and  of 
the  fact  that  all  the  molecules  of  each  are  as 
near  alike  as  we  can  judge. 

The  position  is  this.  We  know,  with  great 
probability,  that  wherever  there  is  an  atom 
there  is  a  small  electric  current.  Very  many 
of  the  properties  of  atoms  are  explained  by 
means  of  this  current :  we  have  vague  hopes 
that  all  the  rest  will  likewise  be  explained.  If 
these  hopes  should  be  realised,  we  shall  say 
that  an  atom  is  a  small  current.  If  not,  we 


THE  UNSEEN  UNIVERSE  287 

shall  have  to  say  that  it  is  a  small  current  and 
something  else  besides. 

Of  course,  after  all  this,  there  is  room  for 
vortex-motion  or  other  such  hypothesis  to  ex- 
plain the  observed  properties  of  the  ether  ;  but 
in  the  last  resort  all  these  questions  of  physical 
speculation  abut  upon  a  metaphysical  question. 
We  are  describing  phenomena  in  terms  of 
phenomena  ;  the  objects  we  observe  are  groups 
of  perceptions,  and  exist  only  in  our  minds  ; 
the  molecules  and  ether,  in  terms  of  which  we 
describe  them,  are  only  still  more  complex 
mental  images.  Is  there  anything  that  is  not 
in  our  minds  of  which  these  things  are  pictures 
or  symbols  ?  and  if  so,  what  ? 

Our  authors  reply  that  matter  and  energy 
possess  this  external  reality,  because  they  can- 
not be  created  or  destroyed  by  us  ;  the  quantity 
of  each  is  fixed  and  invariable.  The  argument 
is  better  than  most  that  belong  to  this  question, 
but  it  will  not  hold  water  for  a  moment.  Every 
quantitative  relation  among  phenomena  can  be 
put  into  a  form  which  asserts  the  constancy  of 
some  quantity  which  can  be  calculated  from 
the  phenomena.  "  Gravitation  is  inversely  as 
the  square  of  the  distance  for  the  same  two 
bodies "  ;  this  may  be  also  said  in  the  form, 
"gravitation  multiplied  by  the  square  of  the 
distance  is  constant  for  the  same  two  bodies." 
"  Pressure  varies  as  density,  in  a  perfect  gas  at 


288  LECTURES  AND  ESSAYS 

the  same  temperature,"  may  be  also  expressed, 
"  pressure  divided  by  density  is  constant  in  a 
perfect  gas  at  the  same  temperature."  But 
this  does  not  make  the  quotient  of  pressure  by 
density  to  be  an  external  reality  transcending 
phenomena.  It  is  entirely  beside  the  question, 
as  we  may  see  in  another  way.  A  dream  is  a 
succession  of  phenomena  having  no  external 
reality  to  correspond  to  them.  Do  we  never 
dream  of  things  that  we  cannot  destroy? 

So  the  fact  that  matter,  as  a  phenomenon,  is 
not  to  be  increased  or  diminished  in  quantity, 
has  nothing  to  say  to  the  question  about  the 
existence  of  something  which  is  not  matter,  not 
phenomenon  at  all,  but  of  which  matter  is  the 
symbol  or  representative.  The  answer  to  this 
question  is  only  to  be  found  in  the  theory  of 
sensation  ;  which  tells  us  not  merely  that  there 
is  a  non-phenomenal  counterpart  of  the  material 
or  phenomenal  world,  but  also  in  some  measure 
what  it  is  made  of.  Namely,  the  reality  cor- 
responding to  our  perception  of  the  motion  of 
matter  is  an  element  of  the  complex  thing  we 
call  feeling.  What  we  might  perceive  as  a 
plexus  of  nerve-disturbances  is  really  in  itself  a 
feeling  ;  and  the  succession  of  feelings  which 
constitutes  a  man's  consciousness  is  the  reality 
which  produces  in  our  minds  the  perception  of 
the  motions  of  his  brain.  These  elements  of 
feeling  have  relations  of  ne.rtness  or  contiguity 


THE  UNSEEN  UNIVERSE  289 

in  space,  which  are  exemplified  by  the  sight- 
perceptions  of  contiguous  points  ;  and  relations 
of  succession  in  time,  which  are  exemplified  by 
all  perceptions.  Out  of  these  two  relations  the 
future  theorist  has  to  build  up  the  world  as 
best  he  may.  Two  things  may,  perhaps,  help 
him.  There  are  many  lines  of  mathematical 
thought  which  indicate  that  distance  or  quantity 
may  come  to  be  expressed  in  terms  of  position 
in  the  wide  sense  of  the  analysis  situs.  And 
the  theory  of  space-curvature  hints  at  a  possi- 
bility of  describing  matter  and  motion  in  terms 
of  extension  only. 

So  much  for  the  vortex-atom,  its  relation  to 
the  present  state  of  science,  and  the  prospects 
of  physical  speculation.  We  propose  now  to 
follow  our  authors  farther  ;  to  examine  their 
hypothesis  of  a  second  ether,  and  to  see  what 
good  it  can  do  them. 

There  are  four  ways  of  accounting  for  the 
too  small  number  of  stars  of  low  magnitudes 
without  assuming  that  light  is  absorbed  by  the 
ether.  In  the  first  place,  the  calculation  as- 
sumes that  stars  are  distributed  with  approxi- 
mate uniformity  over  infinite  space.  So  far  is 
this  from  being  true,  that  we  know  the  vast 
majority  of  stars  that  we  can  see  to  belong  to 
a  single  system,  of  which  the  nebulae  also  are 
members,  and  which  occupies  a  finite  portion 
of  space.  It  is  very  probable  that  around  and 

VOL.  I  U 


290  LECTURES  AND  ESSAYS 

beyond  this,  to  distances  vaster  even  than  its 
vast  dimensions,  there  are  regions  nearly  devoid 
of  stars.  If  other  such  systems  do  anywhere 
exist,  they  may  well  be  too  far  off  to  be  seen 
at  all.  The  method  of  Struve  has,  indeed,  been 
beautifully  applied  by  Mr.  Charles  S.  Peirce  to 
the  richer  materials  now  at  hand  with  the  view 
of  determining  approximately  the  shape  of  the 
solar  galaxy  and  the  mode  of  distribution  of 
stars  in  it.  Secondly,  a  great  amount  of  light 
must  be  stopped  by  the  dark  bodies  of  burnt- 
out  suns.  Thirdly,  space  contains  gaseous 
matter  in  a  state  of  extreme  diffusion — not  too 
rare,  however,  to  produce  an  effect  in  distances 
so  enormous  as  we  have  here  to  consider. 
Lastly,  the  possible  curvature  and  finite  extent 
of  space  have  been  suggested  by  Zollner  as  an 
escape  from  the  reasoning  of  Olbers  and 
Struve.  Of  these  four  the  first  is  undoubtedly 
the  true  account  of  the  matter,  and  will  supply 
us  with  trustworthy  knowledge  of  the  contents 
of  surrounding  space. 

But  if  the  ether  did  absorb  light  what  would 
this  mean  ?  Vibratory  motion  of  solids,  which 
is  really  a  molecular  disturbance,  is  absorbed 
by  being  transformed  into  other  kinds  of  mole- 
cular motion,  and  so  may  finally  be  transferred 
to  the  ether.  There  is  no  reason  why  vibratory 
motion  of  the  ether  should  not  be  transformed 
into  other  kinds  of  ethereal  motion  ;  in  fact, 


THE  UNSEEN  UNIVERSE  291 

there  is  no  reason  why  it  should  not  go  to  the 
making  of  atoms.  Of  course  there  is  equally 
no  reason  why  it  should  ;  but  we  present  this 
speculation  to  anybody  who  wants  the  universe 
to  go  on  for  ever. 

Apart  from  this,  however,  the  laws  of  motion 
and  the  conservation  of  energy  are  very  general 
propositions  which  are  as  nearly  true  as  we  can 
make  out  for  gross  bodies,  and  which,  being 
tentatively  applied  to  certain  motions  of  mole- 
cules and  the  ether,  are  found  to  fit.  There  is 
nothing  to  tell  us  that  they  are  absolutely  ex- 
act in  any  particular  case,  or  that  they  are 
everywhere  and  always  true.  If  it  were  shown 
conclusively  that  energy  was  lost  from  the 
ether,  it  would  not  at  all  follow  that  it  was 
handed  on  to  anything  else.  The  right  state- 
ment might  be  that  the  conservation  of  energy 
was  only  a  very  near  approximation  to  the 
facts. 

It  is  perhaps  hardly  necessary  to  say  that 
the  experiment  of  Tait  and  Balfour  Stewart, 
who  found  that  a  disc  was  heated  by  rapid 
rotation  in  vacuo,  though  of  the  first  importance 
in  itself,  by  no  means  bears  upon  the  question 
of  the  internal  friction  of  the  ether.  That  a 
molecule  in  travelling  through  the  ether  should 
be  made  to  vibrate  is  just  what  we  might 
expect ;  the  only  wonder  is  that  it  gets  through 
with  so  little  resistance.  But  this  is  a  transfer 


292  LECTURES  AND  ESSAYS 

of  energy  of  translation  of  a  molecule  into 
energy  of  vibration  ;  a  task  to  which  one  ether 
is  entirely  competent. 

Far  greater,  indeed,  is  the  work  which  the 
second  ether  has  to  perform  :  nothing  less  than 
the  fashioning  of  a  "  spiritual  body."  While 
our  consciousness  proceeds  part  passu  with 
molecular  disturbance  in  our  brains,  this  mole- 
cular disturbance  agitates  the  first  ether,  which 
transfers  a  part  of  its  energy  to  the  second. 
Thus  is  gradually  elaborated  an  organism  in 
that  second  or  unseen  universe,  with  whose 
motions  our  consciousness  is  as  much  connected 
as  it  is  with  our  material  bodies.  When  the 
marvellous  structure  of  the  brain  decays,  and  it 
can  no  more  receive  or  send  messages,  then  the 
spiritual  body  is  replete  with  energy,  and  starts 
off  through  the  unseen,  taking  consciousness 
with  it,  but  leaving  its  molecules  behind. 
Having  grown  with  the  growth  of  our  mortal 
frame,  and  preserving  in  its  structure  a  record 
of  all  that  has  befallen  us,  it  becomes  an  organ 
of  memory,  linking  the  future  with  the  past, 
and  securing  a  personal  immortality. 

Can  another  body,  then,  avail  to  stay  the 
hand  of  death,  and  shall  man  by  a  second 
nervous  system  escape  scot  free  from  the  ruin 
of  the  first?  We  think  not.  The  laws  con- 
necting consciousness  with  changes  in  the  brain 
are  very  definite  and  precise,  and  their  necessary 


THE  UNSEEN  UNIVERSE  293 

consequences  are  not  to  be  evaded  by  any  such 
means.  Consciousness  is  a  complex  thing 
made  up  of  elements,  a  stream  of  feelings. 
The  action  of  the  brain  is  also  a  complex  thing 
made  up  of  elements,  a  stream  of  nerve- 
messages.  For  every  feeling  in  consciousness 
there  is  at  the  same  time  a  nerve-message  in 
the  brain.  This  correspondence  of  feeling  to 
nerve-message  does  not  depend  on  the  feeling 
being  part  of  a  consciousness,  and  the  nerve- 
message  part  of  the  action  of  a  brain.  How 
do  we  know  this  ?  Because  the  nervous  system 
of  animals  grows  more  and  more  simple  as  we 
go  down  the  scale,  and  yet  there  is  no  break 
that  we  can  point  to  and  say,  "  above  this  there 
is  consciousness  or  something  like  it ;  below 
there  is  nothing  like  it."  Even  to  those  nerve- 
messages  which  do  not  form  part  of  the  con- 
tinuous action  of  our  brains,  there  must  be 
simultaneous  feelings  which  do  not  form  part 
of  our  consciousness.  Here,  then,  is  a  law 
which  is  true  throughout  the  animal  king- 
dom ;  nerve-message  exists  at  the  same  time 
with  feeling.  Consciousness  is  not  a  simple 
thing,  but  a  complex  ;  it  is  the  combination  of 
feelings  into  a  stream.  It  exists  at  the  same 
time  with  the  combination  of  nerve-messages 
into  a  stream.  If  individual  feeling  always 
goes  with  individual  nerve-message,  if  combina- 
tion or  stream  of  feelings  always  goes  with 


294  LECTURES  AND  ESSAYS 

stream  of  nerve-messages,  does  it  not  follow 
that  when  the  stream  of  nerve -messages  is 
broken  up,  the  stream  of  feelings  will  be  broken 
up  also,  will  no  longer  form  a  consciousness  ? 
does  it  not  follow  that  when  the  messages 
themselves  are  broken  up,  the  individual  feel- 
ings will  be  resolved  into  still  simpler  elements  ? 
The  force  of  this  evidence  is  not  to  be  weakened 
by  any  number  of  spiritual  bodies.  Inexorable 
facts  connect  our  consciousness  with  this  body 
that  we  know  ;  and  that  not  merely  as  a  whole, 
but  the  parts  of  it  are  connected  severally  with 
parts  of  our  brain  -  action.  If  there  is  any 
similar  connection  with  a  spiritual  body,  it  only 
follows  that  the  spiritual  body  must  die  at  the 
same  time  with  the  natural  one. 

Consider  a  mountain  rill.  It  runs  down  in 
the  sunshine,  and  its  water  evaporates  ;  yet  it 
is  fed  by  thousands  of  tiny  tributaries,  and  the 
stream  flows  on.  The  water  may  be  changed 
again  and  again,  yet  still  there  is  the  same 
stream.  It  widens  over  plains,  or  is  prisoned 
and  fouled  by  towns  ;  always  the  same  stream  ; 
but  at  last 

"even  the  weariest  river 
Winds  somewhere  safe  to  sea." 

When  that  happens  no  drop  of  the  water  is 
lost,  but  the  stream  is  dead. 


THE  UNSEEN  UNIVERSE  295 

3ttS    Vf,' 

III. 

Our  authors  "  assume,  as  absolutely  self- 
evident,  the  existence  of  a  Deity  who  is  the 
Creator  of  all  things."  They  must  both  have 
had  enough  to  do  with  examinations  to  be 
aware  that  "  it  is  evident "  means  "  I  do  not 
know  how  to  prove."  The  creation,  however, 
was  not  necessarily  a  direct  process  ;  the  great 
likeness  of  atoms  gives  them  the  "  stamp  of  the 
manufactured  article,"  and  so  they  must  have 
been  made  by  intelligent  agency,  but  this  may 
have  been  the  agency  of  finite  and  conditioned 
beings.  As  such  beings  would  have  bodies 
made  of  one  or  other  of  the  ethers,  this  form  of 
the  argument  escapes  at  least  one  difficulty  of 
the  more  common  form,  which  may  be  stated 
as  follows : — "  Because  atoms  are  exactly  alike 
and  apparently  indestructible,  they  must  at 
one  time  have  come  into  existence  out  of 
nothing.  This  can  only  have  been  effected  by 
the  agency  of  a  conscious  mind  not  associated 
with  a  material  organism."  Forasmuch  as  the 
momentous  character  of  the  issue  is  apt  to 
blind  us  to  the  logic  of  such  arguments  as  these, 
it  may  not  be  useless  to  offer  for  consideration 
the  following  parody  :  "  Because  the  sea  is  salt 
and  will  put  out  a  fire,  there  must  at  one  time 
have  been  a  large  fire  lighted  at  the  bottom  of 


296  LECTURES  AND  ESSAYS 

it.  This  can  only  have  been  effected  by  the 
agency  of  the  whale  who  lives  in  the  middle  of 
Sahara."  But  let  us  return  to  our  finite  in- 
telligences having  ethereal  bodies,  who  made 
the  atomic  vortex-rings  out  of  ether.  With 
such  a  machinery  it  seems  a  needless  simplifica- 
tion to  adopt  Prout's  hypothesis,  and  suppose 
that  the  sixty-three  elements  are  compounded 
of  one  simpler  form  of  matter.  Rather  let  us 
contemplate  the  reposeful  picture  of  the  uni- 
versal divan,  where  these  intelligent  beings 
whiled  away  the  tedium  of  eternity  by  blowing 
smoke-rings  from  sixty-three  different  kinds  of 
mouths.  We  may  suppose,  if  we  like,  that  the  in- 
telligent beings  were  all  alike,  and  each  had  sixty- 
three  mouths  ;  or  that  each  was  so  constituted  in 
his  physical  or  moral  nature  that  he  could  or 
would  pull  only  sixty-three  faces.  How  lofty 
must  have  been  the  existence  of  such  a  makerand 
master  of  grimace  !  How  fertile  of  resource  is 
the  theologic  method,  when  it  once  has  clay  for 
its  wheel ! 

As  the  permanence  of  matter  proves  the 
existence  of  an  external  reality,  a  substance  in 
which  all  things  consist,  so  the  conservation  of 
energy  points  to  a  principle  of  motion,  coming 
out  of  the  unconditioned,  entering  into  the 
visible  universe  and  obeying  its  laws,  to  pass 
back  finally  into  the  unseen  world.  But, 
further,  the  fact  that  organisms  large  enough  to 


THE  UNSEEN  UNIVERSE  297 

be  visible  have  not  yet  under  the  conditions  of 
the  laboratory  been  produced  from  inorganic 
matter,  shows  that  life  is  a  great  mystery,  pene- 
trating into  the  depths  of  the  arcana  of  the 
universe,  proceeding  from  substance  and  energy 
and  yet  not  identical  with  either.  The  reader 
will  see  what  this  points  to.  It  is  clear  that 
the  good  old  gods  of  our  race  —  sun,  sky, 
thunder,  and  beauty — are  to  be  replaced  by 
philosophic  abstractions  —  substance,  energy, 
and  life,  under  the  patronage  respectively  of 
the  persons  of  the  Christian  Trinity.  But  why 
are  we  to  stay  here?  Is  not  neurility,  the 
universal  function  of  nerves,  as  much  a  special 
and  distinct  form  of  life  as  life  is  a  distinct 
form  of  energy  ?  And  over  against  these 
physical  principles,  absolutely  separate  and 
distinct  from  them,  stands  Consciousness,  which 
cannot  be  left  out  of  a  fair  estimate  of  the 
world.  It  would  seem  fitting  that  the  presi- 
dency and  patronage  of  the  nerves  should  be 
assigned  to  the  modern  Isis  as  her  portion. 
While  if,  as  Von  Hartmann  says,  Conscious- 
ness is  the  great  mistake  of  the  universe,  it  will 
not  unsuitably  fall  to  the  care  of  the  devil. 
In  this  way  we  shall  save  the  odd  number 
(numero  deus  impare  gaudet\  and  give  a 
certain  historical  completeness  to  our  repre- 
sentation. 

But  why  does  a  material  so  plastic  present 


298  LECTURES  AND  ESSAYS 

itself  in  this  identical  shape  ?  Why  this 
particular  trinity  of  the  great  Ptah,  Horus  the 
Son,  and  Kneph  the  Wind-god,  retained  and 
refurbished  by  bishops  of  Alexandria  and 
Carthage  out  of  the  wrecks  of  Egyptian  super- 
stition ?  Not  because  it  is  contained  in  the 
unseen  universe,  but  because  we  were  born  in  a 
particular  place.  If  you,  however,  choose  to 
find  one  thing  in  the  chain  of  ethers,  we  may 
quite  lawfully  find  another.  If  there  is  room 
in  the  unseen  universe  for  the  harmless  pan- 
theistic deities  which  our  authors  have  put  there, 
room  may  also  be  found  for  the  goddess  Kali, 
with  her  obscene  rites  and  human  sacrifices,  or 
for  any  intermediate  between  these.  Here  is 
the  clay  :  make  your  images  to  your  heart's 
desire  ! 

When  Mohammed  was  conquering  Arabia, 
a  certain  tribe  offered  to  submit  if  they  should 
be  spared  the  tribute  and  service  in  the  holy 
war,  and  if  they  might  keep  their  idol  Lat  for  a 
year.  The  prophet  agreed,  and  began  to 
dictate  to  his  scribe  the  terms  of  the  treaty. 
When  it  came  to  the  permission  of  idolatry  he 
paused  and  looked  on  the  ground.  The  envoys 
were  impatient,  and  repeated  the  article.  Then 
arose  Omar,  and  turned  upon  them  furious. 
"  You  have  soiled  the  heart  of  the  Prophet,"  he 
said  ;  "  may  God  fill  your  hearts  with  fire ! " 
"  I  refuse  the  treaty,"  said  Mohammed,  looking 


THE  UNSEEN  UNIVERSE  299 

up.  "  Let  us  keep  Lat  only  six  months,  then," 
pleaded  the  envoys.  "  Not  another  hour,"  said 
the  Prophet  ;  and  he  drove  them  out  and 
subdued  them. 

"  Only  for  another  half-century  let  us  keep 
our  hells  and  heavens  and  gods."  It  is  a 
piteous  plea ;  and  it  has  soiled  the  heart  of 
these  prophets,  great  ones  and  blessed,  giving 
light  to  their  generation,  and  dear  in  particular 
to  our  mind  and  heart.  These  sickly  dreams 
of  hysterical  women  and  half-starved  men,  what 
have  they  to  do  with  the  sturdy  strength  of  a 
wide-eyed  hero  who  fears  no  foe  with  pen  or 
club  ?  This  sleepless  vengeance  of  fire  upon 
them  that  have  not  seen  and  have  not  believed, 
what  has  it  to  do  with  the  gentle  patience  of 
the  investigator  that  shines  through  every  page 
of  this  book,  that  will  ask  only  consideration 
and  not  belief  for  anything  that  has  not  with 
infinite  pains  been  solidly  established  ?  That 
which  you  keep  in  your  hearts,  my  brothers,  is 
the  slender  remnant  of  a  system  which  has 
made  its  red  mark  on  history,  and  still  lives  to 
threaten  mankind.  The  grotesque  forms  of  its 
intellectual  belief  have  survived  the  discredit  of 
its  moral  teaching.  Of  this  what  the  kings 
could  bear  with,  the  nations  have  cut  down  ; 
and  what  the  nations  left,  the  right  heart  of 
man  by  man  revolts  against  day  by  day.  You 
have  stretched  out  your  hands  to  save  the  dregs 


300  LECTURES  AND  ESSAYS 

of  the  sifted  sediment  of  a  residuum.  Take  heed 
lest  you  have  given  soil  and  shelter  to  the  seed 
of  that  awful  plague  which  has  destroyed  two 
civilisations,  and  but  barely  failed  to  slay  such 
promise  of  good  as  is  now  struggling  to  live 
among  men. 


THE   PHILOSOPHY   OF  THE   PURE 
SCIENCES  l 

I. — STATEMENT   OF   THE   QUESTION 

ON  entering  this  room  and  looking  rapidly 
round,  what  do  I  see  ?  I  see  a  theatre,  with  a 
gallery,  and  with  an  arrangement  of  seats  in 
tiers.  I  see  people  sitting  upon  these  seats, 
people  with  heads  more  or  less  round,  with 
bodies  of  a  certain  shape  ;  sitting  in  various 
positions.  Above  I  see  a  roof  with  a  skylight, 
and  a  round  disc  evidently  capable  of  vertical 
motion.  Below  I  see  the  solid  floor  supporting 
us  all.  In  front  of  me  I  see  a  table,  and  my 
hands  resting  upon  it.  In  the  midst  of  all 
these  things  I  see  a  void  space,  which  I  can 
walk  about  in  if  I  like.  The  different  things 
I  have  mentioned  I  see  at  various  distances 
from  one  another,  and  from  me  ;  and  (now  that 
the  door  is  shut)  I  see  that  they  completely 
enclose  this  void  space,  and  hedge  it  in.  My 

1  Lectures  delivered  at  the  Royal  Institution  in  March  1873. 


302  LECTURES  AND  ESSAYS 

view  is  not  made  of  patches  here  and  there, 
but  is  a  continuous  boundary  going  all  round 
the  void  space  I  have  mentioned.  All  this  I 
see  to  exist  at  the  same  time  ;  but  some  of  you 
are  not  sitting  quite  still,  and  I  see  you  move  ; 
that  is  to  say,  I  see  you  pass  from  one  position 
into  another  by  going  through  an  infinite  series 
of  intermediate  positions.  Moreover,  when  I 
put  my  hands  on  the  table,  I  feel  a  hard  flat 
horizontal  surface  at  rest,  covered  with  cloth. 

Have  I  spoken  correctly  in  making  these 
assertions  ?  Yes,  you  will  say,  this  is  on  the 
whole  just  what  I  ought  to  have  seen  and  felt 
under  the  circumstances.  With  the  exception 
of  one  or  two  points  expressed  in  too  technical 
a  form,  this  is  just  the  sort  of  language  that  a 
witness  might  use  in  describing  any  ordinary 
event,  without  invalidating  his  testimony.  You 
would  not  say  at  once,  "  This  is  absurd  ;  the 
man  must  not  be  listened  to  any  longer."  And 
if,  having  been  precisely  in  my  situation,  you 
wished  to  describe  facts  with  the  view  of  draw- 
ing inferences  from  them — even  important 
inferences — you  would  make  all  these  state- 
ments as  matter  of  your  own  direct  personal 
experience  ;  and  if  need  were,  you  would  even 
testify  to  them  in  a  court  of  law. 

And  yet  I  think  we  shall  find  on  a  little  re- 
flection that  not  one  of  these  statements  can  by 
any  possibility  have  been  strictly  true. 


PHILOSOPHY  OF  THE  PURE  SCIENCES       303 

"  I  see  a  theatre."  I  do  not ;  the  utmost  I 
can  possibly  see  is  two  distinct  curved  pictures 
of  a  theatre.  Upon  the  two  retinas  of  my  eyes 
there  are  made  pictures  of  the  scene  before  me, 
exactly  as  pictures  are  made  upon  the  ground 
glass  in  a  photographer's  camera.  The  sensa- 
tion of  sight  which  I  get  comes  to  me  at  any 
rate  through  those  two  pictures  ;  and  it  cannot 
tell  me  any  more,  or  contain  in  itself  any  more, 
than  is  in  those  two  pictures.  Now  the 
pictures  are  not  solid  ;  each  of  them  is  simply 
a  curved  surface  variously  illuminated  at 
various  parts.  Whereas,  therefore,  I  think  I 
see  a  solid  scene,  having  depth,  and  relief,  and 
distance  in  it,  reflection  tells  me  that  I  see 
nothing  of  the  kind  ;  but  only  (at  the  most) 
two  distinct  surfaces,  having  no  depth  and  no 
relief,  and  only  a  kind  of  distance  which  is 
quite  different  from  that  of  the  solid  figures 
before  me.  You  will  say,  probably,  that  this 
is  only  a  quibble  on  two  senses  of  the  word 
"  see."  Whether  it  is  so  or  not  makes  no 
difference  to  our  subsequent  argument ;  and 
yet  I  think  you  will  admit  that  the  latter  sense, 
in  which  I  do  not  see  the  solid  things,  is  the 
more  correct  one.  For  the  question  is  not 
about  what  is  there,  but  about  what  I  see. 
Now  exactly  the  same  sensation  can  be  pro- 
duced in  me  by  two  slightly  different  pictures 
placed  in  a  stereoscope  —  I  say  exactly  the 


3o4  LECTURES  AND  ESSAYS 

same ;  because  if  I  had  sufficiently  accurate 
coloured  photographs  of  this  room  properly 
illuminated,  the  rays  of  light  converging  on 
every  part  of  each  of  my  retinas  might  be  made 
exactly  the  same  as  they  are  now ;  and  the 
sensation  would  therefore  not  only  appear  to 
be  the  same  but  would  actually  be  the  same. 
I  should  think  I  saw  a  solid  scene;  and  I 
should  not  be  seeing  one.  Now  to  see,  and  to 
see  what  is  actually  there,  are  two  different 
things. 

Again,  "  I  see  people  with  heads  more  or 
less  round." — I  cannot  see  your  heads  ;  I  can 
only  see  your  faces.  I  must  have  imagined  the 
rest.  But  just  consider  what  it  is  that  I  have 
imagined.  It  is  merely  that  besides  what  I  do 
see  I  have  added  something  that  I  might  see 
by  going  round  to  the  other  side  ?  No,  there 
is  more  than  that.  The  complete  sensation 
which  I  have  of  a  human  head  when  I  look  at 
one  is  not  merely  something  which  I  do  not  see 
now,  but  something  which  I  never  could  see  by 
any  possibility.  I  have  the  sensation  of  a  solid 
object,  and  not  of  a  series  of  pictures  of  a  solid 
object.  Although  that  sensation  may  be  really 
constructed  out  of  a  countless  number  of  possible 
pictures,  yet  it  is  not  like  any  of  them.  I  im- 
agine to  myself,  and  seem  to  see  the  other  side  of 
things,  not  as  it  would  look  if  viewed  from  beyond 
them,  but  as  it  would  look  if  viewed  from  here. 


PHILOSOPHY  OF  THE  PURE  SCIENCES       305 

I  seem  to  see  the  back  of  your  head,  not  as  it 
would  look  if  I  got  behind  you,  but  as  if  I  saw 
it  through  your  face  from  the  spot  where  I  am 
standing  ;  and  that,  you  know,  is  impossible. 

I  seem  to  see  all  these  objects  as  exist- 
ing together.  But  really  as  a  matter  of  fact 
I  move  my  eyes  about  and  see  a  succes- 
sion of  small  pictures  very  rapidly  changed. 
Each  of  my  eyes  has  six  muscles  which  pull  it 
about,  and  if  I  knew  which  of  these  muscles 
were  moving,  and  how  fast,  at  any  moment, 
I  should  get  information  about  the  direction 
in  which  my  eye  was  looking  at  the  time. 
Now  it  is  only  a  very  small  part  of  the  scene 
before  me  that  I  can  really  see  distinctly  at 
once  ;  so  that  I  have  really  seen  a  panorama, 
and  not  the  one  large  picture  that  I  imagined  ; 
and  yet  while  looking  at  the  small  portion 
which  I  can  really  see  distinctly,  I  think  I  see 
distinctly  the  whole  room. 

Again,  I  seem  to  see  that  in  some  directions, 
at  least,  this  void  space  in  the  middle  is  com- 
pletely bounded — the  surface  of  the  floor,  for 
example,  which  bounds  it,  appears  to  be  com- 
pletely filled  up  and  continuous,  to  have  no 
breaks  in  it.  And  when  you  move  I  seem  to 
see  you  go  continuously  from  one  position  to 
another  through  an  infinite  series  of  intermediate 
positions.  Now,  quite  apart  from  the  question 
whether  these  conclusions  are  true  or  not,  it 

VOL.  I  X 


306  LECTURES  AND  ESSAYS 

can  be  made  out  distinctly  that  I  could  not 
possibly  see  either  the  surface  of  a  thing,  or  a 
motion,  as  continuous  ;  for  the  sensitive  portion 
of  my  retina,  which  receives  impressions,  is  not 
itself  a  continuous  surface,  but  consists  of  an 
enormously  large  but  still  finite  number  of 
nerve  filaments  distributed  in  a  sort  of  network. 
And  the  messages  that  go  along  my  nerves  do 
not  consist  in  any  continuous  action,  but  in  a 
series  of  distinct  waves  succeeding  one  another 
at  very  small  but  still  finite  intervals.  All  I 
can  possibly  have  seen  therefore  at  any  moment 
is  a  picture  made  of  a  very  large  number  of 
very  small  patches,  exceedingly  near  to  one 
another,  but  not  actually  touching.  And  all 
I  can  have  seen  as  time  passed  is  a  succession 
of  such  distinct  pictures  coming  rapidly  after 
one  another.  You  know  that  precisely  as  the 
stereoscope  is  made  to  imitate  the  property  of 
my  two  eyes  out  of  which  I  imagine  solid 
things,  so  another  instrument  has  been  con- 
structed to  imitate  that  property  of  my  nerves 
out  of  what  I  imagine  continuous  motion.  The 
instrument  is  called  the  Zoetrope,  or  Wheel  of 
Life.  It  presents  to  you  a  succession  of  distinct 
pictures  coming  after  one  another  at  small  in- 
tervals ;  and  the  impression  produced  by  that 
series  is  precisely  the  impression  of  one  thing 
in  continuous  motion. 

Let  us  now  put  shortly  together  what  we 


PHILOSOPHY  OF  THE  PURE  SCIENCES       307 

have  said  about  this  sensation  of  sight  I  shall 
use  the  word  mosaic  to  represent  a  few  discon- 
nected patches  which  a  painter  might  put  down 
with  a  view  of  remembering  a  scene  he  had  no 
time  to  sketch.  Then,  I  seem  to  see  a  large 
collection  of  solid  objects  in  continuous  motion. 
The  utmost  I  can  really  see  is  a  panorama 
painted  in  mosaic  and  shown  in  a  wheel  of  life. 
I  do  not  know  that  my  direct  perception 
amounts  to  so  much  ;  but  it  cannot  possibly 
amount  to  more.  What  it  really  does  amount 
to  must  be  reserved  for  subsequent  discussion. 
At  any  rate  I  must  have  imagined  the  rest. 

Lastly,  when  I  put  my  hands  on  the  table, 
I  feel  a  hard,  flat,  horizontal  surface  at  rest, 
covered  with  cloth.  Now  there  are  three 
things  that  really  happen.  First,  there  is  a 
definite  kind  of  irritation  of  certain  organs  of 
my  skin,  called  papillae.  It  is  that  irritation 
that  makes  me  say  cloth.  Secondly,  certain  of 
my  muscles  are  in  a  state  of  compression,  and 
they  tell  me  that.  Thirdly,  I  make  a  certain 
muscular  effort  which  is  not  followed  by  motion. 
This  is  all  that  I  can  really  feel  ;  but  those 
three  things  do  not  constitute  a  hard,  flat, 
horizontal  surface  covered  with  cloth.  As 
before,  I  must  have  imagined  the  rest 

Do  not  suppose  that  I  am  advocating  any 
change  in  our  common  language  about  sensation. 
I  do  not  want  anybody  to  say,  for  instance, 


308  LECTURES  AND  ESSAYS 

instead  of,  "  I  saw  you  yesterday  on  the  other 
side  of  the  street,"  "  I  saw  a  series  of  panoramic 
pictures  in  a  sort  of  mosaic,  of  such  a  nature 
that  the  imaginations  I  constructed  out  of  them 
were  not  wholly  unlike  the  imaginations  I  have 
constructed  out  of  similar  series  of  panoramic 
pictures  seen  by  me  on  previous  occasions  when 
you  were  present."  This  would  be  clumsy,  and 
it  would  not  be  sufficient.  And  yet  I  cannot 
help  thinking  that  in  certain  assemblies,  when 
some  of  those  who  are  present  are  in  an  exalted 
state  of  emotional  expectation,  and  the  lights 
are  low,  even  this  roundabout  way  of  putting 
things  might  be,  to  say  the  least,  a  salutary 
exercise. 

But  the  conclusion  I  want  you  to  draw  from 
all  this  that  we  have  been  saying  is  that  there 
are  really  two  distinct  parts  in  every  sensation 
that  we  get.  There  is  a  message  that  comes 
to  us  somehow  ;  but  this  message  is  not  all 
that  we  apparently  see  and  hear  and  feel.  In 
every  sensation  there  is,  besides  the  actual 
message,  something  that  we  imagine  and  add 
to  the  message.  This  is  sometimes  expressed 
by  saying  that  there  is  a  part  which  comes 
from  the  external  world  and  a  part  which  is 
supplied  by  the  mind.  But  however  we  ex- 
press it,  the  fact  to  be  remembered  is  that  not 
the  whole  of  a  sensation  is  immediate  experi- 
ence (where  by  immediate  experience  I  mean 


PHILOSOPHY  OF  THE  PURE  SCIENCES       309 

the  actual  message — whatever  it  is — that  comes 
to  us) ;  but  that  this  experience  is  supplemented 
by  something  else  which  is  not  in  it.  And  thus 
you  may  see  that  it  is  a  perfectly  real  question, 
"  Where  does  this  supplement  come  from  ? " 
This  question  has  been  before  philosophers  for 
a  very  long  time  ;  and  it  is  this  question  that 
we  have  to  discuss. 

But  first  of  all  we  must  inquire  a  little 
further  into  the  nature  of  the  supplement  by 
which  we  fill  in  our  experience.  When  I  fill 
in  my  experience  of  this  room  in  the  way  that  I 
have  described,  I  do  not  do  so  at  random,  but 
according  to  certain  rules.  And  in  fact  I 
generally  fill  it  in  right;  that  is  to  say,  from 
the  imaginations  that  I  have  built  up  I  can 
deduce  by  rules  certain  other  experiences  which 
would  follow  from  actions  of  a  definite  sort. 
When  I  seem  to  see  a  solid  floor,  I  conclude 
that  if  I  went  there  I  could  feel  it  as  I  do  the 
table.  And  upon  trial  these  conclusions  in 
general  turn  out  right.  I  cannot  therefore  have 
filled  in  my  experience  at  random,  but  accord- 
ing to  certain  rules.  Let  us  now  consider 
what  are  a  few  of  these  rules. 

In  the  first  place,  out  of  pictures  I  have 
imagined  solid  things.  Out  of  space  of  two 
dimensions,  as  we  call  it,  I  have  made  space 
of  three  dimensions,  and  I  imagine  these  solid 
things  as  existing  in  it  ;  that  is  to  say,  as  having 


310  LECTURES  AND  ESSAYS 

certain  relations  of  distance  to  one  another. 
Now  these  relations  of  distance  are  always  so 
filled  in  as  to  fulfil  a  code  of  rules,  some  called 
common  notions,  and  some  called  definitions, 
and  some  called  postulates,  and  some  assumed 
without  warning,  but  all  somehow  contained  in 
Euclid's  Elements  of  Geometry.  For  example, 
I  sometimes  imagine  that  I  see  two  lines  in  a 
position  which  I  call  parallel.  Parallelism  is 
impossible  on  the  curved  pictures  of  my  retina; 
so  this  is  part  of  the  filling  in.  Now  when- 
ever I  imagine  that  I  see  a  quadrilateral  figure 
whose  opposite  sides  are  parallel,  I  always  fill 
them  in  so  that  the  opposite  sides  are  also 
equal.  This  equality  is  also  a  part  of  the  filling 
in,  and  relates  to  possible  perceptions  other 
than  the  one  immediately  present.  From  this 
example,  then,  you  can  see  that  the  funda- 
mental axioms  and  definitions  of  geometry  are 
really  certain  rules  according  to  which  we 
supplement  or  fill  in  our  experience. 

Now  here  is  a  rather  more  complicated  ex- 
ample. If  I  see  a  train  going  along  and  a 
man  moving  inside  of  it,  I  fill  in  the  motion  of 
the  train  as  continuous  out  of  a  series  of  dis- 
tinct pictures  of  it  ;  and  so  also  I  fill  in  the 
motion  of  the  man  relatively  to  the  train  as 
continuous.  I  imagine  all  motions,  therefore, 
according  to  the  rule  of  continuity  ;  that  is, 
between  the  distinct  pictures  which  I  see,  I 


PHILOSOPHY  OF  THE  PURE  SCIENCES       311 

insert  an  infinite  number  of  intermediate 
pictures.  Moreover,  both  of  these  motions  are 
imagined  in  accordance  with  the  laws  of 
geometry  ;  that  is  to  say,  they  are  imagined  so 
that  the  relations  of  distance  at  any  instant  obey 
those  laws.  But  now  I  may,  if  I  like,  consider, 
besides  the  motion  of  the  train  and  the  motion 
of  the  man  relative  to  it,  the  motion  of  the  man 
relative  to  me,  as  if  there  were  no  train  ;  and 
this  like  the  other  motions  is  part  of  the  filling 
in.  But  I  always  fill  this  in  in  such  a  way  that 
the  three  motions — of  the  train  by  itself,  of  the 
man  by  himself,  and  of  the  man  relatively  to 
the  train — satisfy  certain  rules,  by  which  one 
can  be  found  when  the  other  two  are  given. 
These  rules  are  called  the  laws  of  kinematic,  or 
of  the  pure  science  of  motion. 

Then  we  may  say,  to  begin  with,  that  we 
supplement  our  experience  in  accordance  with 
certain  rules  ;  and  that  some  of  these  rules  are 
the  foundations  of  the  pure  sciences  of  Space 
and  Motion. 

Instead  of  Space  and  Motion,  many  people 
would  like  to  say  Space  and  Time.  But  in  re- 
gard to  the  special  matter  that  we  are  consider- 
ing, it  seems  to  me,  for  reasons  which  I  do  not 
wish  to  give  at  present,  to  be  more  correct  to 
say  that  we  imagine  time  by  putting  together 
space  and  motion,  than  that  we  imagine  motion 
by  putting  together  space  and  time. 


312  LECTURES  AND  ESSAYS 

There  are  other  rules,  besides  those  of  space 
and  motion,  according  to  which  we  fill  in  our 
experience.  One  of  these  rules  I  may  call  the 
continuity  of  things.  I  can  see  this  table,  and 
feel  it,  and  hear  a  sound  when  I  strike  it.  The 
table  is  an  imagination  by  which  I  fill  in  a 
great  variety  of  different  experiences.  It  is 
what  I  call  a  thing.  Now,  if  I  come  into  this 
room  again,  and  have  any  experience  of  the 
table,  I  shall  fill  it  in  in  such  a  way  as  to  imply 
that  the  same  variety  of  experiences  might  be 
combined  again  ;  that  is,  I  shall  imagine  the 
thing  to  be  persistent.  But  this  rule  will  not 
apply  universally,  and  I  do  not  always  observe 
it  Because  I  have  seen  a  tree  without  leaves 
in  the  winter,  I  do  not  in  the  summer  fill  in  my 
experience  of  the  trunk  with  imagination  of 
leafless  branches  above.  But  I  do  fill  in  the 
two  experiences  with  an  imagination  of  an 
infinite  series  of  gradual  intermediate  changes. 
Some  people  divide  this  rule  into  two — the 
persistence  of  substance  and  the  continuity  of 
qualities.  I  prefer  to  make  one  rule,  and  to 
call  it  the  continuity  of  things.  Things — that 
is  to  say,  combinations  of  possible  experience — 
are  not  persistent,  but  they  change  continuously 
in  the  imagination  by  which  we  fill  up  that  ex- 
perience. Or  we  may  say  that  experience  at 
any  one  time  is  always  so  filled  in  as  to  aggre- 
gate together  the  possible  perceptions  implied 


PHILOSOPHY  OF  THE  PURE  SCIENCES       313 

by  the  result  into  groups  which  we  call  things  ; 
and  that  experience  of  a  period  of  time  is 
always  so  filled  in  that  things  change  only  in  a 
continuous  manner. 

Another  rule  of  the  supplement  which  we 
imagine  is  that  which  provides  that  these 
changes  of  things  shall  take  place  according  to 
a  certain  uniformity.  The  simplest  case  of  this 
is  when  the  same  experience  is  repeated,  and 
we  fill  up  the  changes  subsequent  to  the  second 
experience  so  that  they  shall  be  the  same  as 
those  subsequent  to  the  first.  It  is  not  neces- 
sary that  the  experience  should  be  actually 
repeated  ;  it  may  only  be  filled  up  in  the  same 
way.  The  uniformity,  however,  which  is  in- 
volved in  this  law  is  a  much  more  complicated 
thing  than  this  simple  case.  I  can  only  say 
here  that  experience  is  filled  up  always  so 
as  to  make  the  imagined  history  of  things 
exhibit  some  uniformity  ;  but  the  definiteness 
of  this  varies  in  different  individuals  and  at 
different  times.  Some  people  prefer  to  call 
this  the  law  of  causation,  and  to  say  that  we 
always  supplement  our  experiences  in  such  a 
way  that  every  event  has  a  cause  or  causes 
which  determine  it,  and  effects  which  flow 
from  it. 

Now  all  this  filling  up  that  we  have  been 
considering  happens  directly  in  the  sensations 
that  I  get  from  day  to  day,  just  as  I  get  them. 


314  LECTURES  AND  ESSAYS 

(It  is  convenient  to  use  the  word  sensation  as 
meaning  the  whole  phenomenon,  not  only  the 
immediate  experience,  but  also  the  supplement.) 
But  if  I  want  to  talk  to  you  about  them,  or  if, 
advancing  upon  that  practice,  I  talk  to  myself 
about  them,  then  I  am  obliged  to  use  language, 
or  to  represent  them  by  signs  ;  and  this  requires 
me  to  group  them  in  a  new  manner.  I  have 
to  make  imaginations  not  of  things,  but  of  whole 
series  of  things,  of  relations  of  these  to  one 
another,  and  combinations  of  the  relations.  I 
have  to  construct,  in  fact,  what  I  shall  call  for 
shortness  the  apparatus  of  thought — the  means 
by  which  I  talk  to  myself.  For  there  seems 
reason  to  think  that  the  conceptions  which 
correspond  to  general  terms — names  of  a  class, 
or  of  an  abstract  relation — are  first  rendered 
necessary  by  the  language  which  expresses 
them.1  But  however  that  may  be,  this  new 
world  of  conceptions  is  not  made  wholly  at 
random,  but  satisfies  certain  laws.  For  ex- 
ample, in  order  to  describe  a  certain  group  of 
things,  I  introduce  the  very  complicated  concep- 
tion six,  and  say  there  are  six  of  them.  Now, 
whenever  this  is  done  in  the  case  of  two  groups, 
giving  rise  to  the  conceptions  six  and  three,  it 
is  possible  to  apply  the  same  process  to  the 


l  See  this  view  ably  defended  in  Professor  Max  Mullers 
Lectures,  delivered  at  the  Royal  Institution  in  April  1873,  and 
since  published  in  Prater's  Magazine. 


PHILOSOPHY  OF  THE  PURE  SCIENCES       315 

group  compounded  of  those  two,  and  it  always 
gives  rise  to  the  conception  nine.  Here,  then, 
is  a  law  of  combination  to  which  the  world  of 
conceptions  has  to  conform.  And  another  is 
this  :  If  every  individual  which  belongs  to  the 
class  A  belongs  also  to  the  class  B,  and  if  every 
individual  which  belongs  to  the  class  B  belongs 
also  to  the  class  C,  then  always  every  individual 
which  belongs  to  the  class  A  belongs  also  to 
the  class  C.  Rules  like  these  which  regulate 
the  world  of  conceptions,  built  out  of  our 
sensations,  are  also  said  to  belong  to  the  pure 
sciences  ;  and  the  two  examples  which  I  have 
chosen  belong  respectively  to  the  sciences  of 
Number  and  Logic. 

There  may  be  other  kinds  of  rules  according 
to  which  experience  is  supplemented  and  sensa- 
tions are  built  up  into  conceptions  ;  but  I  am 
not  aware  of  any  more  kinds,  and  perhaps  those 
that  I  have  mentioned  will  be  sufficient  for  our 
purpose.  I  will  just  state  again  the  names  of  the 
sciences  which  consist  in  these  three  groups  : — 

The  rules  about  Space  and  Motion  constitute 
the  pure  sciences  of  Geometry  and  Kinematic. 

The  rules  about  Things  and  Uniformity 
have  been  said  to  belong  to  a  pure  science  of 
Nature. 

The  rules  about  Numbers  and  Classes  con- 
stitute the  pure  sciences  of  Arithmetic  and 
Formal  Logic. 


316  LECTURES  AND  ESSAYS 

But  for  the  present  let  us  confine  our  atten- 
tion to  the  first  group  of  rules,  those  which 
relate  to  space  and  motion.  There  is  one  other 
property  of  them  which  we  have  to  consider, 
besides  the  fact  that  our  experience  is  filled  up 
in  accordance  with  them.  I  have  already 
mentioned  this  property,  but  only  in  passing. 
It  is  that  in  general  this  filling  in  of  experience 
is  right:  and  that,  so  far  as  these  rules  are 
concerned,  it  is  not  only  right  in  general,  but 
always  right.  That  is  to  say,  if  from  the  sensa- 
tion which  is  made  by  the  filled-up  experience 
we  predict  certain  other  perceptions  as  con- 
sequent upon  our  actions,  these  predictions  will 
actually  be  fulfilled.  To  take  the  example  we 
considered  before,  I  always  imagine  a  parallelo- 
gram so  that  its  opposite  sides  are  equal.  Now 
the  conclusion  from  this  is  that  if  I  go  to  the 
parallelogram  and  apply  one  of  the  sides  to 
the  other,  I  shall  not  perceive  any  difference. 
The  rule  by  which  I  supplement  my  perception 
is  also  a  true  statement  about  objects  ;  it  is 
capable  of  a  certain  kind  of  verification,  and  it 
always  stands  this  test. 

Here,  however,  I  could  use  the  word  equal 
only  in  its  practical  sense,  in  which  two  things 
are  equal  when  I  cannot  perceive  their  differ- 
ence ;  not  in  its  theoretical  sense,  in  which 
two  things  are  equal  when  they  have  no 
difference  at  all.  But  there  has  been  for  ages 


PHILOSOPHY  OF  THE  PURE  SCIENCES       317 

a  conviction  in  the  minds  of  men  that  these 
rules  about  space  are  true  objectively  in  the 
exact  or  theoretical  sense,  and  under  all 
possible  circumstances.  If  two  -straight  lines 
are  drawn  perpendicular  to  the  same  plane, 
geometers  would  have  told  you  for  more  than 
two  thousand  years  that  these  straight  lines  may 
be  prolonged  for  ever  and  ever  without  getting 
the  least  bit  nearer  to  one  another  or  further 
away  from  one  another  ;  and  that  they  were 
perfectly  certain  of  this.  They  knew  for 
certain  that  the  sum  of  the  angles  of  a  triangle, 
no  matter  how  big  or  how  small  it  was, .  or 
where  it  was  situated,  must  always  be  exactly 
equal  to  two  right  angles,  neither  more  nor  less. 
And  those  who  were  philosophers  as  well  as 
geometers  knew  more  than  this.  They  knew  not 
only  that  the  thing  was  true,  but  that  it  could  not 
possibly  have  been  otherwise  ;  that  it  was  neces- 
sarily true.  And  this  means,  apparently,  not 
merely  that  I  know  that  it  must  be,  but  that  I 
know  that  you  must  know  that  it  must  be. 

The  case  of  arithmetical  propositions  is 
perhaps  more  easily  comprehended  in  this 
respect.  Everybody  knows  that  six  things  and 
three  things  make  nine  things  at  all  possible 
times  and  places  ;  you  cannot  help  seeing  not 
only  that  they  do  always  without  exception 
make  nine  things,  but  that  they  must  do  so, 
and  that  the  world  could  not  have  been  con- 


3i8  LECTURES  AND  ESSAYS 

structed  otherwise.  For  to  those  ingenious 
speculations  which  suppose  that  in  some  other 
planet  there  may  always  be  a  tenth  thing  in- 
evitably suggested  upon  the  union  of  the  six 
and  the  three,  so  that  they  cannot  be  added 
together  without  making  ten  ;  to  these,  I  say, 
it  may  be  replied  that  the  words  number  and 
thing,  if  used  at  all,  must  have  different  mean- 
ings in  that  planet.  The  reply  is  important, 
and  I  shall  return  to  it  in  a  subsequent  lecture. 
Locke  and  Hume  gave  explanations  of  the 
existence  of  two  of  these  general  rules  which  I 
have  put  into  my  second  group.  Locke  ex- 
plained the  notion  of  substance,  the  notion  that 
a  thing  means  something  more  than  an  aggre- 
gate of  possible  perceptions,  by  the  fact  that  we 
are  accustomed  to  get  these  perceptions  all 
together  ;  by  this  custom  they  are  welded  or 
linked  together,  and  our  imagination  of  the 
thing  is  then  this  connected  structure  of  per- 
ceptions, which  is  called  up  as  a  whole  when- 
ever one  or  more  of  the  component  perceptions 
is  called  up.  Having  thus  by  custom  formed 
the  complete  sensation  which  we  have  of  the 
thing,  we  suppose  that  this  is  a  message,  like 
the  actual  perceptions,  and  comes  from  some- 
thing outside.  That  something  is  the  substance. 
Locke  did  not  admit  that  this  supposition  is 
right,  and  that  the  linking  together  of  messages 
is  really  itself  a  message  ;  but  still  he  thought 


PHILOSOPHY  OF  THE  PURE  SCIENCES       319 

there  was  something  outside  to  correspond  to 
this  linking.  Hume  explained  in  the  same 
way  the  rule  of  causation.  He  said  we  get  it 
from  being  accustomed  to  perceive  one  event 
following  another  ;  so  that  these  two  percep- 
tions got  linked  together,  and  when  one  of 
them  occurs  alone,  we  fill  it  in  with  the  other 
one.  And  then,  regarding  this  link,  produced 
only  by  custom,  as  if  it  were  a  message  from 
somewhere,  like  the  simple  perceptions,  we  give 
it  the  name  of  causation. 

These  explanations  agree  in  saying  that  the 
supplement  of  experience  is  made  up  of  past 
experience,  together  with  links  which  bind  to- 
gether perceptions  that  have  been  accustomed 
to  occur  together.  This  fact,  that  perceptions 
and  feelings  which  have  frequently  occurred 
together  get  linked,  so  that  one  calls  up  the 
other,  is  called  the  law  of  Association,  and  has 
been  made  the  basis  of  scientific  Psychology. 
According  to  these  explanations  of  Locke  and 
Hume  (which  extended  to  the  other  two  groups 
of  rules)  all  the  knowledge  we  have  that  the 
rules  are  right,  or  may  be  objectively  verified, 
is  really  derived  from  experience  ;  only  it  is 
past  experience,  which  we  have  had  so  often 
and  got  so  accustomed  to  that  it  is  now  really 
a  part  of  ourselves. 

But  Kant,  after  being  staggered  for  some 
time  by  Hume's  explanation,  at  length  said, 


320  LECTURES  AND  ESSAYS 

"  It  is  impossible  that  all  your  knowledge  can 
have  come  from  experience.  For  you  know 
that  the  axioms  of  mathematics  are  absolutely 
and  universally  true,  and  no  experience  can 
possibly  have  told  you  this.  However  often 
you  may  have  found  the  angles  of  a  triangle 
amount  to  two  right  angles,  however  accustomed 
you  may  have  got  to  this  experience,  you  have 
no  right  to  know  that  the  angles  of  every 
possible  triangle  are  equal  to  two  right  angles, 
nor  indeed  that  those  of  any  one  triangle  are 
absolutely  and  exactly  so  equal.  Now  you  do 
know  this,  and  you  cannot  deny  it.  You  have 
therefore  some  knowledge  which  could  not 
possibly  be  derived  from  experience ;  it  must 
therefore  have  come  in  some  other  way ;  or 
there  is  some  other  source  of  knowledge  besides 
experience." 

At  that  time  there  was  no  answer  whatever 
to  this.  For  men  did  think  that  they  knew  at 
least  the  absolute  universality  if  not  the  neces- 
sity of  the  mathematical  axioms.  To  any  one 
who  admitted  the  necessity,  the  argument  was 
even  stronger ;  for  it  was  clear  that  no  experi- 
ence could  make  any  approach  to  supply 
knowledge  of  this  quality.  But  if  a  man  felt 
absolutely  sure  that  two  straight  lines  per- 
pendicular to  the  same  line  would  never  meet, 
however  far  produced,  he  could  not  maintain 
against  Kant  that  all  knowledge  is  derived 


PHILOSOPHY  OF  THE  PURE  SCIENCES       321 

from  experience.  He  was  obliged  to  admit 
the  existence  of  knowledge  a  priori,  that  is, 
knowledge  lying  ready  in  the  mind  from  the 
first,  antecedent  to  all  experience. 

But  now  here  is  a  difficulty  to  be  explained. 
How  is  it  possible  that  I  can  have  knowledge 
about  objects  which  is  prior  to  all  experience 
of  objects,  and  which  transcends  the  bounds  of 
possible  experience? 

First  of  all,  what  do  I  mean  by  objects  ? 
In  the  answer  to  this  question  lies  really  Kant's 
solution  of  the  problem,  and  I  shall  endeavour 
to  make  this  clear  by  a  comparison. 

If  a  man  had  on  a  pair  of  green  spectacles, 
he  would  see  everything  green.  And  if  he 
found  out  this  property  of  his  spectacles,  he 
might  say  with  absolute  certainty  that  while  he 
had  those  spectacles  on  everything  that  he  saw 
without  exception  would  be  green. 

"  Everything  that  he  saw  ; "  that  is  to  say, 
all  objects  of  sight  to  him.  But  here  it  is  clear 
that  the  word  object  is  relative  ;  it  means  a 
representation  that  he  gets,  and  has  nothing  to 
do  with  the  thing  in  itself.  And  the  assertion 
that  everything  is  green  would  not  be  an 
assertion  about  the  things  in  themselves,  but 
about  the  representations  of  them  which  came 
to  him.  The  colour  of  these  representations 
would  depend  partly  on  the  things  outside  and 
partly  on  his  spectacles.  It  would  vary  for 
VOL.  l  Y 


322  LECTURES  AND  ESSAYS 

different  things,  but  there  would  always  be 
green  in  it. 

Let  us  modify  this  example  a  little.  I 
know  for  certain  that  the  colour  of  every  object 
in  the  universe  is  made  up  of  colours  that  lie 
within  the  range  of  the  visible  spectrum.  This 
is  apparently  a  universal  statement,  and  yet  I 
know  it  to  be  true  of  things  which  it  is  im- 
possible that  I  should  ever  see.  How  is  this  ? 
Why,  simply,  that  my  eyes  are  only  affected 
by  light  which  lies  within  the  range  of  the 
visible  spectrum.  Now  I  say  that  this  case  is 
only  a  little  modified  from  the  previous  one. 
The  green  glass  lets  in  a  certain  range  of  light ; 
the  range  is  very  little  increased  when  you  take 
it  away.  Only  in  the  second  case  it  happens 
that  we  are  all  actually  wearing  very  nearly  the 
same  spectacles.  That  universal  statement 
which  I  made  is  true  not  only  of  objects  as 
they  appear  to  me,  but  also  of  objects  as 
they  appear  to  you.  It  is  a  statement  about 
objects ;  that  is,  about  certain  representations 
which  we  perceive.  It  may  therefore  so  far 
have  its  origin  in  the  things  of  which  these  are 
representations,  or  it  may  have  its  origin  in  us. 
And  we  happen  to  know  that  in  this  case  it 
is  not  a  statement  about  external  things,  but 
about  our  eyes. 

Admitting,  then,  that  the  objects  of  our 
sensations  are  representations  made  to  us  ;  that 


PHILOSOPHY  OF  THE  PURE  SCIENCES       323 

their  character  must  therefore  be  partly 
dependent  upon  our  own  character  ;  what 
properties  of  these  objects  should  we  naturally 
suppose  to  have  this  origin,  to  be  derived  from 
the  constitution  of  our  minds  ?  Why,  clearly, 
those  which  are  necessary  and  universal  ;  for 
only  such  properties  can  be  so  derived,  and 
there  is  no  other  way  in  which  they  can  be 
known  to  be  universal. 

Accordingly,  Kant  supposes  that  Space  and 
Time  are  necessary  forms  of  perception,  imposed 
upon  it  by  the  perceiving  mind  ;  that  things  are 
in  space  and  time  as  they  appear  to  us,  and  not 
in  themselves  ;  and  that  consequently  the  state- 
ment that  all  things  exist  in  space  and  time  is  a 
statement  about  the  nature  of  our  perception 
and  not  about  the  things  perceived. 

The  word  corresponding  to  experience 
(Erfahrung)  is  used  by  Kant  nearly  in  the 
sense  in  which  I  have  used  sensation,  to  mean 
the  whole  phenomenon  consisting  of  the  bare 
message  and  also  of  the  filling-in,  the  complete 
representation  which  we  get  of  objects.  But  it 
is  not  apparently  confined  to  this  ;  it  means 
not  merely  the  sensations  which  I  get,  but  the 
sensations  which  I  talk  about.  Giving  to  the 
word  this  sense  for  the  present,  we  may  say 
that  in  his  theory  the  form,  the  general  char- 
acter, of  experience  is  imposed  upon  it  by  two 
faculties  which  we  all  possess :  Intuition  and 


324  LECTURES  AND  ESSAYS 

Understanding.  Intuition  has  necessarily  the 
forms  of  Space  and  Time  ;  but  we  are  not  to 
say  that  those  properties  of  space  which  are 
expressed  in  the  geometrical  axioms  are  all 
necessitated  by  the  forms  of  intuition  ;  for  it  is 
the  understanding  that  supplies  us  with  the 
pure  notions  of  quantity,  quality,  relation,  and 
modality.  It  is  not  always  easy  to  separate 
the  parts  played  by  these  two  faculties  in 
supplying  the  general  rules  to  which  experience 
conforms ;  but  it  appears,  for  example,  that 
the  three  dimensions  of  space  are  given  by  pure 
intuition  itself,  while  the  equality  of  the  opposite 
sides  of  a  parallelogram  is  only  given  by  help 
of  the  understanding.  It  is  not  to  our  purpose 
to  investigate  the  difference  between  these  two 
faculties,  or  even  to  remember  that  Kant  made 
a  distinction  between  them.  All  that  is  im- 
portant for  us  is  the  theory  that  those  general 
statements  upon  which  the  pure  sciences  are 
founded,  although  really  true  of  objects,  that  is 
of  representations  made  to  me,  are  in  fact  state- 
ments about  me  and  not  about  the  things  in 
themselves  :  just  as  my  general  statement  about 
the  colours  of  things  was  really  a  statement 
about  my  own  eyes  and  not  about  the  things. 
And  it  is  just  because  these  statements  are 
about  me  that  I  know  them  to  be  not  only 
universally,  but  always  necessarily  true  about 
the  objects  I  perceive  ;  for  it  is  always  the 


PHILOSOPHY  OF  THE  PURE  SCIENCES       325 

same  me  that  perceives  them — or  at  any  rate  it 
is  a  me  possessing  always  the  same  faculties  of 
representation. 

Now  observe  what  it  is  that  this  theory  does 
with  general  statements  ;  what  is  the  means  by 
which  it  gets  rid  of  them — for  it  does  get  rid 
of  them.  It  makes  them  into  particular  state- 
ments. Instead  of  being  statements  about  all 
possible  places  and  times  and  things,  they  are 
made  out  to  be  statements  about  me,  and  about 
other  men  in  so  far  as  they  have  the  same 
faculties  that  I  have.  I  want  you  to  notice 
this  transformation  particularly,  because  I  shall 
afterwards  endeavour  to  establish  a  similar 
transformation,  though  in  rather  a  different 
manner. 

In  the  next  place,  observe  that  the  question 
which  was  proposed  by  the  Critical  Philosophy 
is  a  perfectly  real  and  important  question.  It 
is  this  : — "  Are  there  any  properties  of  objects 
in  general  which  are  really  due  to  me  and  to 
the  way  in  which  I  perceive  them,  and  which 
do  not  belong  to  the  things  themselves  ?  "  But 
it  seems  to  me  that  the  method  by  which  Kant 
attempted  to  answer  this  question  was  not  the 
right  method.  It  consisted  in  finding  what  are 
those  characters  of  experience  which  we  know 
to  be  necessary  and  universal ;  and  concluding 
that  these  are  characters  of  me.  It  requires, 
therefore,  some  infallible  way  of  judging  what 


326  LECTURES  AND  ESSAYS 

characters  are  necessary  and  universal.  Now, 
unfortunately,  as  I  hope  to  show  you,  judgments 
of  this  kind  may  very  possibly  be  mistaken. 
If  you  went  up  to  our  man  with  the  green 
spectacles,  and  argued  with  him  that  since  he 
knew  for  certain  that  everything  was  green, 
whereas  no  experience  could  tell  him  so,  this 
greenness  must  be  somewhere  in  the  apparatus 
by  which  he  perceived  things  ;  there  would  be 
just  one  weakness  in  the  argument.  He  might 
be  mistaken  in  thinking  he  knew  that  every- 
thing was  green.  But  the  proper  thing  to  do, 
as  it  appears  to  me,  would  be  to  take  him  to  a 
looking-glass  and  show  him  that  these  spectacles 
were  actually  upon  his  nose.  And  so  also  in 
the  general  question  which  is  proposed  by 
the  Critical  Philosophy.  The  answer  to  that 
question  must  be  sought  not  in  the  subjective 
method,  in  the  conviction  of  universality  and 
necessity,  but  in  the  physiological  method,  in 
the  study  of  the  physical  facts  that  accompany 
sensation,  and  of  the  physical  properties  of  the 
nervous  system.  The  materials  for  this  valid 
criticism  of  knowledge  did  not  exist  in  Kant's 
time.  I  believe  that  they  do  exist  at  present 
to  such  an  extent  at  least  as  to  indicate 
the  nature  of  the  results  which  that  criticism 
is  to  furnish. 

The  Kantian  theory  of  universal  truths  was 
largely,  though    not   completely,   accepted    by 


PHILOSOPHY  OF  THE  PURE  SCIENCES       327 

Whewell,  and  applied  with  considerable  detail 
in  his  Philosophy  of  the  Inductive  Sciences. 
It  is  necessary  to  mention  him  here,  not  on 
account  of  any  important  modification  that  he 
introduced  into  the  theory,  but  because  the  form 
into  which  he  put  it  has  had  great  influence  in 
directing  the  attention  of  scientific  students  to 
the  philosophy  of  science  ;  and  because  by  in- 
telligent controversy  he  contributed  very  much 
to  the  clearing  up  and  development  of  an 
opinion  which  we  have  next  to  consider — that 
of  Mr.  John  Stuart  Mill.  I  can  best,  I  think, 
set  this  opinion  before  you,  if  I  have  permission 
to  quote  a  short  passage. 

"  To  these  arguments  (of  Dr.  Whewell,  con- 
tending that  the  axioms  could  not  be  known 
by  experience)  ...  a  satisfactory  answer  will, 
I  conceive,  be  found,  if  we  advert  to  one  of  the 
characteristic  properties  of  geometrical  forms — 
their  capacity  of  being  painted  in  the  imagina- 
tion with  a  distinctness  equal  to  reality  :  in 
other  words,  the  exact  resemblance  of  our  ideas 
of  form  to  the  sensations  which  suggest  them. 
This,  in  the  first  place,  enables  us  to  make  (at 
least  with  a  little  practice)  mental  pictures  of  all 
possible  combinations  of  lines  and  angles,  which 
resemble  the  realities  quite  as  well  as  any  which 
we  could  make  on  paper  ;  and  in  the  next  place, 
make  those  pictures  just  as  fit  subjects  of  geo- 


328  LECTURES  AND  ESSAYS 

metrical  experimentation  as  the  realities  them- 
selves ;  inasmuch  as  pictures,  if  sufficiently 
accurate,  exhibit  of  course  all  the  properties 
which  would  be  manifested  by  the  realities  at 
one  given  instant,  and  on  simple  inspection  ; 
and  in  geometry  we  are  concerned  only  with 
such  properties,  and  not  with  that  which 
pictures  could  not  exhibit,  the  mutual  action 
of  bodies  upon  one  another.  The  foundations 
of  geometry  would  therefore  be  laid  in  direct 
experience,  even  if  the  experiments  (which  in 
this  case  consist  merely  in  attentive  contempla- 
tion) were  practised  solely  upon  what  we  call 
our  ideas,  that  is,  upon  the  diagrams  in  our 
minds,  and  not  upon  outward  objects.  For  in 
all  systems  of  experimentation  we  take  some 
objects  to  serve  as  representatives  of  all  which 
resemble  them  ;  and  in  the  present  case  the 
conditions  which  qualify  a  real  object  to  be  the 
representative  of  its  class  are  completely  fulfilled 
by  an  object  existing  only  in  our  fancy. 
Without  denying,  therefore,  the  possibility  of 
satisfying  ourselves  that  two  straight  lines 
cannot  enclose  a  space,  by  merely  thinking  of 
straight  lines  without  actually  looking  at  them, 
I  contend  that  we  do  not  believe  this  truth  on 
the  ground  of  the  imaginary  intuition  simply, 
but  because  we  know  that  the  imaginary  lines 
exactly  resemble  real  ones,  and  that  we  may 
conclude  from  them  to  real  ones  with  quite  as 


PHILOSOPHY  OF  THE  PURE  SCIENCES       329 

much  certainty  as  we  could  conclude  from  one 
real  line  to  another.  The  conclusion,  therefore, 
is  still  an  induction  from  observation.  And 
we  should  not  be  authorised  to  substitute 
observation  of  the  image  in  our  mind  for 
observation  of  the  reality,  if  we  had  not  learnt 
by  long -continued  experience  that  the  pro- 
perties of  the  reality  are  faithfully  represented  in 
the  image  ;  just  as  we  should  be  scientifically 
warranted  in  describing  an  animal  which  we 
had  never  seen  from  a  picture  made  of  it  with 
a  daguerreotype  ;  but  not  until  we  had  learnt 
by  ample  experience  that  observation  of  such  a 
picture  is  precisely  equivalent  to  observation  of 
the  original. 

"These  considerations  also  remove  the 
objection  arising  from  the  impossibility  of  our 
ocularly  following  the  lines  in  their  prolonga- 
tion to  infinity.  For  though,  in  order  actually 
to  see  that  two  given  lines  never  meet,  it  would 
be  necessary  to  follow  them  to  infinity ;  yet 
without  doing  so  we  may  know  that  if  they 
ever  do  meet,  or  if,  after  diverging  from  one 
another,  they  begin  again  to  approach,  this 
must  take  place  not  at  an  infinite,  but  at  a 
finite  distance.  Supposing,  therefore,  such  to 
be  the  case,  we  can  transport  ourselves  thither 
in  imagination,  and  can  frame  a  mental  image 
of  the  appearance  which  one  or  both  of  the 
lines  must  present  at  that  point,  which  we  may 


3JO  LECTURES  AND  ESSAYS 

rely  on  as  being  precisely  similar  to  the  reality. 
Now,  whether  we  fix  our  contemplation  upon  this 
imaginary  picture,  or  call  to  mind  the  generali- 
sations we  have  had  occasion  to  make  from 
former  ocular  observation,  we  learn  by  the 
evidence  of  experience  that  a  line  which,  after 
diverging  from  another  straight  line,  begins  to 
approach  to  it,  produces  the  impression  on  our 
senses,  which  we  describe  by  the  expression  '  a 
bent  line,'  not  by  the  expression  '  a  straight 
line.'" — Logic,  Book  ii.  chap.  v.  s.  5. 

Upon  this  argument  I  have  one  very  simple 
remark  to  make.  That  "characteristic  pro- 
perty of  geometrical  forms "  is  derived  from 
experience  ; — we  have  "  learnt  by  long-con- 
tinued experience  that  the  properties  of  the 
reality  are  faithfully  represented  in  the  image." 
Experience  could  only  tell  us  this  of  realities 
and  of  images  both  of  which  we  have  ex- 
perienced. I  must  know  both  of  two  things 
to  know  that  one  faithfully  represents  the  other. 
Experience  then  tells  me  that  my  mental 
images  of  geometrical  figures  are  faithful  repre- 
sentations of  those  realities  which  are  within 
the  bounds  of  experience.  But  what  is  to  tell 
me  that  they  are  faithful  representations  of 
realities  that  are  beyond  the  bounds  of  ex- 
perience? Surely  no  experience  can  tell  me 
that. 


PHILOSOPHY  OF  THE  PURE  SCIENCES       331 

Again,  our  notion  of  straiglit  is  a  combina- 
tion of  several  properties,  an  aggregate  of  im- 
pressions on  our  senses,  which  holds  together 
within  the  limits  of  experience.  But  what  is 
to  tell  us  that  these  impressions  hold  together 
beyond  the  limits  of  experience  ? 

It  seems  to  me,  then,  that  in  admitting  the 
universality  of  certain  statements  Mr.  Mill 
knows  something  which  on  his  own  principles 
he  has  no  right  to  know. 

In  the  following  section  Mr.  Mill  deals  with 
the  supposed  necessity  of  these  truths.  Taking 
this  to  mean  the  inconceivability  of  the  nega- 
tion of  them,  he  explains  it  in  somewhat  the 
same  way  as  Hume  explained  the  idea  of 
cause,  namely,  by  means  of  the  law  of  associa- 
tion. But  that  which  in  Locke  and  Hume  had 
been  merely  a  special  explanation  of  particular 
phenomena  has  in  the  meantime  grown  into 
an  extensive  and  most  successful  science  of 
Psychology.  It  began,  as  you  remember,  in 
the  form  of  a  link  between  two  impressions 
that  occur  frequently  together.  Perhaps  the 
most  important  step  was  Hartley's  idea  of 
"  mental  chemistry "  ;  that  the  result  of  two 
linked  impressions  might  not  put  in  evidence 
either  of  the  components  any  more  than  water 
exhibits  to  us  the  hydrogen  and  the  oxygen 
which  it  contains.  In  the  hands  of  James 
Mill  and  Mr.  Bain  this  mode  of  explanation 


332  LECTURES  AND  ESSAYS 

has  been  applied  with  marked  success  to  a  vast 
number  of  mental  phenomena ;  so  that  when 
Mr.  Mill  makes  use  of  it  to  account  for  the 
inconceivability  of  that  which  has  not  yet  been 
experienced,  he  is  backed  by  an  enormous  mass 
of  similar  and  most  successful  explanations. 

This  view,  that  the  supplementary  part  of 
our  sensations  is  an  accumulation  of  past 
experience,  has  been  further  defended  by  Mr. 
Bain  in  many  excellent  books.  But  there  is 
one  respect  in  which  the  doctrines  of  Mr.  Mill 
and  Mr.  Bain  differ  very  importantly  from  the 
one  which  we  have  next  to  consider — that  of 
Mr.  Herbert  Spencer.  He  also  believes  that 
the  whole  of  our  knowledge  comes  from  ex- 
perience ;  but  while  in  the  former  view  this 
experience  is  our  own,  and  has  been  acquired 
during  the  lifetime  of  the  individual,  in  the 
latter  it  is  not  the  experience  of  you  or  me, 
but  of  all  our  ancestors.  The  perceptions,  not 
only  of  former  generations  of  men,  but  of  those 
lower  organisms  from  which  they  were  originally 
derived,  beginning  even  with  the  first  molecule 
that  was  complex  enough  to  preserve  records 
of  its  own  changes  ;  all  these  have  been  built 
into  the  organism,  have  determined  its  character, 
and  have  been  handed  down  to  us  by  hereditary 
descent.  The  effect  of  this  upon  Kant's  doctrine 
may  be  best  displayed  by  another  quotation  : — 

"  The  universal  law  that,  other  things  equal, 


PHILOSOPHY  OF  THE  PURE  SCIENCES       333 

the  cohesion  of  psychical  states  is  proportionate 
to  the  frequency  with  which  they  have  followed 
one  another  in  experience,  supplies  an  explana- 
tion of  the  so-called  '  forms  of  thought,'  as  soon 
as  it  is  supplemented  by  the  law  that  habitual 
psychical  successions  entail  some  hereditary 
tendency  to  such  successions,  which,  under  per- 
sistent conditions,  will  become  cumulative  in 
generation  after  generation.  We  saw  that  the 
establishment  of  those  compound  reflex  actions 
called  instincts  is  comprehensible  on  the  prin- 
ciple that  inner  relations  are,  by  perpetual 
repetition,  organised  into  correspondence  with 
outer  relations.  We  have  now  to  observe  that 
the  establishment  of  those  consolidated,  those 
indissoluble,  those  instinctive  mental  relations 
constituting  our  ideas  of  Space  and  Time,  is 
comprehensible  on  the  same  principle.  .  .  . 

"In  the  sense,  then,  that  there  exist  in  the 
nervous  system  certain  pre-established  relations 
answering  to  relations  in  the  environment, 
there  is  a  truth  in  the  doctrine  of  '  forms  of  in- 
tuition ' — not  the  truth  which  its  defenders 
suppose,  but  a  parallel  truth.  Corresponding 
to  absolute  external  relations,  there  are 
established  in  the  structure  of  the  nervous 
system  absolute  internal  relations  —  relations 
that  are  potentially  present  before  birth  in  the 
shape  of  definite  nervous  connections  ;  that  are 
antecedent  to,  and  independent  of,  individual 


334  LECTURES  AND  ESSAYS 

experiences ;  and  that  are  automatically  dis- 
closed along  with  the  first  cognitions.  And,  as 
here  understood,  it  is  not  only  these  funda- 
mental relations  which  are  thus  pre-determined ; 
but  also  hosts  of  other  relations  of  a  more  or 
less  constant  kind,  which  are  congenitally 
represented  by  more  or  less  complete  nervous 
connections.  But  these  pre-determined  internal 
relations,  though  independent  of  the  experiences 
of  the  individual,  are  not  independent  of  ex- 
periences in  general :  they  have  been  determined 
by  the  experiences  of  preceding  organisms.  The 
corollary  here  drawn  from  the  general  argument 
is  that  the  human  brain  is  an  organised  register 
of  infinitely  numerous  experiences  received 
during  the  evolution  of  life,  or  rather,  during 
the  evolution  of  that  series  of  organisms  through 
which  the  human  organism  has  been  reached. 
The  effects  of  the  most  uniform  and  frequent 
of  these  experiences  have  been  successively 
bequeathed,  principal  and  interest  ;  and  have 
slowly  mounted  to  that  high  intelligence  which 
lies  latent  in  the  brain  of  the  infant — which 
the  infant  in  after-life  exercises  and  perhaps 
strengthens  or  further  complicates — and  which, 
with  minute  additions,  it  bequeaths  to  future 
generations.  And  thus  it  happens  that  the 
European  inherits  from  twenty  to  thirty  cubic 
inches  more  brain  than  the  Papuan.  Thus  it 
happens  that  faculties,  as  of  music,  which 


PHILOSOPHY  OF  THE  PURE  SCIENCES       335 

scarcely  exist  in  some  inferior  human  races, 
become  congenital  in  superior  ones.  Thus  it 
happens  that  out  of  savages  unable  to  count  up 
to  the  number  of  their  fingers,  and  speaking 
a  language  containing  only  nouns  and  verbs, 
arise  at  length  our  Newtons  and  our  Shake- 
speares." — Principles  of  Psychology,  §  208,  vol. 
i.  pp.  466,  470. 

This  doctrine  of  Mr.  Spencer's  is  what  I 
believe  to  be  really  the  truth  about  the  matter  ; 
and  I  shall  have  to  return  to  it  again  by  and 
by.  But  I  have  a  remark  to  make  here.  It 
seems  to  me  that  the  Kantian  dilemma  about 
universal  propositions  is  just  as  valid  now,  in 
spite  of  these  explanations,  as  it  was  in  his 
time.  How  am  I  to  know  that  the  angles  of  a 
triangle  are  exactly  equal  to  two  right  angles 
under  all  possible  circumstances  ;  not  only  in 
those  regions  of  space  where  the  solar  system 
has  been,  but  everywhere  else  ?  The  accumu- 
lated experience  of  all  my  ancestors  for  a 
hundred  and  fifty  million  years  is  no  more 
competent  to  tell  me  tJiat  than  my  own  experi- 
ence of  the  last  five  minutes.  Either  I  have 
some  source  of  knowledge  other  than  experi- 
ence, and  I  must  admit  the  existence  of  a 
priori  truths,  independent  of  experience  ;  or  I 
cannot  know  that  any  universal  statement  is 
true.  Now  the  doctrine  of  evolution  itself  for- 
bids me  to  admit  any  transcendental  source  of 


336  LECTURES  AND  ESSAYS 

knowledge  ;  so  that  I  am  driven  to  conclude 
in  regard  to  every  apparently  universal  state- 
ment, either  that  it  is  not  really  universal,  but 
a  particular  statement  about  my  nervous 
system,  about  my  apparatus  of  thought ;  or  that 
I  do  not  know  that  it  is  true.  And  to  this 
conclusion,  by  a  detailed  examination  of  various 
apparently  universal  statements,  I  shall  in  sub- 
sequent lectures  endeavour  to  lead  you. 

II. KNOWLEDGE   AND   FEELING 

The  following  fragment  appears  to  represent  what  was  the  con- 
clusion of  the  series  of  Lectures  as  they  were  delivered  in 
March  1873.  It  was  found  among  Professor  Clifford's 
papers  without  any  external  indication  of  its  proper  con- 
text ;  and  as  the  Lectures  now  stand  after  the  author's  re- 
vision, it  seems  to  come  in  better  as  an  appendix  to  the 
first  of  them.  Clifford  himself  regarded  it  apparently  (note 
to  the  Third  Lecture  in  Nineteenth  Century,  March 
1879)  as  superseded  by  his  article  on  "The  Nature  of 
Things-in-themselves " ;  but  it  contains  critical  remarks 
and  illustrations  which  are  not  there,  and  it  has  seemed 
best  to  the  editors  to  let  it  stand  in  this  place. 

IN  order  to  consider  at  this  point  what  it  is 
that  we  have  arrived  at,  we  must  call  to  mind 
the  point  from  which  we  started.  We  said 
that  the  whole  of  our  sensations  could  not 
possibly  be  a  message  from  outside,  but  that 
some  part  at  least  of  them  must  be  a  supple- 
ment or  filling-in  of  this  message,  added  by 
ourselves.  A  theory  came  before  us — that  of 
Mr.  Herbert  Spencer — according  to  which  this 


PHILOSOPHY  OF  THE  PURE  SCIENCES       337 

filling-in  was  accounted  for  as  the  product  of 
past  experience,  which  had  taken  effect  on  the 
brains  of  our  ancestors  and  produced  certain 
changes  in  them.  These  changes  have  gradually 
moulded  the  structure  of  the  nervous  system 
which  was  handed  on  to  us  by  hereditary 
descent.  There  was  one  obstacle  to  our  accept- 
ance of  that  theory  as  a  sufficient  account  of 
the  matter ;  namely,  that  we  apparently  had 
some  knowledge  which  could  not  possibly  have 
been  got  in  that  way — knowledge  that  certain 
general  statements  are  absolutely  and  universally 
true.  This  obstacle  I  shall  endeavour  to  re- 
move, by  showing  that  such  general  statements 
may  be  divided  into  two  classes  ;  of  which 
those  in  the  first  class  may  for  all  we  know  be 
false,  while  those  in  the  second  class  are  general 
statements  only  in  form,  and  really  are  judg- 
ments about  the  apparatus  of  thought.  If  this 
be  so,  we  are  at  liberty  to  accept  the  view  that 
all  human  knowledge  is  derived  from  experi- 
ence ;  and  that  of  the  two  factors  in  sensation, 
that  supplement  which  we  provide  of  ourselves 
is  a  giving  out  again  of  what  has  originally  be- 
longed to  the  other  factor,  to  experience  proper, 
But  here  a  doubt  suggests  itself  which  appears 
exactly  to  reverse  all  that  we  have  done.  We 
said  there  were  two  factors  of  experience  :  that 
all  of  it  could  not  be  direct  message  ;  and  we 
have  come  to  the  conclusion  that  the  two  factors 
VOL.  I  Z 


338  LECTURES  AND  ESSAYS 

are  really  of  the  same  kind.  But  we  did  not 
show  that  any  of  it  was  direct  message  from 
outside  ;  we  only  showed  that  some  portions  of 
it  were  not  Suppose  it  is  all  supplement,  and 
there  is  no  message  at  all !  In  that  case  our 
two  factors  will  indeed  be  reduced  to  one  ;  but 
in  what  sense  can  we  say  that  our  knowledge 
is  derived  from  experience  ?  It  will  of  course 
be  derived  from  experience  in  the  large  sense, 
that  is,  from  sensation  ;  but  in  the  sense  in 
which  we  have  used  the  term,  as  meaning  that 
part  of  sensation  which  is  not  supplied  by  our- 
selves, there  will  be'  no  experience  for  us  to 
derive  knowledge  from.  This  question  then  is 
an  extremely  important  one  ;  for  if  we  have  to 
admit  that  there  is  no  real  message  from  with- 
out, all  the  sciences  will  become  pure  sciences, 
all  knowledge  will  be  a  priori  knowledge  ;  and 
we  may  construct  the  universe  by  sitting  down 
and  thinking  about  it.  It  is  this  question  then 
that  I  propose  to  consider  for  a  short  time,  a 
time  very  much  too  short  for  the  consideration 
of  it,  but  perhaps  long  enough  to  let  me  indicate 
in  some  way  the  kind  of  answer  which  is  given 
by  an  extension  of  that  Physiological  Method 
which  we  began  by  using. 

We  traced  the  message  of  sight  to  the  retina 
of  the  eye,  saying  that  the  only  direct  message 
possible  is  contained  in  the  picture  there  drawn. 
But  we  may  go  a  little  farther.  The  picture 


PHILOSOPHY  OF  THE  PURE  SCIENCES       339 

consists  in  an  aggregate  of  forms  and  colours 
having  a  certain  mode  of  connection.  It  is 
carried  inwards  by  the  optic  nerve  ;  but  in 
order  to  be  so  carried,  it  has  to  undergo  a  still 
further  transformation.  The  optic  nerve  is  a 
great  bundle  of  telegraph  wires,  each  carrying 
its  own  message  undisturbed  by  the  rest.  Each 
wire  only  tells  what  is  happening  at  a  particular 
point  of  the  retina  ;  that  is  to  say,  what  colour 
and  what  intensity  the  light  impinging  on  the 
point  has.  Now  in  order  to  tell  the  colour  and 
intensity,  it  appears  that  it  must  consist  of  three 
distinct  strands  ;  for  it  has  been  made  out  that 
every  sensation  of  colour  is  composed  of  three 
simple  sensations  combined  in  a  certain  propor- 
tion, this  proportion  varying  from  colour  to 
colour.  Does  then  the  optic  nerve  carry  the 
picture  itself  as  a  message  ?  It  is  clear  that  it 
cannot ;  but  it  may  take  an  account  of  every 
point  in  it,  and  of  their  relations  of  contiguity  ; 
that  is,  it  carries  an  aggregate  of  elementary 
messages,  which  has  a  point-for-point  connec- 
tion with  the  picture,  of  such  a  nature  as  to 
retain  the  relations  of  nextness  or  contiguity. 
But  the  point  to  notice  is  that  two  messages 
carried  by  the  optic  nerve  differ  only  as  two 
chords  played  upon  the  same  organ,  or  as  two 
books  written  in  the  same  alphabet ;  they  are 
combinations  or  connected  aggregates  of  the 
same  elementary  messages,  selected  and  fastened 


340  LECTURES  AND  ESSAYS 

together  in  different  ways.     The  difference  is  a 
matter  of  arrangement  and  building  up  ;  not  a 
difference  of  the   elements    that    are    built  up. 
This    very    important    step    in    the   theory    of 
sensation  was  made  by  Helmholtz,  following  in 
the  steps  of  Miiller,  equally  in  the  case  of  sight 
and  sound.      It  was  he  who  made  out  clearly 
that  the  special  nerves  of  the  senses  had  not 
absolutely  special  functions  of  transmitting  their 
particular   sensation  as  a  whole,  but   that  the 
difference    consisted    in    the   various    ways    of 
combining  together  the  same  elementary  nerve- 
message.      Where,    then,    are    these    messages 
taken  ?     They  are  taken  to  the  gray  corpuscles 
within  the  brain  ;    and  apparently  each  nerve 
goes  to  its  own  corpuscle,  and  sets  it  in  com- 
motion with  the  message.     Finally  we  get  this 
result :  that  the  presence  of  a  picture  on  the 
retina    involves    the    commotion    of   a    certain 
number   of  gray  corpuscles  ;    the   selection  of 
which  and  the  amount  of  excitement  given  to 
each  are  determined  by  the  picture.     And  the 
same   thing   happens   for  every  other  kind   of 
sensation.      Now  the  direct  knowledge  that  we 
get  can  only  be  knowledge  of  this  commotion 
in  the  gray  matter.     For  we  can  tap  the  tele- 
graph, so  to  speak,  and  transmit  a  false  message 
by  it  ;  and  it  is  found   that  if  the  optic  nerve 
be  excited  either  by  pressure  of  the  eye  or  by 
an  electric  shock,  the  sensation  of  sight  is  pro- 


PHILOSOPHY  OF  THE  PURE  SCIENCES       341 

duced,  although  no  light  has  been  present. 
The  difference,  then,  of  different  sensations  is 
made  by  the  difference  of  the  gray  corpuscles 
excited  ;  and  the  immediate  knowledge  that  is 
given  to  us  by  experience  can  only  be  know- 
ledge of  more  or  less  excitement  of  certain 
parts  of  the  gray  matter.  This  applies  equally 
to  touch,  taste,  smell,  muscular  action,  the 
organic  sensations  of  pain  or  pleasure.  If  you 
and  I,  then,  choose  to  contemplate  another 
person,  we  shall  say  that  the  world  which  he 
directly  perceives  is  really  inside  his  brain,  and 
not  outside  ;  but  that  corresponding  to  these 
changes  that  go  on  in  his  brain  there  are 
certain  changes  going  on  outside  of  him,  and 
that  in  many  cases  there  is  such  a  correspond- 
ence of  the  relations  of  contiguity  in  one  case 
to  the  relation  of  contiguity  in  the  other,  that 
conclusions  about  the  outer  world  may  fairly 
be  drawn  from  the  world  in  his  brain. 

But  now,  if  instead  of  considering  this  other 
person,  I  consider  myself,  the  case  is  rather 
altered.  I  shall  conclude  by  analogy  that  this 
world  which  I  directly  perceive  is  not  really 
outside  of  me ;  that  the  things  which  are 
apparently  made  known  to  me  by  my  percep- 
tions are  really  themselves  only  groups  of  my 
perceptions  ;  that  the  universe  which  I  perceive 
is  made  up  of  my  feelings  ;  that  in  fact  it  is 
really  me.  And — by  analogy  also — I  shall 


342  LECTURES  AND  ESSAYS 

conclude  that  there  is  something  besides  this, 
different  from  it ;  the  changes  in  which  corre- 
spond in  a  certain  way  to  the  changes  in  my 
universe.  Is  it  then  possible  for  me  to  know 
what  that  is  ?  or  is  there  nothing  at  all  except 
my  feelings  ? 

If,  instead  of  approaching  this  question  from 
the  physiological  side,  we  adopt  another  point 
of  view,  it  is  not  unlikely  that  we  shall  be  led 
to  the  latter  conclusion.  If  I  consider  merely 
my  own  feelings,  and  ask  what  evidence  they 
give  of  anything  beyond  them,  it  seems  to  me 
that  I  must  answer,  no  evidence  at  all.  This 
at  least  was  the  answer  given  by  Berkeley  in  a 
passage  which  has  been  quoted  here  before  by 
Professor  Huxley,  but  will  bear  quoting  again : — 

"  Some  truths  there  are  so  near  and  obvious 
to  the  mind  that  a  man  need  only  open  his 
eyes  to  see  them.  Such  I  take  this  important 
one  to  be,  viz.  that  all  the  choir  of  heaven  and 
furniture  of  the  earth,  in  a  word,  all  those 
bodies  which  compose  the  mighty  frame  of  the 
world,  have  not  any  subsistence  without  a 
mind,  that  their  being  is  to  be  perceived  or 
known  ;  that  consequently  so  long  as  they  are 
not  actually  perceived  by  me,  or  do  not  exist 
in  my  mind  or  that  of  any  other  created  spirit, 
they  must  either  have  no  existence  at  all,  or 
else  subsist  in  the  mind  of  some  Eternal  Spirit." 
— Principles  of  Human  Knowledge,  §  6. 


PHILOSOPHY  OF  THE  PURE  SCIENCES       343 

If  I  say  that  such  and  such  things  existed 
at  some  previous  time,  I  mean  that  if  I  had 
been  there  I  could  have  perceived  them  ;  if  I 
say  that  there  is  hydrogen  in  the  sun,  I  mean 
that  if  I  could  get  any  of  that  gas  I  should  be 
able  to  burn  it  in  oxygen  and  produce  exactly 
the  same  impressions  on  my  senses  as  those 
which,  in  the  aggregate,  I  call  water. 

This  doctrine,  that  the  essence  of  things 
consists  in  my  perceiving  them,  is  called 
Idealism.  The  form  of  it  held  by  Berkeley, 
however,  is  not  altogether  pure.  He  believed 
that  no  material  external  world  exists  ;  but 
only  spirits  exist,  thinking  beings  whose  nature 
consists  of  conception  and  volition.  Now, 
from  this  point  of  view,  fairly  accepted,  you  are 
only  phenomena  of  my  consciousness  as  much 
as  the  rest  of  the  world  ;  I  cannot  allow  the 
existence  of  any  spirits,  but  only  of  one  spirit, 
myself.  And  even  this  language  is  hardly 
suitable  ;  for  why  should  I  give  myself  a  class- 
name  like  spirit  when  I  am  really  the  sum- 
total  of  the  universe  ?  Notwithstanding  this 
failure  to  reach  complete  idealism,  the  doctrine 
of  Berkeley,  in  its  positive  aspect,  is  a  distinct 
and  most  important  step  in  philosophy ;  it 
established  in  a  security  that  has  never  yielded 
to  attack  the  subjective  character  of  the  world 
of  phenomena  ;  that  this  world  which  I  per- 
ceive is  my  perceptions  and  nothing  more. 


344  LECTURES  AND  ESSAYS 

Whether  there  is  anything  else  quite  different 
which  corresponds  to  it  in  a  certain  way,  is 
another  question  ;  Berkeley  said  there  were 
also  spirits. 

According  to  Berkeley,  moreover,  there 
exists,  besides  this  world  of  my  perceptions, 
a  particular  spirit,  mey  that  perceives  them. 
To  get  rid  of  this  imaginary  soul  or  substance, 
underlying  the  succession  of  my  feelings,  was 
the  work  of  Hume.  Just  as  an  object,  in 
Berkeley's  theory,  is  merely  a  bundle  of  per- 
ceptions which  always  occur  together,  a  linked 
aggregate  of  feelings  ;  so,  said  Hume,  out  of 
the  swift  current  of  ideas  that  succeed  one 
another  we  construct  a  unity  which  we  call 
Self  or  Ego.  But  this,  he  said,  is  a  pure 
illusion  ;  and  the  ego,  when  analysed,  turns 
out  to  be  only  the  whole  complex  of  my  feel- 
ings. This,  you  see,  is  a  step  towards  simpli- 
fication ;  we  had  to  begin  with  an  external 
thing  which  is  perceived  ;  then  the  perception 
or  feeling ;  then  the  soul  or  self  which  per- 
ceives. With  Berkeley  we  get  rid  of  the 
thing  perceived  ;  it  is  reduced  to  a  bundle  of 
perceptions.  With  Hume  we  get  rid  also  of 
the  perceiving  self ;  it  is  reduced  to  the  whole 
aggregate  of  feelings,  linked  together  and 
succeeding  one  another  in  a  certain  manner. 

The  step  made  by  Mill  is  a  more  complete 
definition  of  the  same  view,  and  an  explanation 


PHILOSOPHY  OF  THE  PURE  SCIENCES       345 

by  means  of  the  law  of  association  of  the  way  in 
which  we  come  to  believe  in  an  external  world. 
He  says  that  objects  are  completely  described 
by  the  phrase,  "  permanent  possibilities  of 
sensation." 

"  The  Psychological  Theory  maintains  that 
there  are  associations  naturally  and  even  neces- 
sarily generated  by  the  order  of  our  sensations 
and  of  our  reminiscences  of  sensation,  which, 
supposing  no  intuition  of  an  external  world  to 
have  existed  in  consciousness,  would  inevitably 
generate  the  belief,  and  would  cause  it  to  be 
regarded  as  an  intuition.  .  .  .  The  conception 
I  form  of  the  world  existing  at  any  moment 
comprises,  along  with  the  sensations  I  am 
feeling,  a  countless  variety  of  possibilities  of 
sensation  :  namely,  the  whole  of  those  which 
past  observation  tells  me  that  I  could,  under 
any  supposable  circumstances,  experience  at 
this  moment,  together  with  an  indefinite  and 
illimitable  multitude  of  others  which  though  I 
do  not  know  that  I  could,  yet  it  is  possible 
that  I  might,  experience  in  circumstances  not 
known  to  me.  These  various  possibilities  are 
the  important  thing  to  me  in  the  world.  My 
present  sensations  are  generally  of  little  im- 
portance, and  are  moreover  fugitive  :  the 
possibilities,  on  the  contrary,  are  permanent, 
which  is  the  character  that  mainly  distinguishes 


346  LECTURES  AND  ESSAYS 

our  idea  of  Substance  or  Matter  from  our 
notion  of  sensation.  .  .  .  Matter,  then,  may  be 
defined,  a  Permanent  Possibility  of  Sensation."1 

In  the  meanwhile,  you  observe,  the  associa- 
tion-theory of  the  mind  had  been  created  ;  and 
it  is  here  applied  to  defend  the  position  of 
Hume.  It  is  worth  while  to  notice  now  where 
we  are.  The  universe  consists  of  feelings.  A 
certain  cable  of  feelings,  linked  together  in  a 
particular  manner,  constitutes  me.  Similar 
cables  constitute  you.  That  is  all  there  is. 
But  in  the  cable  of  feelings  that  make  up  me 
there  are  certain  persistent  bundles  or  strands, 
which  occasionally  come  to  the  outside ;  there 
are  similar  strands  in  the  cables  of  which  you 
are  constituted.  These  correspond  to  external 
objects  ;  we  only  think  them  external  for  the 
reasons  assigned. 

Now,  when  we  pass  to  Mr.  Herbert  Spencer, 
we  come  into  the  presence  of  another  great  de- 
partment of  science  that  has  not  had  so  strong 
an  action  upon  Mr.  Mill ;  and  that  is  the 
anatomy  of  the  nervous  system.  The  effect 
of  investigations  in  this  subject  is  to  analyse 
all  the  various  kinds  of  nervous  action  into 
different  combinations  of  two  simple  elements  ; 
the  transmission  of  messages  along  nerve- 

1  J.  S.  Mill,   Examination  of  Sir    W.  Hamilton's  Philosophy, 
pp.  192,  193,  198,  2nd  edit. 


PHILOSOPHY  OF  THE  PURE  SCIENCES       347 

threads  of  white  matter,  and  the  excitement  of 
nerve-cells  of  gray  matter.  Apparently  all  the 
nerve-threads  are  alike,  and  all  the  nerve-cells 
are  alike.  The  only  thing  that  remains  to 
produce  the  very  different  effects  that  we 
observe  is  the  variety  of  ways  in  which  selec- 
tions may  be  made  from  the  nerve-cells  to  be 
excited  at  any  moment.  The  direct  effects  of 
nerve-action  are  the  effect  on  muscular  tissue 
of  contraction  or  release,  and  the  effect  on 
glands  of  secretion. 

Here,  then,  were  two  great  branches  of 
analysis  present  to  Mr.  Spencer  :  the  analysis 
of  mental  action  given  by  the  association-theory, 
which  reduced  everything  to  the  linking -to- 
gether of  feelings,  and  the  analysis  of  nervous 
action  supplied  by  the  histologists.  It  was  his 
business  to  supply  not  merely  the  link  between 
the  two,  but  an  account  of  their  simultaneous 
evolution.  If  we  find  that  certain  complicated 
forms  of  mental  action  always  accompany 
certain  forms  of  nervous  action  ;  if  each  of  these 
can  be  reduced  into  elements,  and  the  relation 
of  each  compound  to  its  elements  is  the  same 
— the  bricks  different,  but  the  mode  of  putting 
them  together  identical  in  these  two  houses — 
there  is  a  very  strong  presumption  that  the 
element  of  mental  action  always  accompanies 
the  element  of  nervous  action.  But  this  pre- 
sumption is  converted  into  knowledge  when  we 


348  LECTURES  AND  ESSAYS 

have  an  account  of  their  origin.  When  the 
evolution  of  the  living  organism  is  traced  up- 
wards from  the  simplest  forms  to  the  most 
complex,  and  it  is  found  that  the  evolution  of 
mind  proceeds  part  passu  with  it,  following  the 
same  laws  and  passing  through  the  same  stages, 
either  evolution  being  expressed  as  a  continual 
building  up  with  the  same  element,  we  have 
actual  evidence  that  the  one  element  goes  with 
the  other. 

Hence,  then,  is  the  great  advantage  of  Mr. 
Herbert  Spencer  in  the  study  of  both  orders  of 
facts.  He  can  make  any  step  in  analysis  of 
the  one  help  in  the  analysis  of  the  other.  And 
accordingly  he  has  carried  both  to  an  extent 
which  leaves  all  previous  investigators  far  be- 
hind. But  you  will  see  at  once  that  we  must 
look  at  the  question  of  idealism  from  the 
physiological  point  of  view.  And  accordingly 
he  considers  that  there  is  something  different 
from  our  perceptions,  the  changes  in  which 
correspond  in  a  certain  way  to  the  changes  in 
the  worlds  we  perceive.  He  thinks,  however, 
that  we  can  never  know  what  it  is ;  and  he 
says : — 

"  We  can  think  of  Matter  only  in  terms  of 
Mind.  We  can  think  of  Mind  only  in  terms 
of  Matter.  When  we  have  pushed  our  explora- 
tions of  the  first  to  the  uttermost  limit,  we  are 
referred  to  the  second  for  a  final  answer ;  and 


PHILOSOPHY  OF  THE  PURE  SCIENCES       349 

when  we  have  got  the  final  answer  of  the 
second,  we  are  referred  back  to  the  first  for  an 
interpretation  of  it.  We  find  the  value  of  x  in 
terms  of  y ;  then  we  find  the  value  of  y  in 
terms  of  x ;  and  so  on  we  may  continue  for 
ever  without  coming  nearer  to  a  solution.  The 
antithesis  of  subject  and  object,  never  to  be 
transcended  while  consciousness  lasts,  renders 
impossible  all  knowledge  of  that  Ultimate 
Reality  in  which  subject  and  object  are 
united." — Principles  of  Psychology,  §  272  (vol. 
i.  p.  627). 

Now,  the  singular  character  of  this  realism 
is  that  it  is  defended  from  the  idealistic  point 
of  view,  namely,  Mr.  Spencer  attempts  to  make 
my  feelings  give  me  evidence  of  something 
which  is  not  included  among  them.  A  careful 
study  of  all  his  arguments  to  that  effect  has 
only  convinced  me  over  again  that  the  attempt 
is  hopeless.  In  this  respect  he  differs  consider- 
ably from  Mr.  Shadworth  Hodgson,  who  must 
be  regarded  as  an  advance,  within  the  British 
school,  in  the  direction  of  Berkeley  and  Hume. 
He  accepts  the  analysis  of  the  individual  ego 
or  self  into  a  complex  of  feeling  ;  and,  like 
Hume  or  Mill,  makes  the  universe  to  consist  of 
feelings  variously  bound  together.  But  this  is 
only  one  aspect  of  it  and  of  all  contained 
phenomena.  Every  phenomenon  has  two 
aspects  ;  in  its  subjective  aspect  it  is  a  feeling 


350  LECTURES  AND  ESSAYS 

in  its  objective  aspect  a  quality.  But  it  is  not 
necessarily  a  feeling  of  my  consciousness  or  of 
your  consciousness  ;  it  may  be  a  feeling  of  the 
general  or  universal  consciousness,  which  is 
coextensive  with  all  existence.  The  universal 
consciousness  bears  the  same  relation  to  the 
universal  Ego  of  Schelling  or  Hegel  that  the 
stream  of  feelings  does  to  the  soul ;  it  is  an 
analysis  of  it  into  elements. 

The  important  thing  here  is  the  conclusion 
that  there  is  only  one  world,  combined  with 
the  analysis  of  mental  phenomena.  The 
German  Idealist  attempted  to  construct  the 
world  out  of  very  abstract  ideas,  which  are  the 
most  complex  of  all  forms  of  mental  action. 
In  this  way  we  did  get  one  world,  a  mental 
world  ;  but  the  bricks  of  which  it  was  built 
were  made  by  the  ingenious  piling  together  of 
houses.  I  do  not  think  that  that  process  is 
likely  to  produce  serviceable  bricks.  Now  Mr. 
Hodgson's  element,  feeling,  although  it  seems 
to  imply  something  too  complicated,  is  yet  at 
least  a  step  in  the  way  of  analysis,  an  indica- 
tion that  analysis  is  desired. 

Can  we  now  get  out  of  our  hobble,  and 
arrive  at  real  knowledge  derived  from  external 
experience,  from  messages  and  not  from  im- 
agination? I  think  we  can.  But  it  is  necessary 
to  say  first  what  is  the  character  of  the  know- 
ledge we  desire.  It  will  be  of  the  nature  of 


PHILOSOPHY  OF  THE  PURE  SCIENCES       351 

inference,  and  not  of  absolute  certainty.  Now 
inference  depends  on  the  assumption  of  the 
uniformity  of  nature  ;  and  what  does  this  rest 
on  ?  We  cannot  infer  that  which  is  the  ground 
of  all  inference ;  but  although  I  cannot  give 
you  a  logical  reason  for  believing  it,  I  can  give 
you  a  physical  explanation  of  the  fact  that  we 
all  do  believe  it.  We  believe  a  thing  when  we 
are  prepared  to  act  as  if  it  were  true.  Now,  if 
you  and  I  had  not  habitually  acted  on  the 
assumption  of  the  uniformity  of  nature  from 
the  time  when  we  could  act  at  all,  we  should 
not  be  here  to  discuss  the  question.  Nature  is 
selecting  for  survival  those  individuals  and 
races  who  act  as  if  she  were  uniform  ;  and 
hence  the  gradual  spread  of  that  belief  over 
the  civilised  world. 

This  uniformity  may  be  merely  a  uniformity 
of  phenomena,  a  law  relating  to  my  feelings. 
So  long  as  I  only  am  concerned,  it  seems  to 
me  that  the  idealist  theory  is  perfectly  sufficient. 
It  is  quite  capable  of  explaining  me  ;  but  when 
you  come  into  the  question,  it  is  utterly  at  a 
loss.  The  distinction  between  the  universal 
and  the  individual  ego  seems  to  me  a  merely 
useless  abstraction  that  throws  dust  in  our  eyes. 
I  do  believe  that  you  are  conscious  in  the  same 
way  as  I  am  ;  and  once  that  is  conceded,  the 
whole  idealist  theory  falls  to  pieces.  For  there 
are  feelings  which  are  not  my  feelings,  which 


35*  LECTURES  AND  ESSAYS 

are  entirely  outside  my  consciousness  ;  so  that 
there  is  at  least  an  external  world.  But  let  us 
consider  now  in  what  way  we  infer  it ;  why  do 
I  believe  that  there  are  feelings  which  are  not 
mine  ?  Because,  as  I  belong  to  a  gregarious 
race,  the  greater  part  of  my  life  consists  in  act- 
ing upon  the  supposition  that  it  is  true. 

But  now  further,  have  I  reason  for  believing 
that  the  changes  in  this  external  world  corre- 
spond in  any  way  with  the  changes  in  my  world 
which  I  perceive  ?  I  think  so.  The  complex 
of  feelings  which  constitutes  you  corresponds  in 
a  definite  way  with  the  changes  which  I  might 
perceive  in  your  brain.  By  inferences  that  I 
have  previously  indicated,  I  conclude  that  the 
ultimate  element  into  which  your  feeling  can 
be  analysed  goes  with  the  ultimate  element  out 
of  which  the  changes  of  the  nerve-matter  in 
your  brain  are  built  up.  But  physiological 
action  is  complicated  chemistry  in  the  same 
way  that  chemistry  is  complicated  mechanics. 
The  actions  that  take  place  in  the  brain  differ 
in  no  way  from  other  material  actions,  except 
in  their  complexity.  Conjoin  with  this  the 
doctrine  of  Evolution,  and  you  will  see  evidence 
that  the  simplest  mental  change  goes  always 
with  the  simplest  material  change,  whether  in 
the  brain  or  not.  The  external  world,  then,  is 
a  complex  of  mental  changes  ;  the  ultimate 
elements  into  which  feeling  can  be  analysed  ; 


PHILOSOPHY  OF  THE  PURE  SCIENCES       353 

so  simple  that  the  simplest  feeling  which  we 
can  experience  is  an  enormously  complex  mass 
of  them.  Some  of  these  are  built  up  into 
sufficiently  complicated  forms  to  constitute 
what  we  call  personality,  will,  consciousness. 
They  all  succeed  one  another  according  to 
certain  laws ;  and  in  virtue  of  these  any  conscious 
aggregate  of  them  is  acted  upon  by  the  rest ; 
the  changes  so  produced  in  it  are  what  we  call 
a  material  world. 

There  is  thus  only  one  world  of  elementary 
feelings ;  which  is  perceived  by  me  as  my 
material  world.  And  I  am  not  to  look  for 
those  complex  forms  of  mental  action  called 
intelligence  and  consciousness,  except  where 
I  can  perceive  a  correspondingly  complex 
aggregation  of  matter. 


III. THE    POSTULATES    OF    THE   SCIENCE 

OF   SPACE 

IN  my  first  lecture  I  said  that,  out  of  the 
pictures  which  are  all  that  we  can  really  see, 
we  imagine  a  world  of  solid  things  ;  and  that 
this  world  is  constructed  so  as  to  fulfil  a  certain 
code  of  rules,  some  called  axioms,  and  some 
called  definitions,  and  some  called  postulates, 
and  some  assumed  in  the  course  of  demonstra- 
tion, but  all  laid  down  in  one  form  or  another 
in  Euclid's  Elements  of  Geometry.  It  is  this 
VOL.  I  2  A 


354  *  LECTURES  AND  ESSAYS 

code  of  rules  that  we  have  to  consider  to-day. 
I  do  not,  however,  propose  to  take  this  book 
that  I  have  mentioned,  and  to  examine  one 
after  another  the  rules  as  Euclid  has  laid  them 
down  or  unconsciously  assumed  them  ;  not- 
withstanding that  many  things  might  be  said 
in  favour  of  such  a  course.  This  book  has  been 
for  nearly  twenty-two  centuries  the  encourage- 
ment and  guide  of  that  scientific  thought  which 
is  one  thing  with  the  progress  of  man  from  a 
worse  to  a  better  state.  The  encouragement  ; 
for  it  contained  a  body  of  knowledge  that  was 
really  known  and  could  be  relied  on,  and  that 
moreover  was  growing  in  extent  and  applica- 
tion. For  even  at  the  time  this  book  was 
written — shortly  after  the  foundation  of  the 
Alexandrian  Museum — Mathematic  was  no 
longer  the  merely  ideal  science  of  the  Platonic 
school,  but  had  started  on  her  career  of  con- 
quest over  the  whole  world  of  Phenomena. 
The  guide  ;  for  the  aim  of  every  scientific 
student  of  every  subject  was  to  bring  his  know- 
ledge of  that  subject  into  a  form  as  perfect  as 
that  which  geometry  had  attained.  Far  up  on 
the  great  mountain  of  Truth,  which  all  the 
sciences  hope  to  scale,  the  foremost  of  that 
sacred  sisterhood  was  seen,  beckoning  to  the 
rest  to  follow  her.  And  hence  she  was  called, 
in  the  dialect  of  the  Pythagoreans,  "  the  purifier 
of  the  reasonable  soul."  Being  thus  in  itself  at 


PHILOSOPHY  OF  THE  PURE  SCIENCES       355 

once  the  inspiration  and  the  aspiration  of 
scientific  thought,  this  Book  of  Euclid's  has 
had  a  history  as  chequered  as  that  of  human 
progress  itself.  It  embodied  and  systematised 
the  truest  results  of  the  search  after  truth  that 
was  made  by  Greek,  Egyptian,  and  Hindu.  It 
presided  for  nearly  eight  centuries  over  that 
promise  of  light  and  right  that  was  made  by 
the  civilised  Aryan  races  on  the  Mediterranean 
shores  ;  that  promise,  whose  abeyance  for  nearly 
as  long  an  interval  is  so  full  of  warning  and  of 
sadness  for  ourselves.  It  went  into  exile  along 
with  the  intellectual  activity  and  the  goodness 
of  Europe.  It  was  taught,  and  commented 
upon,  and  illustrated,  and  supplemented,  by 
Arab  and  Nestorian,  in  the  Universities  of 
Bagdad  and  of  Cordova.  From  these  it  was 
brought  back  into  barbaric  Europe  by  terrified 
students  who  dared  tell  hardly  any  other  thing 
of  what  they  had  learned  among  the  Saracens. 
Translated  from  Arabic  into  Latin,  it  passed 
into  the  schools  of  Europe,  spun  out  with  ad- 
ditional cases  for  every  possible  variation  of 
the  figure,  and  bristling  with  words  which  had 
sounded  to  Greek  ears  like  the  babbling  of 
birds  in  a  hedge.  At  length  the  Greek  text 
appeared  and  was  translated  ;  and,  like  other 
Greek  authors,  Euclid  became  an  authority. 
There  had  not  yet  arisen  in  Europe  "  that  fruit- 
ful faculty,"  as  Mr.  Winwood  Reade  calls  it, 


356  LECTURES  AND  ESSAYS 

"  with  which  kindred  spirits  contemplate  each 
other's  works  ;  which  not  only  takes,  but  gives ; 
which  produces  from  whatever  it  receives ; 
which  embraces  to  wrestle,  and  wrestles  to  em- 
brace." Yet  it  was  coming ;  and  though  that 
criticism  of  first  principles  which  Aristotle  and 
Ptolemy  and  Galen  underwent  waited  longer 
in  Euclid's  case  than  in  theirs,  it  came  for  him 
at  last.  What  Vesalius  was  to  Galen,  what 
Copernicus  was  to  Ptolemy,  that  was  Lobat- 
chewsky  to  Euclid.  There  is,  indeed,  a  some- 
what instructive  parallel  between  the  last  two 
cases.  Copernicus  and  Lobatchewsky  were 
both  of  Slavic  origin.  Each  of  them  has 
brought  about  a  revolution  in  scientific  ideas  so 
great  that  it  can  only  be  compared  with  that 
wrought  by  the  other.  And  the  reason  of  the 
transcendent  importance  of  these  two  changes 
is  that  they  are  changes  in  the  conception  of 
the  Cosmos.  Before  the  time  of  Copernicus, 
men  knew  all  about  the  Universe.  They  could 
tell  you  in  the  schools,  pat  off  by  heart,  all  that 
it  was,  and  what  it  had  been,  and  what  it  would 
be.  There  was  the  flat  earth,  with  the  blue 
vault  of  heaven  resting  on  it  like  the  dome  of 
a  cathedral,  and  the  bright  cold  stars  stuck  into 
it ;  while  the  sun  and  planets  moved  in  crystal 
spheres  between.  Or,  among  the  better  in- 
formed, the  earth  was  a  globe  in  the  centre  of  the 
universe,  heaven  a  sphere  concentric  with  it ; 


PHILOSOPHY  OF  THE  PURE  SCIENCES       357 

intermediate  machinery  as  before.  At  any 
rate,  if  there  was  anything  beyond  heaven,  it 
was  a  void  space  that  needed  no  further  de- 
scription. The  history  of  all  this  could  be  traced 
back  to  a  certain  definite  time,  when  it  began  ; 
behind  that  was  a  changeless  eternity  that 
needed  no  further  history.  Its  future  could  be 
predicted  in  general  terms  as  far  forward  as  a 
certain  epoch,  about  the  precise  determination 
of  which  there  were,  indeed,  differences  among 
the  learned.  But  after  that  would  come  again 
a  changeless  eternity,  which  was  fully  accounted 
for  and  described.  But  in  any  case  the  Uni- 
verse was  a  known  thing.  Now  the  enormous 
effect  of  the  Copernican  system,  and  of  the 
astronomical  discoveries  that  have  followed  it, 
is  that,  in  place  of  this  knowledge  of  a  little, 
which  was  called  knowledge  of  the  Universe,  of 
Eternity  and  Immensity,  we  have  now  got 
knowledge  of  a  great  deal  more  ;  but  we  only 
call  it  the  knowledge  of  Here  and  Now.  We 
can  tell  a  great  deal  about  the  solar  system  ; 
but,  after  all,  it  is  our  house,  and  not  the  city. 
We  can  tell  something  about  the  star-system 
to  which  our  sun  belongs  ;  but,  after  all,  it  is 
our  star-system,  and  not  the  Universe.  We 
are  talking  about  Here  with  the  consciousness 
of  a  There  beyond  it,  which  we  may  know 
some  time,  but  do  not  at  all  know  now.  And 
though  the  nebular  hypothesis  tells  us  a  great 


35«  LECTURES  AND  ESSAYS 

deal  about  the  history  of  the  solar  system,  and 
traces  it  back  for  a  period  compared  with  which 
the  old  measure  of  the  duration  of  the  Universe 
from  beginning  to  end  is  not  a  second  to  a 
century,  yet  we  do  not  call  this  the  history  of 
eternity.  We  may  put  it  all  together  and  call 
it  Now,  with  the  consciousness  of  a  Then  before 
it,  in  which  things  were  happening  that  may 
have  left  records ;  but  we  have  not  yet  read 
them.  This,  then,  was  the  change  effected  by 
Copernicus  in  the  idea  of  the  Universe.  But 
there  was  left  another  to  be  made.  For  the 
laws  of  space  and  motion,  that  we  are  presently 
going  to  examine,  implied  an  infinite  space  and 
an  infinite  duration,  about  whose  properties 
as  space  and  time  everything  was  accurately 
known.  The  very  constitution  of  those  parts 
of  it  which  are  at  an  infinite  distance  from  us, 
"  geometry  upon  the  plane  at  infinity,"  is  just 
as  well  known,  if  the  Euclidean  assumptions 
are  true,  as  the  geometry  of  any  portion  of  this 
room.  In  this  infinite  and  thoroughly  well- 
known  space  the  Universe  is  situated  during  at 
least  some  portion  of  an  infinite  and  thoroughly 
well-known  time.  So  that  here  we  have  real 
knowledge  of  something  at  least  that  concerns 
the  Cosmos  ;  something  that  is  true  throughout 
the  Immensities  and  the  Eternities.  That 
something  Lobatchewsky  and  his  successors 
have  taken  away.  The  geometer  of  to-day 


PHILOSOPHY  OF  THE  PURE  SCIENCES      359 

knows  nothing  about  the  nature  of  actually 
existing  space  at  an  infinite  distance  ;  he 
knows  nothing  about  the  properties  of  this 
present  space  in  a  past  or  a  future  eternity. 
He  knows,  indeed,  that  the  laws  assumed  by 
Euclid  are  true  with  an  accuracy  that  no  direct 
experiment  can  approach,  not  only  in  this  place 
where  we  are,  but  in  places  at  a  distance  from 
us  that  no  astronomer  has  conceived  ;  but  he 
knows  this  as  of  Here  and  Now  ;  beyond  his 
range  is  a  There  and  Then  of  which  he  knows 
nothing  at  present,  but  may  ultimately  come  to 
know  more.  So,  you  see,  there  is  a  real  parallel 
between  the  work  of  Copernicus  and  his  suc- 
cessors on  the  one  hand,  and  the  work  of 
Lobatchewsky  and  his  successors  on  the  other. 
In  both  of  these  the  knowledge  of  Immensity 
and  Eternity  is  replaced  by  knowledge  of  Here 
and  Now.  And  in  virtue  of  these  two  revolu- 
tions the  idea  of  the  Universe,  the  Macrocosm, 
the  All,  as  subject  of  human  knowledge,  and 
therefore  of  human  interest,  has  fallen  to  pieces. 
It  will  now,  I  think,  be  clear  to  you  why  it 
will  not  do  to  take  for  our  present  considera- 
tion the  postulates  of  geometry  as  Euclid  has 
laid  them  down.  While  they  were  all  certainly 
true,  there  might  be  substituted  for  them  some 
other  group  of  equivalent  propositions  ;  and  the 
choice  of  the  particular  set  of  statements  that 
should  be  used  as  the  groundwork  of  the  science 


360  LECTURES  AND  ESSAYS 

was  to  a  certain  extent  arbitrary,  being  only 
guided  by  convenience  of  exposition.  But  from 
the  moment  that  the  actual  truth  of  these 
assumptions  becomes  doubtful,  they  fall  of 
themselves  into  a  necessary  order  and  classifica- 
tion ;  for  we  then  begin  to  see  which  of  them 
may  be  true  independently  of  the  others.  And 
for  the  purpose  of  criticising  the  evidence  for 
them,  it  is  essential  that  this  natural  order 
should  be  taken  ;  for  I  think  you  will  see 
presently  that  any  other  order  would  bring 
hopeless  confusion  into  the  discussion. 

Space  is  divided  into  parts  in  many  ways. 
If  we  consider  any  material  thing,  space  is  at 
once  divided  into  the  part  where  that  thing  is 
and  the  part  where  it  is  not.  The  water  in 
this  glass,  for  example,  makes  a  distinction 
between  the  space  where  it  is  and  the  space 
where  it  is  not.  Now,  in  order  to  get  from  one 
of  these  to  the  other  you  must  cross  the  surface 
of  the  water  ;  this  surface  is  the  boundary  of  the 
space  where  the  water  is  which  separates  it  from 
the  space  where  it  is  not  Every  thing,  con- 
sidered as  occupying  a  portion  of  space,  has  a 
surface  which  separates  the  space  where  it  is  from 
the  space  where  it  is  not.  But,  again,  a  surface 
may  be  divided  into  parts  in  various  ways. 
Part  of  the  surface  of  this  water  is  against  the 
air,  and  part  is  against  the  glass.  If  you  travel 
over  the  surface  from  one  of  these  parts  to  the 


PHILOSOPHY  OF  THE  PURE  SCIENCES       361 

other,  you  have  to  cross  the  line  which  divides 
them  ;  it  is  this  circular  edge  where  water,  air, 
and  glass  meet.  Every  part  of  a  surface  is 
separated  from  the  other  parts  by  a  line  which 
bounds  it.  But  now  suppose,  further,  that  this 
glass  had  been  so  constructed  that  the  part 
towards  you  was  blue  and  the  part  towards  me 
was  white,  as  it  is  now.  Then  this  line,  divid- 
ing two  parts  of  the  surface  of  the  water,  would 
itself  be  divided  into  two  parts  ;  there  would 
be  a  part  where  it  was  against  the  blue  glass, 
and  a  part  where  it  was  against  the  white  glass. 
If  you  travel  in  thought  along  that  line,  so  as  to 
get  from  one  of  these  two  parts  to  the  other,  you 
have  to  cross  a  point  which  separates  them,  and 
is  the  boundary  between  them.  Every  part  of 
a  line  is  separated  from  the  other  parts  by 
points  which  bound  it.  So  we  may  say 
altogether — 

The  boundary  of  a  solid  (i.e.  of  a  part  of 
space)  is  a  surface. 

The  boundary  of  a  part  of  a  surface  is  a  line. 

The  boundaries  of  a  part  of  a  line  are 
points. 

And  we  are  only  settling  the  meanings  in 
which  words  are  to  be  used.  But  here  we  may 
make  an  observation  which  is  true  of  all  space 
that  we  are  acquainted  with :  it  is  that  the 
process  ends  here.  There  are  no  parts  of  a 
point  which  are  separated  from  one  another  by 


362  LECTURES  AND  ESSAYS 

the  next  link  in   the  series.     This  is  also   in- 
dicated by  the  reverse  process. 

For  I  shall  now  suppose  this  point — the  last 
thing  that  we  got  to  —  to  move  round  the 
tumbler  so  as  to  trace  out  the  line,  or  edge, 
where  air,  water,  and  glass  meet.  In  this  way 
I  get  a  series  of  points,  one  after  another ;  a 
series  of  such  a  nature  that,  starting  from  any 
one  of  them,  only  two  changes  are  possible  that 
will  keep  it  within  the  series  :  it  must  go 
forwards  or  it  must  go  backwards,  and  each  of 
these  is  perfectly  definite.  The  line  may  then 
be  regarded  as  an  aggregate  of  points.  Now 
let  us  imagine,  further,  a  change  to  take  place 
in  this  line,  which  is  nearly  a  circle.  Let  us 
suppose  it  to  contract  towards  the  centre  of  the 
circle,  until  it  becomes  indefinitely  small,  and 
disappears.  In  so  doing  it  will  trace  out  the 
upper  surface  of  the  water,  the  part  of  the 
surface  where  it  is  in  contact  with  the  air.  In 
this  way  we  shall  get  a  series  of  circles  one 
after  another — a  series  of  such  a  nature  that, 
starting  from  any  one  of  them,  only  two  changes 
are  possible  that  will  keep  it  within  the  series  : 
it  must  expand  or  it  must  contract.  This  series, 
therefore,  of  circles,  is  just  similar  to  the  series 
of  points  that  make  one  circle  ;  and  just  as  the 
line  is  regarded  as  an  aggregate  of  points,  so 
we  may  regard  this  surface  as  an  aggregate  of 
lines.  But  this  surface  is  also  in  another  sense 


PHILOSOPHY  OF  THE  PURE  SCIENCES       363 

an  aggregate  of  points,  in  being  an  aggregate 
of  aggregates  of  points.  But,  starting  from  a 
point  in  the  surface,  more  than  two  changes  are 
possible  that  will  keep  it  within  the  surface,  for 
it  may  move  in  any  direction.  The  surface, 
then,  is  an  aggregate  of  points  of  a  different 
kind  from  the  line.  We  speak  of  the  line  as  a 
point  -  aggregate  of  one  dimension,  because, 
starting  from  one  point,  there  are  only  two 
possible  directions  of  change  ;  so  that  the  line 
can  be  traced  out  in  one  motion.  In  the  same 
way,  a  surface  is  a  line-aggregate  of  one  dimen- 
sion, because  it  can  be  traced  out  by  one  motion 
of  the  line  ;  but  it  is  a  point-aggregate  of  two 
dimensions,  because,  in  order  to  build  it  up  of 
points,  we  have  first  to  aggregate  points  into  a 
line,  and  then  lines  into  a  surface.  It  requires 
two  motions  of  a  point  to  trace  it  out. 

Lastly,  let  us  suppose  this  upper  surface  of 
the  water  to  move  downwards,  remaining 
always  horizontal  till  it  becomes  the  under 
surface.  In  so  doing  it  will  trace  out  the  part 
of  space  occupied  by  the  water.  We  shall  thus 
get  a  series  of  surfaces  one  after  another,  pre- 
cisely analogous  to  the  series  of  points  which 
make  a  line,  and  the  series  of  lines  which  make 
a  surface.  The  piece  of  solid  space  is  an  aggre- 
gate of  surfaces,  and  an  aggregate  of  the  same 
kind  as  the  line  is  of  points  ;  it  is  a  surface- 
aggregate  of  one  dimension.  But  at  the  same 


364  LECTURES  AND  ESSAYS 

time  it  is  a  line-aggregate  of  two  dimensions, 
and  a  point  -  aggregate  of  three  dimensions. 
For  if  you  consider  a  particular  line  which  has 
gone  to  make  this  solid,  a  circle  partly  con- 
tracted and  part  of  the  way  down,  there  are 
more  than  two  opposite  changes  which  it  can 
undergo.  For  it  can  ascend  or  descend,  or 
expand  or  contract,  or  do  both  together  in  any 
proportion.  It  has  just  as  great  a  variety  of 
changes  as  a  point  in  a  surface.  And  the  piece 
of  space  is  called  a  point-aggregate  of  three 
dimensions,  because  it  takes  three  distinct 
motions  to  get  it  from  a  point.  We  must  first 
aggregate  points  into  a  line,  then  lines  into  a 
surface,  then  surfaces  into  a  solid. 

At  this  step  it  is  clear,  again,  that  the  process 
must  stop  in  all  the  space  we  know  of.  For  it 
is  not  possible  to  move  that  piece  of  space  in 
such  a  way  as  to  change  every  point  in  it 
When  we  moved  our  line  or  our  surface,  the 
new  line  or  surface  contained  no  point  what- 
ever that  was  in  the  old  one  ;  we  started  with 
one  aggregate  of  points,  and  by  moving  it  we 
got  an  entirely  new  aggregate,  all  the  points  of 
which  were  new.  But  this  cannot  be  done  with 
the  solid  ;  so  that  the  process  is  at  an  end. 
We  arrive,  then,  at  the  result  that  space  is  of 
three  dimensions. 

Is  this,  then,  one  of  the  postulates  of  the 
science  of  space  ?  No ;  it  is  not.  The  science 


PHILOSOPHY  OF  THE  PURE  SCIENCES      365 

of  space,  as  we  have  it,  deals  with  relations  of 
distance  existing  in  a  certain  space  of  three 
dimensions,  but  it  does  not  at  all  require  us  to 
assume  that  no  relations  of  distance  are  possible 
in  aggregates  of  more  than  three  dimensions. 
The  fact  that  there  are  only  three  dimensions 
does  regulate  the  number  of  books  that  we  write, 
and  the  parts  of  the  subject  that  we  study  :  but  it 
is  not  itself  a  postulate  of  the  science.  We  in- 
vestigate a  certain  space  of  three  dimensions, 
on  the  hypothesis  that  it  has  certain  elementary 
properties  ;  and  it  is  the  assumptions  of  these 
elementary  properties  that  are  the  real  postu- 
lates of  the  science  of  space.  To  these  I  now 
proceed. 

The  first  of  them  is  concerned  with  points, 
and  with  the  relation  of  space  to  them.  We 
spoke  of  a  line  as  an  aggregate  of  points. 
Now  there  are  two  kinds  of  aggregates,  which 
are  called  respectively  continuous  and  discrete. 
If  you  consider  this  line,  the  boundary  of  part 
of  the  surface  of  the  water,  you  will  find  yourself 
believing  that  between  any  two  points  of  it  you 
can  put  more  points  of  division,  and  between 
any  two  of  these  more  again,  and  so  on  ;  and 
you  do  not  believe  there  can  be  any  end  to  the 
process.  We  may  express  that  by  saying  you 
believe  that  between  any  two  points  of  the  line 
there  is  an  infinite  number  of  other  points. 
But  now  here  is  an  aggregate  of  marbles,  which, 


366  LECTURES  AND  ESSAYS 

regarded  as  an  aggregate,  has  many  characters 
of  resemblance  with  the  aggregate  of  points. 
It  is  a  series  of  marbles,  one  after  another  ;  and 
if  we  take  into  account  the  relations  of  nextness 
or  contiguity  which  they  possess,  then  there  are 
only  two  changes  possible  from  one  of  them  as 
we  travel  along  the  series  :  we  must  go  to  the 
next  in  front,  or  to  the  next  behind.  But  yet 
it  is  not  true  that  between  any  two  of  them 
there  is  an  infinite  number  of  other  marbles  ; 
between  these  two,  for  example,  there  are  only 
three.  There,  then,  is  a  distinction  at  once 
between  the  two  kinds  of  aggregates.  But 
there  is  another,  which  was  pointed  out  by 
Aristotle  in  his  Physics  and  made  the  basis  of 
a  definition  of  continuity.  I  have  here  a  row 
of  two  different  kinds  of  marbles,  some  white 
and  some  black.  This  aggregate  is  divided 
into  two  parts,  as  we  formerly  supposed  the  line 
to  be.  In  the  case  of  the  line  the  boundary 
between  the  two  parts  is  a  point  which  is  the 
element  of  which  the  line  is  an  aggregate.  In 
this  case  before  us,  a  marble  is  the  element  ; 
but  here  we  cannot  say  that  the  boundary 
between  the  two  parts  is  a  marble.  The 
boundary  of  the  white  parts  is  a  white  marble, 
and  the  boundary  of  the  black  parts  is  a  black 
marble  ;  these  two  adjacent  parts  have  different 
boundaries.  Similarly,  if  instead  of  arranging 
my  marbles  in  a  series,  I  spread  them  out  on  a 


PHILOSOPHY  OF  THE  PURE  SCIENCES       367 

surface,  I  may  have  this  aggregate  divided  into 
two  portions — a  white  portion  and  a  black 
portion  ;  but  the  boundary  of  the  white  portion 
is  a  row  of  white  marbles,  and  the  boundary  of 
the  black  portion  is  a  row  of  black  marbles. 
And  lastly,  if  I  made  a  heap  of  white  marbles, 
and  put  black  marbles  on  the  top  of  them,  I 
should  have  a  discrete  aggregate  of  three 
dimensions  divided  into  two  parts  :  the  bound- 
ary of  the  white  part  would  be  a  layer  of  white 
marbles,  and  the  boundary  of  the  black  part 
would  be  a  layer  of  black  marbles.  In  all  these 
cases  of  discrete  aggregates,  when  they  are 
divided  into  two  parts,  the  two  adjacent  parts 
have  different  boundaries.  But  if  you  come  to 
consider  an  aggregate  that  you  believe  to  be 
continuous,  you  will  see  that  you  think  of  two 
adjacent  parts  as  having  the  same  boundary. 
What  is  the  boundary  between  water  and  air 
here  ?  Is  it  water  ?  No  ;  for  there  would  still 
have  to  be  a  boundary  to  divide  that  water  from 
the  air.  For  the  same  reason  it  cannot  be  air. 
I  do  not  want  you  at  present  to  think  of  the 
actual  physical  facts  by  the  aid  of  any  mole- 
cular theories ;  I  want  you  only  to  think  of 
what  appears  to  be,  in  order  to  understand 
clearly  a  conception  that  we  all  have.  Suppose 
the  things  actually  in  contact  If,  however 
much  we  magnified  them,  they  still  appeared  to 
be  thoroughly  homogeneous,  the  water  filling 


368  LECTURES  AND  ESSAYS 

up  a  certain  space,  the  air  an  adjacent  space  ; 
if  this  held  good  indefinitely  through  all  degrees 
of  conceivable  magnifying,  then  we  could  not 
say  that  the  surface  of  the  water  was  a  layer  of 
water  and  the  surface  of  air  a  layer  of  air  ;  we 
should  have  to  say  that  the  same  surface  was 
the  surface  of  both  of  them,  and  was  itself 
neither  one  nor  the  other  —  that  this  surface 
occupied  no  space  at  all.  Accordingly,  Aristotle 
defined  the  continuous  as  that  of  which  two 
adjacent  parts  have  the  same  boundary  ;  and 
the  discontinuous  or  discrete  as  that  of  which 
two  adjacent  parts  have  direct  boundaries.1 

Now  the  first  postulate  of  the  science  of 
space  is  that  space  is  a  continuous  aggregate 
of  points,  and  not  a  discrete  aggregate.  And 
this  postulate  —  which  I  shall  call  the  postulate 
of  continuity  —  is  really  involved  in  those  three 
of  the  six2  postulates  of  Euclid  for  which 
Robert  Simson  has  retained  the  name  of 
postulate.  You  will  see,  on  a  little  reflection, 
that  a  discrete  aggregate  of  points  could  not  be 

1  Phys.  Ausc.  V.  3,  p.  227,  ed.  Bekker.  Ti  6£  <nive\^  ftm 
fifr  Strep  fx^^"^"  TL>  ^yw  8'  flvai  ffwex^i  Srav  ravrb  yfvtjTat  ical 
tr  TO  ina.Ttpov  irtpat  olt  dVroircu,  Kal  Gxrirep  ffrjfjLalvfi  roCvofM 
ffw^xn™'  ToOro  8'  oux  olbv  re  Svoly  dvroiv  dvai  TO'IV  t<r)(6.TOt.v. 

A  little  farther  on  he  makes  the  important  remark  that  on  the 
hypothesis  of  continuity  a  line  is  not  made  up  of  points  in  the  same 
way  that  a  whole  is  made  up  of  parts,  VI.  i,  p.  231.  'ASiWrov 
££  ASiaipiruv  etc  at  ri  trwex^*,  olov  ypa,fjip.i)v  tic  ffTtyfi&v,  ftwep  ij 


2  See  De  Morgan,  in  Smith's  Diet,  of  Biography  and  Mythology, 
Art.  "  Euclid  "  ;  and  in  the  English  Cyclopedia,  Art.  "  Axiom." 


PHILOSOPHY  OF  THE  PURE  SCIENCES      369 

so  arranged  that  any  two  of  them  should  be 
relatively  situated  to  one  another  in  exactly  the 
same  manner,  so  that  any  two  points  might  be 
joined  by  a  straight  linet  which  should  always 
bear  the  same  definite  relation  to  them.  And 
the  same  difficulty  occurs  in  regard  to  the 
other  two  postulates.  But  perhaps  the  most 
conclusive  way  of  showing  that  this  postulate 
is  really  assumed  by  Euclid  is  to  adduce  the 
proposition  he  proves,  that  every  finite  straight 
line  may  be  bisected.  Now  this  could  not  be 
the  case  if  it  consisted  of  an  odd  number  of 
separate  points.  As  the  first  of  the  postulates 
of  the  science  of  space,  then,  we  must  reckon 
this  postulate  of  Continuity ;  according  to 
which  two  adjacent  portions  of  space,  or  of  a 
surface,  or  of  a  line,  have  the  same  boundary, 
viz. — a  surface,  a  line,  or  a  point ;  and  between 
every  two  points  on  a  line  there  is  an  infinite 
number  of  intermediate  points. 

The  next  postulate  is  that  of  Elementary 
Flatness.  You  know  that  if  you  get  hold  of  a 
small  piece  of  a  very  large  circle,  it  seems  to 
you  nearly  straight.  So,  if  you  were  to  take 
any  curved  line,  and  magnify  it  very  much, 
confining  your  attention  to  a  small  piece  of  it, 
that  piece  would  seem  straighter  to  you  than 
the  curve  did  before  it  was  magnified.  At 
least,  you  can  easily  conceive  a  curve  possess- 
ing this  property,  that  the  more  you  magnify 
VOL.  I  2  B 


370  LECTURES  AND  ESSAYS 

it,  the  straighter  it  gets.  Such  a  curve  would 
possess  the  property  of  elementary  flatness. 
In  the  same  way,  if  you  perceive  a  portion  of 
the  surface  of  a  very  large  sphere,  such  as  the 
earth,  it  appears  to  you  to  be  flat.  If,  then, 
you  take  a  sphere  of  say  a  foot  diameter,  and 
magnify  it  more  and  more,  you  will  find  that 
the  more  you  magnify  it  the  flatter  it  gets. 
And  you  may  easily  suppose  that  this  process 
would  go  on  indefinitely  ;  that  the  curvature 
would  become  less  and  less  the  more  the 
surface  was  magnified.  Any  curved  surface 
which  is  such  that  the  more  you  magnify  it 
the  flatter  it  gets,  is  said  to  possess  the  property 
of  elementary  flatness.  But  if  every  succeed- 
ing power  of  our  imaginary  microscope  disclosed 
new  wrinkles  and  inequalities  without  end,  then 
we  should  say  that  the  surface  did  not  possess 
the  property  of  elementary  flatness. 

But  how  am  I  to  explain  how  solid  space 
can  have  this  property  of  elementary  flatness  ? 
Shall  I  leave  it  as  a  mere  analogy,  and  say 
that  it  is  the  same  kind  of  property  as  this  of 
the  curve  and  surface,  only  in  three  dimensions 
instead  of  one  or  two  ?  I  think  I  can  get  a 
little  nearer  to  it  than  that  ;  at  all  events  I 
will  try. 

If  we  start  to  go  out  from  a  point  on  a  surface, 
there  is  a  certain  choice  of  directions  in  which 
we  may  go.  These  directions  make  certain 


PHILOSOPHY  OF  THE  PURE  SCIENCES      371 

angles  with  one  another.  We  may  suppose 
a  certain  direction  to  start  with,  and  then 
gradually  alter  that  by  turning  it  round  the 
point :  we  find  thus  a  single  series  of  directions 
in  which  we  may  start  from  the  point.  Accord- 
ing to  our  first  postulate,  it  is  a  continuous 
series  of  directions.  Now  when  I  speak  of  a 
direction  from  the  point,  I  mean  a  direction  of 
starting  ;  I  say  nothing  about  the  subsequent 
path.  Two  different  paths  may  have  the  same 
direction  at  starting ;  in  this  case  they  will 
touch  at  the  point ;  and  there  is  an  obvious 
difference  between  two  paths  which  touch  and 
two  paths  which  meet  and  form  an  angle. 
Here,  then,  is  an  aggregate  of  directions,  and 
they  can  be  changed  into  one  another.  More- 
over, the  changes  by  which  they  pass  into  one 
another  have  magnitude,  they  constitute  dis- 
tance-relations ;  and  the  amount  of  change 
necessary  to  turn  one  of  them  into  another  is 
called  the  angle  between  them.  It  is  involved 
in  this  postulate  that  we  are  considering,  that 
angles  can  be  compared  in  respect  of  magni- 
tude. But  this  is  not  all.  If  we  go  on  changing 
a  direction  of  start,  it  will,  after  a  certain  amount 
of  turning,  come  round  into  itself  again,  and  be 
the  same  direction.  On  every  surface  which 
has  the  property  of  elementary  flatness,  the 
amount  of  turning  necessary  to  take  a  direction 
all  round  into  its  first  position  is  the  same  for 


37«  LECTURES  AND  ESSAYS 

all  points  of  the  surface.  I  will  now  show  you 
a  surface  which  at  one  point  of  it  has  not  this 
property.  I  take  this  circle  of  paper  from 
which  a  sector  has  been  cut  out,  and  bend  it 
round  so  as  to  join  the  edges  ;  in  this  way  I 
form  a  surface  which  is  called  a  cone.  Now  on 
all  points  of  this  surface  but  one,  the  law  of 
elementary  flatness  holds  good.  At  the  vertex 
of  the  cone,  however,  notwithstanding  that  there 
is  an  aggregate  of  directions  in  which  you  may 
start,  such  that  by  continuously  changing  one  of 
them  you  may  get  it  round  into  its  original  posi- 
tion, yet  the  whole  amount  of  change  necessary  to 
effect  this  is  not  the  same  at  the  vertex  as  it  is  at 
any  other  point  of  the  surface.  And  this  you 
can  see  at  once  when  I  unroll  it  ;  for  only  part 
of  the  directions  in  the  plane  have  been  included 
in  the  cone.  At  this  point  of  the  cone,  then, 
it  does  not  possess  the  property  of  elementary 
flatness  ;  and  no  amount  of  magnifying  would 
ever  make  a  cone  seem  flat  at  its  vertex. 

To  apply  this  to  solid  space,  we  must  notice 
that  here  also  there  is  a  choice  of  directions  in 
which  you  may  go  out  from  any  point  ;  but  it 
is  a  much  greater  choice  than  a  surface  gives 
you.  Whereas  in  a  surface  the  aggregate  of 
directions  is  only  of  one  dimension,  in  solid 
space  it  is  of  two  dimensions.  But  here  also 
there  are  distance-relations,  and  the  aggregate 
of  directions  may  be  divided  into  parts  which 


PHILOSOPHY  OF  THE  PURE  SCIENCES       373 

have  quantity.  For  example,  the  directions 
which  start  from  the  vertex  of  this  cone  are 
divided  into  those  which  go  inside  the  cone, 
and  those  which  go  outside  the  cone.  The 
part  of  the  aggregate  which  is  inside  the  cone 
is  called  a  solid  angle.  Now  in  those  spaces 
of  three  dimensions  which  have  the  property  of 
elementary  flatness,  the  whole  amount  of  solid 
angle  round  one  point  is  equal  to  the  whole 
amount  round  another  point.  Although  the 
space  need  not  be  exactly  similar  to  itself  in 
all  parts,  yet  the  aggregate  of  directions  round 
one  point  is  exactly  similar  to  the  aggregate 
of  directions  round  another  point,  if  the  space 
has  the  property  of  elementary  flatness. 

How  does  Euclid  assume  this  postulate  of 
Elementary  Flatness  ?  In  his  fourth  postulate 
he  has  expressed  it  so  simply  and  clearly  that 
you  will  wonder  how  anybody  could  make  all 
this  fuss.  He  says,  "  All  right  angles  are  equal." 

Why  could  I  not  have  adopted  this  at  once, 
and  saved  a  great  deal  of  trouble  ?  Because  it 
assumes  the  knowledge  of  a  surface  possessing 
the  property  of  elementary  flatness  in  all  its 
points.  Unless  such  a  surface  is  first  made 
out  to  exist,  and  the  definition  of  a  right  angle 
is  restricted  to  lines  drawn  upon  it — for  there 
is  no  necessity  for  the  word  straight  in  that 
definition  —  the  postulate  in  Euclid's  form  is 
obviously  not  true.  I  can  make  two  lines  cross 


374  LECTURES  AND  ESSAYS  ; 

at  the  vertex  of  a  cone  so  that  the  four  adjacent 
angles  shall  be  equal,  and  yet  not  one  of  them 
equal  to  a  right  angle. 

I  pass  on  to  the  third  postulate  of  the 
science  of  space  —  the  postulate  of  Super- 
position. According  to  this  postulate  a  body 
can  be  moved  about  in  space  without  altering 
its  size  or  shape.  This  seems  obvious  enough, 
but  it  is  worth  while  to  examine  a  little  closely 
into  the  meaning  of  it.  We  must  define  what 
we  mean  by  size  and  by  shape.  When  we  say 
that  a  body  can  be  moved  about  without 
altering  its  size,  we  mean  that  it  can  be  so 
moved  as  to  keep  unaltered  the  length  of  all 
the  lines  in  it.  This  postulate  therefore  in- 
volves that  lines  can  be  compared  in  respect  of 
magnitude,  or  that  they  have  a  length  in- 
dependent of  position  ;  precisely  as  the  former 
one  involved  the  comparison  of  angular  magni- 
tudes. And  when  we  say  that  a  body  can  be 
moved  about  without  altering  its  shape,  we 
mean  that  it  can  be  so  moved  as  to  keep 
unaltered  all  the  angles  in  it.  It  is  not 
necessary  to  make  mention  of  the  motion  of  a 
body,  although  that  is  the  easiest  way  of 
expressing  and  of  conceiving  this  postulate  ; 
but  we  may,  if  we  like,  express  it  entirely  in 
terms  which  belong  to  space,  and  that  we 
should  do  in  this  way.  Suppose  a  figure  to 
have  been  constructed  in  some  portion  of 


PHILOSOPHY  OF  THE  PURE  SCIENCES       375 

space ;  say  that  a  triangle  has  been  drawn 
whose  sides  are  the  shortest  distances  between 
its  angular  points.  Then  if  in  any  other 
portion  of  space  two  points  are  taken  whose 
shortest  distance  is  equal  to  a  side  of  the 
triangle,  and  at  one  of  them  an  angle  is  made 
equal  to  one  of  the  angles  adjacent  to  that  side, 
and  a  line  of  shortest  distance  drawn  equal  to 
the  corresponding  side  of  the  original  triangle, 
the  distance  from  the  extremity  of  this  to  the 
other  of  the  two  points  will  be  equal  to  the 
third  side  of  the  original  triangle,  and  the  two 
will  be  equal  in  all  respects ;  or  generally,  if  a 
figure  has  been  constructed  anywhere,  another 
figure,  with  all  its  lines  and  all  its  angles  equal 
to  the  corresponding  lines  and  angles  of  the 
first,  can  be  constructed  anywhere  else.  Now 
this  is  exactly  what  is  meant  by  the  principle 
of  superposition  employed  by  Euclid  to  prove 
the  proposition  that  I  have  just  mentioned. 
And  we  may  state  it  again  in  this  short  form — 
All  parts  of  space  are  exactly  alike. 

But  this  postulate  carries  with  it  a  most 
important  consequence.  It  enables  us  to  make 
a  pair  of  most  fundamental  definitions — those 
of  the  plane  and  of  the  straight  line.  In  order 
to  explain  how  these  come  out  of  it  when  it  is 
granted,  and  how  they  cannot  be  made  when 
it  is  not  granted,  I  must  here  say  something 
more  about  the  nature  of  the  postulate  itself, 


376  LECTURES  AND  ESSAYS 

which  might  otherwise  have  been  left  until  we 
come  to  criticise  it. 

We  have  stated  the  postulate  as  referring  to 
solid  space.  But  a  similar  property  may  exist 
in  surfaces.  Here,  for  instance,  is  part  of  the 
surface  of  a  sphere.  If  I  draw  any  figure  I 
like  upon  this,  I  can  suppose  it  to  be  moved 
about  in  any  way  upon  the  sphere,  without 
alteration  of  its  size  or  shape.  If  a  figure  has 
been  drawn  on  any  part  of  the  surface  of  a 
sphere,  a  figure  equal  to  it  in  all  respects  may 
be  drawn  on  any  other  part  of  the  surface. 
Now  I  say  that  this  property  belongs  to  the 
surface  itself,  is  a  part  of  its  own  internal 
economy,  and  does  not  depend  in  any  way 
upon  its  relation  to  space  of  three  dimensions. 
For  I  can  pull  it  about  and  bend  it  in  all 
manner  of  ways,  so  as  altogether  to  alter  its 
relation  to  solid  space  ;  and  yet,  if  I  do  not 
stretch  it  or  tear  it,  I  make  no  difference 
whatever  in  the  length  of  any  lines  upon  it,  or 
in  the  size  of  any  angles  upon  it.1  I  do  not  in 
any  way  alter  the  figures  drawn  upon  it,  or  the 
possibility  of  drawing  figures  upon  it,  so  far  as 

1  This  figure  was  made  of  linen,  starched  upon  a  spherical 
surface,  and  taken  off  when  dry.  That  mentioned  in  the  next 
paragraph  was  similarly  stretched  upon  the  irregular  surface  of  the 
head  of  a  bust.  For  durability  these  models  should  be  made  of 
two  thicknesses  of  linen  starched  together  in  such  a  way  that  the 
fibres  of  one  bisect  the  angles  between  the  fibres  of  the  other,  and 
the  edge  should  be  bound  by  a  thin  slip  of  paper.  They  will  then 
retain  their  curvature  unaltered  for  a  long  time. 


PHILOSOPHY  OF  THE  PURE  SCIENCES       377 

their  relations  with  the  stirface  itself  are  con- 
cerned. This  property  of  the  surface,  then, 
could  be  ascertained  by  people  who  lived 
entirely  in  it,  and  were  absolutely  ignorant  of  a 
third  dimension.  As  a  point-aggregate  of  two 
dimensions,  it  has  in  itself  properties  deter- 
mining the  distance-relations  of  the  points  upon 
it,  which  are  absolutely  independent  of  the 
existence  of  any  points  which  are  not  upon  it. 

Now  here  is  a  surface  which  has  not  that 
property.  You  observe  that  it  is  not  of  the 
same  shape  all  over,  and  that  some  parts  of  it 
are  more  curved  than  other  parts.  If  you 
drew  a  figure  upon  this  surface,  and  then  tried 
to  move  it  about,  you  would  find  that  it  was 
impossible  to  do  so  without  altering  the  size 
and  shape  of  the  figure.  Some  parts  of  it 
would  have  to  expand,  some  to  contract,  the 
lengths  of  the  lines  could  not  all  be  kept  the 
same,  the  angles  would  not  hit  off  together. 
And  this  property  of  the  surface — that  its 
parts  are  different  from  one  another — is  a 
property  of  the  surface  itself,  a  part  of  its 
internal  economy,  absolutely  independent  of 
any  relations  it  may  have  with  space  outside 
of  it.  For,  as  with  the  other  one,  I  can  pull 
it  about  in  all  sorts  of  ways,  and,  so  long  as  I 
do  not  stretch  it  or  tear  it,  I  make  no  alteration 
in  the  length  of  lines  drawn  upon  it  or  in  the 
size  of  the  angles. 


378  LECTURES  AND  ESSAYS 

Here,  then,  is  an  intrinsic  difference  between 
these  two  surfaces,  as  surfaces.  They  are  both 
point-aggregates  of  two  dimensions  ;  but  the 
points  in  them  have  certain  relations  of  distance 
(distance  measured  always  on  the  surface),  and 
these  relations  of  distance  are  not  the  same  in 
one  case  as  they  are  in  the  other. 

The  supposed  people  living  in  the  surface 
and  having  no  idea  of  a  third  dimension  might, 
without  suspecting  that  third  dimension  at  all, 
make  a  very  accurate  determination  of  the 
nature  of  their  locus  in  quo.  If  the  people  who 
lived  on  the  surface  of  the  sphere  were  to 
measure  the  angles  of  a  triangle,  they  would  find 
them  to  exceed  two  right  angles  by  a  quantity 
proportional  to  the  area  of  the  triangle.  This 
excess  of  the  angles  above  two  right  angles, 
being  divided  by  the  area  of  the  triangle,  would 
be  found  to  give  exactly  the  same  quotient  at 
all  parts  of  the  sphere.  That  quotient  is  called 
the  curvature  of  the  surface  ;  and  we  say  that 
a  sphere  is  a  surface  of  uniform  curvature. 
But  if  the  people  living  on  this  irregular  surface 
were  to  do  the  same  thing,  they  would  not  find 
quite  the  same  result.  The  sum  of  the  angles 
would,  indeed,  differ  from  two  right  angles,  but 
sometimes  in  excess  and  sometimes  in  defect, 
according  to  the  part  of  the  surface  where  they 
were.  And  though  for  small  triangles  in  any 
one  neighbourhood  the  excess  or  defect  would 


PHILOSOPHY  OF  THE  PURE  SCIENCES      379 

be  nearly  proportional  to  the  area  of  the 
triangle,  yet  the  quotient  obtained  by  dividing 
this  excess  or  defect  by  the  area  of  the  triangle 
would  vary  from  one  part  of  the  surface  to 
another.  In  other  words,  the  curvature  of  this 
surface  varies  from  point  to  point ;  it  is  some- 
times positive,  sometimes  negative,  sometimes 
nothing  at  all. 

But  now  comes  the  important  difference. 
When  I  speak  of  a  triangle,  what  do  I  suppose 
the  sides  of  that  triangle  to  be  ? 

If  I  take  two  points  near  enough  together 
upon  a  surface,  and  stretch  a  string  between 
them,  that  string  will  take  up  a  certain  definite 
position  upon  the  surface,  marking  the  line  of 
shortest  distance  from  one  point  to  the  other. 
Such  a  line  is  called  a  geodesic  line.  It  is  a 
line  determined  by  the  intrinsic  properties  of 
the  surface,  and  not  by  its  relations  with  ex- 
ternal space.  The  line  would  still  be  the 
shortest  line,  however  the  surface  were  pulled 
about  without  stretching  or  tearing.  A  geodesic 
line  may  be  produced,  when  a  piece  of  it  is 
given  ;  for  we  may  take  one  of  the  points,  and, 
keeping  the  string  stretched,  make  it  go  round 
in  a  sort  of  circle  until  the  other  end  has  turned 
through  two  right  angles.  The  new  position 
will  then  be  a  prolongation  of  the  same  geodesic 
line. 

In  speaking  of  a  triangle,  then,   I   meant  a 


380  LECTURES  AND  ESSAYS 

triangle  whose  sides  are  geodesic  lines.  But 
in  the  case  of  a  spherical  surface — or,  more 
generally,  of  a  surface  of  constant  curvature — 
these  geodesic  lines  have  another  and  most  im- 
portant property.  They  are  straight,  so  far  as 
the  surface  is  concerned.  On  this  surface  a 
figure  may  be  moved  about  without  altering  its 
size  or  shape.  It  is  possible,  therefore,  to 
draw  a  line  which  shall  be  of  the  same  shape 
all  along  and  on  both  sides.  That  is  to  say,  if 
you  take  a  piece  of  the  surface  on  one  side  of 
such  a  line,  you  may  slide  it  all  along  the  line 
and  it  will  fit ;  and  you  may  turn  it  round  and 
apply  it  to  the  other  side,  and  it  will  fit  there 
also.  This  is  Leibnitz's  definition  of  a  straight 
line,  and,  you  see,  it  has  no  meaning  except  in 
the  case  of  a  surface  of  constant  curvature,  a 
surface  all  parts  of  which  are  alike. 

Now  let  us  consider  the  corresponding 
things  in  solid  space.  In  this  also  we  may 
have  geodesic  lines  ;  namely,  lines  formed  by 
stretching  a  string  between  two  points.  But 
we  may  also  have  geodesic  surfaces  ;  and  they 
are  produced  in  this  manner.  Suppose  we 
have  a  point  on  a  surface,  and  this  surface 
possesses  the  property  of  elementary  flatness. 
Then  among  all  the  directions  of  starting  from 
the  point,  there  are  some  which  start  in  the 
surface,  and  do  not  make  an  angle  with  it. 
Let  all  these  be  prolonged  into  geodesies  ;  then 


PHILOSOPHY  OF  THE  PURE  SCIENCES       381 

we  may  imagine  one  of  these  geodesies  to 
travel  round  and  coincide  with  all  the  others  in 
turn.  In  so  doing  it  will  trace  out  a  surface 
which  is  called  a  geodesic  surface.  Now  in 
the  particular  case  where  a  space  of  three 
dimensions  has  the  property  of  superposition, 
or  is  all  over  alike,  these  geodesic  surfaces  are 
planes.  That  is  to  say,  since  the  space  is  all 
over  alike,  these  surfaces  are  also  of  the  same 
shape  all  over  and  on  both  sides  ;  which  is 
Leibnitz's  definition  of  a  plane.  If  you  take 
a  piece  of  space  on  one  side  of  such  a  plane, 
partly  bounded  by  the  plane,  you  may  slide  it 
all  over  the  plane  and  it  will  fit ;  and  you  may 
turn  it  round  and  apply  it  to  the  other  side, 
and  it  will  fit  there  also.  Now  it  is  clear  that 
this  definition  will  have  no  meaning  unless  the 
third  postulate  be  granted.  So  we  may  say 
that  when  the  postulate  of  Superposition  is 
true,  then  there  are  planes  and  straight  lines  ; 
and  they  are  defined  as  being  of  the  same  shape 
throughout  and  on  both  sides. 

It  is  found  that  the  whole  geometry  of  a 
space  of  three  dimensions  is  known  when  we 
know  the  curvature  of  three  geodesic  surfaces 
at  every  point.  The  third  postulate  requires 
that  the  curvature  of  all  geodesic  surfaces 
should  be  everywhere  equal  to  the  same  quantity. 

I  pass  to  the  fourth  postulate,  which  I  call 
the  postulate  of  Similarity.  According  to  this 


382  LECTURES  AND  ESSAYS 

postulate,  any  figure  may  be  magnified  or 
diminished  in  any  degree  without  altering  its 
shape.  If  any  figure  has  been  constructed  in 
one  part  of  space,  it  may  be  reconstructed  to 
any  scale  whatever  in  any  other  part  of  space, 
so  that  no  one  of  the  angles  shall  be  altered, 
though  all  the  lengths  of  lines  will  of  course  be 
altered.  This  seems  to  be  a  sufficiently  obvious 
induction  from  experience  ;  for  we  have  all 
frequently  seen  different  sizes  of  the  same 
shape  ;  and  it  has  the  advantage  of  embodying 
the  fifth  and  sixth  of  Euclid's  postulates  in  a 
single  principle,  which  bears  a  great  resemblance 
in  form  to  that  of  Superposition,  and  may  be 
used  in  the  same  manner.  It  is  easy  to  show 
that  it  involves  the  two  postulates  of  Euclid  : 
"  Two  straight  lines  cannot  enclose  a  space," 
and  "  Lines  in  one  plane  which  never  meet 
make  equal  angles  with  every  other  line." 

This  fourth  postulate  is  equivalent  to  the 
assumption  that  the  constant  curvature  of  the 
geodesic  surfaces  is  zero ;  or  the  third  and 
fourth  may  be  put  together,  and  we  shall  then 
say  that  the  three  curvatures  of  space  are  all  of 
them  zero  at  every  point. 

The  supposition  made  by  Lobatchewsky 
was,  that  the  three  first  postulates  were  true, 
but  not  the  fourth.  Of  the  two  Euclidean 
postulates  included  in  this,  he  admitted  one, 
viz.  that  two  straight  lines  cannot  enclose  a 


PHILOSOPHY  OF  THE  PURE  SCIENCES      383 

space,  or  that  two  lines  which  once  diverge  go 
on  diverging  for  ever.  But  he  left  out  the 
postulate  about  parallels,  which  may  be  stated 
in  this  form.  If  through  a  point  outside  of  a 
straight  line  there  be  drawn  another,  indefinitely 
produced  both  ways  ;  and  if  we  turn  this  second 
one  round  so  as  to  make  the  point  of  intersec- 
tion travel  along  the  first  line,  then  at  the  very 
instant  that  this  point  of  intersection  disappears 
at  one  end  it  will  reappear  at  the  other,  and 
there  is  only  one  position  in  which  the  lines  do 
not  intersect  Lobatchewsky  supposed,  instead, 
that  there  was  a  finite  angle  through  which  the 
second  line  must  be  turned  after  the  point  of 
intersection  had  disappeared  at  one  end,  before 
it  reappeared  at  the  other.  For  all  positions 
of  the  second  line  within  this  angle  there  is 
then  no  intersection.  In  the  two  limiting 
positions,  when  the  lines  have  just  done  meet- 
ing at  one  end,  and  when  they  are  just  going 
to  meet  at  the  other,  they  are  called  parallel ; 
so  that  two  lines  can  be  drawn  through  a  fixed 
point  parallel  to  a  given  straight  line.  The 
angle  between  these  two  depends  in  a  certain 
way  upon  the  distance  of  the  point  from  the 
line.  The  sum  of  the  angles  of  a  triangle  is 
less  than  two  right  angles  by  a  quantity  pro- 
portional to  the  area  of  the  triangle.  The 
whole  of  this  geometry  is  worked  out  in  the 
style  of  Euclid,  and  the  most  interesting  con- 


384  LECTURES  AND  ESSAYS 

elusions  are  arrived  at ;  particularly  in  the 
theory  of  solid  space,  in  which  a  surface  turns 
up  which  is  not  plane  relatively  to  that  space, 
but  which,  for  purposes  of  drawing  figures  upon 
it,  is  identical  with  the  Euclidean  plane. 

It  was  Riemann,  however,  who  first  accom- 
plished the  task  of  analysing  all  the  assump- 
tions of  geometry,  and  showing  which  of  them 
were  independent.  This  very  disentangling 
and  separation  of  them  is  sufficient  to  deprive 
them  for  the  geometer  of  their  exactness  and 
necessity;  for  the  process  by  which  it  is 
effected  consists  in  showing  the  possibility  of 
conceiving  these  suppositions  one  by  one  to  be 
untrue ;  whereby  it  is  clearly  made  out  how 
much  is  supposed.  But  it  may  be  worth  while 
to  state  formally  the  case  for  and  against  them. 

When  it  is  maintained  that  we  know  these 
postulates  to  be  universally  true,  in  virtue  of 
certain  deliverances  of  our  consciousness,  it  is 
implied  that  these  deliverances  could  not  exist, 
except  upon  the  supposition  that  the  postulates 
are  true.  If  it  can  be  shown,  then,  from  ex- 
perience that  our  consciousness  would  tell  us 
exactly  the  same  things  if  the  postulates  are 
not  true,  the  ground  of  their  validity  will  be 
taken  away.  But  this  is  a  very  easy  thing  to 
show. 

That  same  faculty  which  tells  you  that 
space  is  continuous  tells  you  that  this  water  is 


PHILOSOPHY  OF  THE  PURE  SCIENCES       385 

continuous,  and  that  the  motion  perceived  in  a 
wheel  of  life  is  continuous.  Now  we  happen 
to  know  that  if  we  could  magnify  this  water  as 
much  again  as  the  best  microscopes  can  magnify 
it,  we  should  perceive  its  granular  structure. 
And  what  happens  in  a  wheel  of  life  is  dis- 
covered by  stopping  the  machine.  Even  apart, 
then,  from  our  knowledge  of  the  way  nerves 
act  in  carrying  messages,  it  appears  that  we 
have  no  means  of  knowing  anything  more 
about  an  aggregate  than  that  it  is  too  fine- 
grained for  us  to  perceive  its  discontinuity,  if  it 
has  any. 

Nor  can  we,  in  general,  receive  a  conception 
as  positive  knowledge  which  is  itself  founded 
merely  upon  inaction.  For  the  conception  of 
a  continuous  thing  is  of  that  which  looks  just 
the  same  however  much  you  magnify  it.  We 
may  conceive  the  magnifying  to  go  on  to  a 
certain  extent  without  change,  and  then,  as  it 
were,  leave  it  going  on,  without  taking  the 
trouble  to  doubt  about  the  changes  that  may 
ensue. 

In  regard  to  the  second  postulate,  we  have 
merely  to  point  to  the  example  of  polished 
surfaces.  The  smoothest  surface  that  can  be 
made  is  the  one  most  completely  covered  with 
the  minutest  ruts  and  furrows.  Yet  geometrical 
constructions  can  be  made  with  extreme  accuracy 
upon  such  a  surface,  on  the  supposition  that  it 
VOL.  I  2  C 


386  LECTURES  AND  ESSAYS 

is  an  exact  plane.  If,  therefore,  the  sharp 
points,  edges,  and  furrows  of  space  are  only 
small  enough,  there  will  be  nothing  to  hinder 
our  conviction  of  its  elementary  flatness.  It 
has  even  been  remarked  by  Riemann  that  we 
must  not  shrink  from  this  supposition  if  it  is 
found  useful  in  explaining  physical  phenomena. 

The  first  two  postulates  may  therefore  be 
doubted  on  the  side  of  the  very  small.  We 
may  put  the  third  and  fourth  together,  and 
doubt  them  on  the  side  of  the  very  great.  For 
if  the  property  of  elementary  flatness  exist  on 
the  average,  and  the  deviations  from  it  being, 
as  we  have  supposed,  too  small  to  be  perceived, 
then,  whatever  were  the  true  nature  of  space, 
we  should  have  exactly  the  conceptions  of  it 
which  we  now  have,  if  only  the  regions  we  can 
get  at  were  small  in  comparison  with  the 
areas  of  curvature.  If  we  suppose  the  curvature 
to  vary  in  an  irregular  manner,  the  effect  of  it 
might  be  very  considerable  in  a  triangle  formed 
by  the  nearest  fixed  stars  ;  but  if  we  suppose 
it  approximately  uniform  to  the  limit  of  tele- 
scopic reach,  it  will  be  restricted  to  very  much 
narrower  limits.  I  cannot  perhaps  do  better 
than  conclude  by  describing  to  you  as  well  as 
I  can  what  is  the  nature  of  things  on  the 
supposition  that  the  curvature  of  all  space  is 
nearly  uniform  and  positive. 

In  this  case  the  Universe,  as  known,  becomes 


PHILOSOPHY  OF  THE  PURE  SCIENCES       387 

again  a  valid  conception  ;  for  the  extent  of 
space  is  a  finite  number  of  cubic  miles.1  And 
this  comes  about  in  a  curious  way.  If  you 
were  to  start  in  any  direction  whatever,  and 
move  in  that  direction  in  a  perfect  straight  line 
according  to  the  definition  of  Leibnitz  ;  after 
travelling  a  most  prodigious  distance,  to  which 
the  parallactic  unit  —  200,000  times  the 
diameter  of  the  earth's  orbit — would  be  only  a 
few  steps,  you  would  arrive  at — this  place. 
Only,  if  you  had  started  upwards,  you  would 
appear  from  below.  Now,  one  of  two  things 
would  be  true.  Either,  when  you  had  got  half- 
way on  your  journey,  you  came  to  a  place  that 
is  opposite  to  this,  and  which  you  must  have 
gone  through,  whatever  direction  you  started 
in  ;  or  else  all  paths  you  could  have  taken 
diverge  entirely  from  each  other  till  they  meet 
again  at  this  place.  In  the  former  case,  every 
two  straight  lines  in  a  plane  meet  in  two  points, 
in  the  latter  they  meet  only  in  one.  Upon 
this  supposition  of  a  positive  curvature,  the 
whole  of  geometry  is  far  more  complete  and 
interesting ;  the  principle  of  duality,  instead  of 
half  breaking  down  over  metric  relations, 
applies  to  all  propositions  without  exception. 
In  fact,  I  do  not  mind  confessing  that  I 

1  The  assumptions  here  made  about  the  Zusammenhang  of 
space  are  the  simplest  ones,  but  even  the  finite  extent  does  not 
follow  necessarily  from  uniform  positive  curvature,  as  Riemann 
seems  to  have  supposed. 


388  LECTURES  AND  ESSAYS 

personally  have  often  found  relief  from  the 
dreary  infinities  of  homaloidal  space  in  the 
consoling  hope  that,  after  all,  this  other  may 
be  the  true  state  of  things. 


IV. THE    UNIVERSAL    STATEMENTS    OF 

ARITHMETIC 

WE  have  now  to  consider  a  series  of  alleged 
universal  statements,  the  truth  of  which  nobody 
has  ever  doubted.  They  are  statements  be- 
longing to  arithmetic,  to  the  science  of  quantity, 
to  pure  logic,  and  to  a  branch  of  the  science  of 
space  which  is  of  quite  recent  origin,  which 
applies  to  other  objects  besides  space,  and 
is  called  the  analysis  of  position.  I  shall 
endeavour  to  show  that  the  case  of  these  state- 
ments is  entirely  different  from  that  of  the  state- 
ments about  space  which  I  examined  in  my  last 
lecture.  There  were  four  of  those  statements  : 
that  the  space  of  three  dimensions  which  we 
perceive  is  a  continuous  aggregate  of  points, 
that  it  is  flat  in  its  smallest  parts,  that  figures 
may  be  moved  in  it  without  alteration  of  size 
or  shape,  and  that  similar  figures  of  different 
sizes  may  be  constructed  in  it.  And  the 
conclusion  which  I  endeavoured  to  establish 
about  these  statements  was  that,  for  all  we 
know,  any  or  all  of  them  may  be  false.  In 
regard  to  the  statements  we  have  now  to 


PHILOSOPHY  OF  THE  PURE  SCIENCES       389 

examine,  I  shall  not  maintain  a  similar  doctrine ; 
I  shall  only  maintain  that,  for  all  we  know, 
there  may  be  times  and  places  where  they  are 
unmeaning  and  inapplicable.  If  I  am  asked 
what  two  and  two  make  I  shall  not  reply  that 
it  depends  upon  circumstances,  and  that  they 
make  sometimes  three  and  sometimes  five ;  but 
I  shall  endeavour  to  show  that  unless  our 
experience  had  certain  definite  characters  there 
would  be  no  such  conception  as  two,  or  three, 
or  four,  and  still  less  such  a  conception  as  the 
adding  together  of  two  numbers  ;  and  that  we 
have  no  warrant  for  the  absolute  universality  of 
these  definite  characters  of  experience. 

In  the  first  place  it  is  clear  that  the  moment 
we  use  language  at  all,  we  may  make  state- 
ments which  are  apparently  universal,  but  which 
really  only  assign  the  meaning  of  words. 
Whenever  we  have  called  a  thing  by  two 
names,  so  that  every  individual  of  a  certain 
class  bears  the  name  A  and  also  the  name  B, 
then  we  may  affirm  the  apparently  universal 
proposition  that  every  A  is  B.  But  it  is  really 
only  the  particular  proposition  that  the  name 
A  has  been  conventionally  settled  to  have  the 
same  meaning  as  the  name  B.  I  may,  for 
example,  enunciate  the  proposition  that  all 
depth  is  profundity,  and  all  profundity  is 
depth.  This  statement  appears  to  be  of 
universal  generality ;  and  nobody  doubts  that 


390  LECTURES  AND  ESSAYS 

itjis  true.  But  for  all  that  it  is  not  a  statement 
of  some  fact  which  is  true  of  nature  as  a  whole  ; 
it  is  only  a  statement  about  the  use  of  certain 
words  in  the  English  language.  In  this  case 
the  meaning  of  the  two  words  is  co-extensive  ; 
one  means  exactly  as  much  as,  and  no  more 
than,  the  other.  But  if  we  suppose  the  word 
crow  to  mean  a  black  bird  having  certain 
peculiarities  of  structure,  the  statement,  "All 
crows  are  black,"  is  in  a  similar  case.  For  the 
word  black  has  part  of  the  meaning  of  the  word 
crow ;  and  the  proposition  only  states  this 
connection  between  the  two  words.  Are  the 
propositions  of  arithmetic,  then,  mere  statements 
about  the  meanings  of  words  ?  No  ;  but  these 
examples  will  help  us  to  understand  them. 
Language  is  part  of  the  apparatus  of  thought  ; 
it  is  that  by  which  I  am  able  to  talk  to  myself. 
But  it  is  not  all  of  the  apparatus  of  thought ; 
and  just  as  these  apparently  general  pro- 
positions, "All  crows  are  black,"  "All  depth 
is  profundity,"  are  really  statements  about 
language,  so  I  shall  endeavour  to  show  that 
the  statements  of  arithmetic  are  really  state- 
ments about  certain  other  apparatus  of  thought. 
We  know  that  six  and  three  are  nine. 
Wherever  we  find  six  things,  if  we  put  three 
things  to  them  there  are  nine  things  altogether. 
The  terms  are  so  simple  and  so  familiar  that  it 
seems  as  if  there  were  no  more  to  be  said,  as  if 


PHILOSOPHY  OF  THE  PURE  SCIENCES      391 

we  could  not  examine  into  the  nature  of  these 
statements  any  further. 

No  more  there  is,  if  we  are  obliged  to  take 
words  as  they  stand,  with  the  complex  mean- 
ings which  at  present  belong  to  them.  But  the 
real  fact  is  that  the  meanings  of  six  and  three 
are  already  complex  meanings,  and  are  capable 
of  being  resolved  into  their  elements.  This 
resolution  is  due  —  I  believe  equally  and  in- 
dependently —  to  two  great  living  mathema- 
ticians, by  whose  other  achievements  this 
country  has  retained  the  scientific  position 
which  Newton  won  for  her  at  a  time  of  fierce 
competition  when  no  ordinary  genius  could 
possibly  have  attained  it.  The  conception  of 
number,  as  represented  by  that  word  and  also 
by  the  particular  signs,  three,  six,  and  so  on, 
has  been  shown  to  embody  in  itself  a  certain 
proposition,  upon  the  repetition  of  which  the 
whole  science  of  arithmetic  is  based.  By  means 
of  this  remark  of  Cayley  and  Sylvester,  we  are 
able  to  assign  the  true  nature  of  arithmetical 
propositions,  and  to  pass  from  thence  by  an 
obvious  analogy  to  those  other  cases  that  we 
have  to  consider. 

What  do  I  do  to  find  out  that  a  certain  set 
of  things  are  six  in  number  ?  I  count  them  ; 
and  all  counting,  like  the  names  of  numbers, 
belongs  first  to  the  fingers.  Now  this  is  the 
operation  of  counting ;  I  take  my  fingers  in  a 


392  LECTURES  AND  ESSAYS 

certain  definite  order — say  I  begin  with  the 
thumb  of  each  hand,  and  with  the  right  hand. 
Then  I  lay  my  fingers  in  this  order  upon  the 
things  to  be  counted  ;  or  if  they  are  too  far 
away,  I  imagine  that  I  lay  them.  And  I 
observe  what  finger  it  is  that  is  laid  upon  the 
last  thing,  and  call  the  things  by  the  name  of 
this  finger.  In  the  present  case  it  is  the  thumb 
of  my  left  hand  ;  and  if  we  were  savages  that 
thumb  would  be  called  six.  At  any  rate,  if 
the  order  of  my  fingers  is  settled  beforehand, 
and  known  to  everybody,  I  can  quite  easily 
make  the  statement,  "  Here  are  six  things,"  by 
holding  up  the  thumb  of  my  left  hand. 

But,  if  I  have  only  gone  through  this  process 
once,  there  is  already  a  great  assumption  made. 
For,  although  the  order  in  which  I  used  my 
fingers  is  fixed,  there  is  nothing  at  all  said 
about  the  order  in  which  the  things  are  touched 
by  them.  It  is  assumed  that  if  the  things  are 
taken  in  any  other  order  and  applied  to  my 
fingers,  the  last  one  so  touched  will  be  the 
thumb  of  my  left  hand.  If  this  were  not  true, 
or  were  not  assumed,  the  word  "  number  "  could 
not  have  its  meaning.  There  is  implied  and 
bound  up  in  that  word  the  assumption  that  a 
group  of  things  comes  ultimately  to  the  same 
finger  in  whatever  order  they  are  counted. 
This  is  the  proposition  of  which  I  spoke  as 
the  foundation  of  the  whole  science  of  number. 


PHILOSOPHY  OF  THE  PURE  SCIENCES       393 

It -.is  involved  not  only  in  the  general  term 
"  number,"  but  also  in  all  the  particular  names 
of  numbers ;  and  not  only  in  these  words,  but 
in  the  sign  of  holding  up  a  finger  to  indicate 
how  many  things  there  are. 

Let  us  now  look  in  this  light  at  the  state- 
ment that  six  and  three  are  nine.  I  have 
counted  a  group  of  things  and  come  to  the 
conclusion  that  there  are  six  of  them.  I  have 
already  said,  therefore,  that  they  may  be  counted 
in  any  order  whatever  and  will  come  to  the 
same  number,  six.  I  have  counted  another 
distinct  group,  and  come  to  the  conclusion  that 
there  are  three  of  them.  Then  I  put  them 
all  together  and  count  them.  Now,  without 
seeing  or  knowing  any  more  of  the  things  than 
is  implied  in  the  previous  statements,  I  can 
already  count  them  in  a  certain  order  with  my 
fingers.  For  I  will  first  suppose  the  six  to  be 
counted  ;  the  last  of  them,  by  hypothesis,  is 
attached  in  thought  to  the  thumb  of  my  left 
hand.  Now  I  will  count  the  other  three  ;  they 
are  then  attached,  by  hypothesis,  to  the  first 
three  fingers  of  my  right  hand.  I  can  now  go 
on  counting  the  aggregate  group  by  attaching 
to  these  three  fingers  the  successive  fingers  of 
my  left  hand  ;  for  thus  I  shall  attach  the 
remaining  three  things  to  those  fingers.  I  find 
in  this  way  that  the  last  of  them  comes  to  the 
fourth  finger  of  my  left  hand,  counting  the 


394  LECTURES  AND  ESSAYS 

thumb  as  first ;  and  I  know,  therefore,  that  if 
the  aggregate  group  has  any  number  at  all,  that 
number  must  be  nine. 

But  this  is  an  operation  performed  on  my 
fingers ;  and  the  statement  that  we  have 
founded  on  it  must  therefore  be,  at  least  in 
part,  a  statement  about  my  counting  apparatus. 
We  may  easily  understand  what  is  meant  by 
saying  that  six  and  three  are  nine  on  my  fingers, 
independently  of  any  other  things  than  those  ; 
this  is  a  particular  statement  only.  The  state- 
ment we  want  to  examine  is  that  this  is  equally 
true  of  any  two  distinct  groups  whatever  of  six 
things  and  three  things,  which  appears  to  be  a 
universal  statement.  Now  I  say  that  this 
latter  statement  can  be  resolved  into  two  as 
follows  : — 

1.  The  particular  statement  aforesaid  :  six 
and  three  are  nine  on  my  fingers. 

2.  If  there  is  a  group  of  things  which  can 
be  attached   to   certain  of  my   fingers,  one  to 
each,  and   another  group  of  things  which  can 
be  attached  to  certain  other  of  my  fingers,  one 
to   each,   then    the    compound    group    can    be 
attached  to  the  whole  set  of  my  fingers  that 
have  been  used,  one  to  each. 

Now  this  latter,  it  seems  to  me,  is  a  tautology 
or  identical  proposition,  depending  merely  upon 
the  properties  of  language.  The  arithmetical 
proposition,  then,  is  resolved  or  analysed  in 


PHILOSOPHY  OF  THE  PURE  SCIENCES       395 

this  way  into  two  parts — a  particular  statement 
about  my  counting  apparatus,  and  a  particular 
statement  about  language  ;  and  it  is  not  really 
general  at  all.  But  this,  it  is  important  to 
notice,  is  not  the  complete  solution  of  the 
problem  ;  there  is  a  certain  part  of  it  reserved. 
For  I  only  arrive  at  the  number  nine  by  certain 
definite  ways  of  counting  ;  I  must  count  the 
six  things  first  and  then  the  three  things  after 
them.  And  I  only  arrive  at  the  result  that  if 
the  aggregate  group  of  things  has  any  number 
at  all,  that  number  is  nine.  It  is  not  yet 
proved  that  they  may  be  counted  in  any  order 
whatever,  and  will  always  come  to  that  number. 
Here,  then,  we  are  driven  back  to  consider  the 
nature  of  that  fundamental  assumption  that  the 
number  of  any  finite  group  of  distinct  things  is 
independent  of  the  order  of  counting.  Here  is 
a  proposition  apparently  still  more  general  than 
any  statement  about  the  sum  of  two  numbers. 
Do  I  or  do  I  not  know  that  this  is  true  of  very 
large  numbers  ?  Consider,  for  example,  the 
molecules  of  water  in  this  glass.  According 
to  Sir  William  Thomson,  if  a  drop  of  water 
were  magnified  to  the  size  of  the  earth  it  would 
appear  coarser-grained  than  a  heap  of  small 
shot,  and  finer-grained  than  a  heap  of  cricket- 
balls.  We  may  therefore  soon  find  that  the 
number  of  molecules  in  this  glass  very  far 
transcends  our  powers  of  conception.  Do  I 


396  LECTURES  AND  ESSAYS 

know  that  if  these  molecules  were  counted  in 
a  certain  order,  and  then  counted  over  again  in 
a  certain  other  order,  the  results  of  these  two 
countings  would  be  the  same  ?  For  the  opera- 
tions are  absolutely  impossible  in  anybody's 
lifetime.  Can  I  know  anything  about  the 
equivalence  of  two  impossible  operations,  neither 
of  which  can  be  conceived  except  in  a  symbolic 
way  ?  And  if  I  do,  how  is  it  possible  for  this 
knowledge  to  come  from  experience  ? 

I  reply  that  I  do  know  it ;  that  such  know- 
ledge of  things  as  there  is  in  it  has  come  from 
experience  ;  and  that,  in  fact,  it  is  made  up  of 
a  particular  statement  and  a  conventional  use  of 
words.  These  views  will  appear  paradoxical  ; 
but  the  justification  of  them  is  to  be  found  in 
the  analysis  of  that  fundamental  assumption 
which  lies  at  the  basis  of  the  idea  of  number. 

In  the  first  place  I  shall  prove  this  funda- 
mental assumption  in  the  case  of  the  number  six 
— that  is  to  say,  I  shall  show  that  it  is  involved 
in  suppositions  which  are  already  made  before 
there  is  any  question  of  it.  The  proposition 
we  have  to  prove  is  :  if  a  group  of  distinct 
things  comes  to  six  when  counted  in  a  certain 
order,  it  will  come  to  six  when  counted  in  any 
other  order.  I  say  that  the  proposition  is 
involved  in  the  meaning  of  the  phrase  distinct 
things,  and  may  be  got  out  of  it  by  help  of  a 
particular  observation. 


PHILOSOPHY  OF  THE  PURE  SCIENCES       397 

What,  then,  is  meant  by  "  a  group  of  distinct 
things  "  ?  That  they  are  all  distinct  from  one 
another,  or  that  any  one  and  any  other  of  them 
make  two.  That  is,  if  they  are  attached  to 
two  of  my  fingers  in  a  certain  order,  they  can 
also  be  attached  to  the  same  two  fingers  in  the 
other  order.  Now,  for  simplicity,  let  us  take 
the  letters  in  the  word  spring,  and  count  them 
first  as  they  occur  in  that  word  and  then  in 
the  alphabetical  order.  I  say  that,  merely  on 
the  supposition  that  they  are  distinct  from  one 
another,  I  can  change  one  order  into  the  other 
while  I  use  the  same  fingers  to  attach  them  to. 

123456 
SPRING 
G  P  R  I  N  S 
G  I  R  P  N  S 
G  I  N  P  R  S 

In  the  new  order  I  want  G  to  be  first ;  now 
the  letters  G  and  S  are  by  hypothesis  distinct, 
they  are  two  letters.  I  can  therefore  inter- 
change the  fingers  to  which  they  are  attached 
without  using  more  or  fewer  fingers  than  before. 
The  same  thing  is  true  by  hypothesis  of  I  and 
P,  and  finally  of  N  and  R.  By  these  steps, 
then,  I  have  changed  one  order  into  the  other 
without  altering  the  fingers  used  in  counting — 
that  is,  without  altering  the  number.  And 
each  of  these  steps  is  involved  in  the  meaning 


398  LECTURES  AND  ESSAYS 

of  the  words  distinct  things — that  is,  it  is  made 
possible  by  the  assumptions  which  these  words 
involve.  But  now  observe  further  :  how  do  I 
know  that  I  can  make  enough  steps  to  effect 
the  whole  change  required  ?  In  this  way.  It 
is  given  to  me  in  the  hypothesis  that  the  things 
have  been  counted  once  ;  I  can  therefore  go  to 
them  one  by  one  till  I  come  to  the  end.  But 
as  I  go  to  each  one  I  can  substitute  by  this 
process  the  new  one  which  is  wanted  in  its 
stead  in  such  a  way  that  the  required  new 
order  shall  hold  good  behind  me.  Thus  you 
see  that  all  the  steps  are  involved  in  the  word 
distinct,  by  the  help  of  an  observation  on  two 
of  my  fingers  ;  and  that  the  possibility  of  a 
sufficient  number  of  them  to  effect  the  change 
is  involved  in  the  hypothesis  that  the  things 
have  been  once  counted.  Here  I  have  two 
distinct  statements  :  the  first  is  that  the  things 
are  distinct,  and  have  been  once  counted  as 
six  ;  the  second  is  that  in  another  order  they 
come  to  the  same.  When  I  examine  into  the 
meaning  of  these,  I  find  that  they  are  not 
statements  of  different  facts,  but  different  state- 
ments of  the  same  facts.  That  one  statement 
is  true,  or  that  the  other  statement  is  true, — 
that  is  a  matter  of  experience  ;  but  that  if  one 
is  true  the  other  is  true,  that  is  a  matter  of 
language. 

I  have  only  spoken,  however,  of  the  particular 


PHILOSOPHY  OF  THE  PURE  SCIENCES       399 

number  six  ;  how  am  I  to  extend  these  remarks 
to  numbers  which  cannot  be  counted,  like  the 
number  of  molecules  in  this  glass  of  water  ? 
In  the  first  place  we  all  know  that  cultivated 
races  do  not  count  directly  with  their  fingers, 
but  with  the  names  of  them — with  the  words 
one,  two,  three,  four.  Next,  this  system  of 
names  has  been  extended  indefinitely,  by  a 
process  to  which  no  end  can  be  conceived. 
But  the  remarks  that  we  have  made  about 
finger-counting  will  hold  good  in  every  case  in 
which  the  actual  counting  can  be  performed. 
Now  in  those  cases  in  which  this  is  not  true — 
in  the  case  of  a  billion,  for  example — we  have 
two  statements  made,  neither  of  which  can  be 
adequately  represented  in  thought,  but  which, 
in  so  far  as  they  can  be  represented,  are  identi- 
cal statements.  That  there  are  a  billion  grains 
of  sand  in  a  certain  heap,  provided  they  be 
counted  in  a  certain  order — this  is  a  supposi- 
tion which  can  only  be  made  symbolically. 
But  in  so  far  as  it  can  be  made,  it  is  the  same 
supposition  as  that  they  also  come  to  a  billion 
in  any  other  order.  Any  step  towards  the 
representation  in  thought  of  the  one  statement 
is  the  same  step  towards  the  representation  in 
thought  of  the  other  ;  and  I  do  not  know  any 
other  way  in  which  two  symbolic  statements 
can  be  statements  of  the  same  facts.  Pure 
water  is  the  same  thing  as  aqua  pura ;  and 


400  LECTURES  AND  ESSAYS 

wherever  there  are  seventy  thousand  million 
tons  of  pure  water  there  are  seventy  thousand 
million  tons  of  aqua  pura.  I  know  that  to  be 
true,  but  it  is  not  a  statement  of  fact :  it  is  a 
statement  about  language,  notwithstanding  that 
the  language  is  used  to  symbolise  that  which 
cannot  be  actually  represented  in  thought.  So 
when  I  say  of  these  molecules  of  water,  "  If 
they  are  distinct  things,  the  number  of  them 
counted  in  one  order  is  equal  to  the  number  of 
them  counted  in  any  other  order,"  I  make  a 
supposition  which  I  cannot  realise  in  thought. 
I  cannot  possibly  call  up  those  molecules  two 
and  two  to  observe  their  distinctness.  The 
supposition  is  only  represented  symbolically  by 
language  ;  but  the  statement  that  follows  it  is 
the  same  supposition  represented  symbolically 
by  other  language  ;  and  the  equivalence  of  the 
two  is,  after  all,  a  statement  about  language 
and  not  about  facts. 

But  you  will  say,  I  do  know  that  these 
molecules  are  distinct  things;  and  so  I  am 
able  to  make  these  equivalent  statements  about 
them.  I  know  that  they  have  a  definite  number, 
which  is  the  same  however  they  are  counted. 

I.  Yes,  I  know  that  they  are  distinct  things  ; 
but  only  by  inference,  on  the  assumption  of  the 
uniformity  of  nature  ;  and  about  that  there  is 
more  to  be  said.  The  distinctness  of  things—- 
the fact  that  one  thing  and  one  thing  make  two 


PHILOSOPHY  OF  THE  PURE  SCIENCES      401 

— this  belongs  to  our  experience.  It  is  a  fact 
that  impressions  hang  together  in  groups  which 
persist  as  groups,  and  in  virtue  of  this  persistence 
we  call  them  things.  So  long  as  our  experience 
consists  of  things,  we  may  build  out  of  it  the 
conceptions  of  number  ;  and  the  nature  and 
connection  of  these  conceptions  are  determined 
by  the  primary  sensation  of  things  as  individuals. 
Now  there  can,  I  think,  be  no  doubt  that  the 
experience  of  a  hundred  or  a  hundred  and  fifty 
million  years  has  so  modified  our  nervous 
systems  that  without  total  disruption  of  them 
we  cannot  cease  to  aggregate  our  perceptions 
into  more  or  less  persistent  groups  ;  the  con- 
tinuity of  things  has  become  a  form  of  sense. 
If  we  were  placed  in  circumstances  where  these 
aggregations  of  feeling  were  not  naturally  pro- 
duced, where  perceptible  things  were  not  con- 
tinuous in  their  changes,  we  should  go  on 
perceiving  chaos  as  made  of  individual  things 
for  at  least  some  time.  But  the  perception 
would  be  a  false  one,  and  in  acting  upon  it  we 
should  come  to  grief.  Meanwhile,  however, 
the  science  of  number  would  be  perfectly  true 
of  our  perceptions,  though  practically  inappli- 
cable to  the  world. 

To  sum  up,  then,  we  carry  about  with  us 

a   certain    apparatus   of   counting,   which    was 

primarily  our  fingers,  but  is  now  extended  into 

a  series  of  signs  which  we  can  remember  in  a 

VOL.  I  2  D 


402  LECTURES  AND  ESSAYS 

certain  order — the  names  of  numbers.  Our 
language  is  so  formed  as  to  make  us  able  to 
talk  to  ourselves  about  the  results  of  counting. 
The  propositions  of  arithmetic  are  compounded 
in  general  of  two  parts ;  a  statement  about  the 
counting  apparatus,  and  a  statement  about  the 
different  ways  of  describing  its  results. 

But  before  quite  leaving  this  let  us  fix  our 
attention  for  a  short  time  on  the  mode  of  use 
of  the  counting  apparatus.  The  operation  of 
counting  a  certain  group  of  things  consists  in 
assigning  one  of  these  numeral  words  to  each 
of  them  ;  in  establishing  a  correspondence 
between  two  groups,  so  that  to  every  thing  or 
element  of  the  one  group  is  assigned  one 
particular  thing  or  element  of  the  other.  There 
is  here  a  one-to-one  correspondence  of  two 
aggregates,  one  of  which  is  carried  about  as  a 
standard  ;  and  the  propositions  arrived  at  are 
always  of  this  kind  : — if  a  group  of  things  can 
have  this  correspondence  with  the  standard 
group,  then  those  properties  of  the  standard 
group  which  are  carried  over  by  the  corre- 
spondence will  belong  to  the  new  group.  Now 
this  establishment  of  correspondence  between 
two  aggregates  and  investigation  of  the  pro- 
perties that  are  carried  over  by  the  correspond- 
ence may  be  called  the  central  idea  of  modern 
mathematics  ;  it  runs  through  the  whole  of  the 
pure  science  and  of  its  applications.  It  may 


PHILOSOPHY  OF  THE  PURE  SCIENCES      403 

be  conceived,  therefore,  that  propositions  which 
are  apparently  as  general  and  certain  as  those 
we  have  discussed  to-day  may  be  analysed  in 
the  same  manner,  and  shown  to  be  really  state- 
ments about  the  apparatus  of  thought 

In  my  second  lecture  I  endeavoured  to 
explain  the  difference  between  a  discrete  and  a 
continuous  aggregate.  In  a  row  of  marbles, 
which  is  a  discrete  aggregate,  we  can  find 
between  any  two  marbles  only  a  finite  number 
of  others,  and  sometimes  none  at  all.  But  if 
two  points  are  taken  on  a  line,  the  hypothesis 
of  continuity  supposes  that  there  is  no  end  to 
the  number  of  intermediate  points  that  we  can 
find.  Precisely  the  same  difference  holds  good 
between  number  and  continuous  quantity.  The 
several  marbles,  beginning  at  any  one  of  them, 
may  be  numbered  one,  two,  three,  etc. ;  and  the 
number  attached  to  each  marble  will  be  the 
number  of  marbles  from  the  starting-point  to 
that  marble  inclusive.  If  the  points  on  a  line 
are  regarded  as  forming  a  continuous  aggregate, 
then  lengths  measured  along  the  line  from  an 
arbitrary  point  on  it  are  called  continuous  quanti- 
ties. So  also,  if  the  instants  of  time  are  re- 
garded as  forming  a  continuous  aggregate — 
that  is,  if  we  suppose  that  between  any  two 
instants  there  is  no  end  to  the  number  of  inter- 
mediate ones  that  might  be  found  —  then 
intervals  or  lengths  of  time  will  be  continuous 


404  LECTURES  AND  ESSAYS 

quantities.  And  just  as  we  may  attach  our 
numbers  one  by  one  to  the  marbles  which  form 
a  discrete  aggregate,  so  we  may  attach  continu- 
ous quantities  (or  shortly  quantities}  one  by 
one  to  the  points  which  form  a  continuous 
aggregate.  Thus  to  the  point  P  will  be  attached 
the  quantity  or  length  A  P.  And  we  see  thus  that 

A"  j,  B 

between  any  two  quantities  there  may  be  found 
an  infinite  number  of  intermediate  quantities, 
while  between  two  numbers  there  can  only  be 
found  a  finite  number  of  intermediate  numbers, 
and  sometimes  none  at  all.  That  is  to  say, 
continuous  quantities  form  a  continuous  aggre- 
gate, while  numbers  form  a  discrete  aggregate. 
Thus  the  science  of  quantity  is  a  totally  different 
thing  from  the  science  of  number. 

Notwithstanding  that  this  difference  was 
clearly  perceived  by  the  ancients,  attempts  have 
constantly  been  made  by  the  moderns  to  treat 
the  two  sciences  as  one,  and  to  found  the  science 
of  quantity  upon  the  science  of  number.  The 
method  is  to  treat  rational  fractions  as  a  neces- 
sary extension  of  numerical  division,  and  then 
to  deal  with  incommensurable  quantities  by 
way  of  continual  approximation.  In  the  science 
of  number,  while  five-sevenths  of  fourteen  has  a 
meaning,  namely,  ten,  five-sevenths  of  twelve  is 


PHILOSOPHY  OF  THE  PURE  SCIENCES      405 

nonsense.  Let  us  then  treat  it  as  if  it  were  sense, 
and  see  what  comes  of  it  A  repetition  of  this 
process  with  every  impossible  operation  that 
occurs  is  supposed  to  lead  in  time  to  continuous 
quantities.  The  results  of  such  attempts  are 
the  substitution  of  algebra  for  the  fifth  book  of 
Euclid  or  some  equivalent  doctrine  of  continuous 
ratios,  and  the  substitution  of  the  differential 
calculus  for  the  method  of  fluxions.  For  my 
own  part,  I  believe  this  method  to  be  logically 
false  and  educationally  mischievous.  For 
reasons  too  long  to  give  here,  I  do  not  believe 
that  the  provisional  use  of  unmeaning  arith- 
metical symbols  can  ever  lead  to  the  science  of 
quantity  ;  and  I  feel  sure  that  the  attempt  to 
found  it  on  such  abstractions  obscures  its  true 
physical  nature.  The  science  of  number  is 
founded  on  the  hypothesis  of  the  distinctness 
of  things  ;  the  science  of  quantity  is  founded 
on  the  totally  different  hypothesis  of  continuity. 
Nevertheless,  the  relations  between  the  two 
sciences  are  very  close  and  extensive.  The 
scale  of  numbers  is  used,  as  we  shall  see,  in 
forming  the  mental  apparatus  of  the  scale  of 
quantities,  and  the  fundamental  conception  of 
equality  of  ratios  is  so  defined  that  it  can  be 
reasoned  about  in  the  terms  of  arithmetic.1 

1  Defining  a  fraction  as  the  ratio  of  two  numbers,  Euclid's 
definition  of  proportion  is  equivalent  to  the  following  :  —  Two 
quantity-ratios  are  equal  if  every  fraction  is  either  le$s  than  both, 
equal  to  both,  or  greater  than  both  of  them. 


406  LECTURES  AND  ESSAYS 

The  operations  of  addition  and  subtraction  of 
quantities  are  closely  analogous  to  the  operations 
of  the  same  name  performed  on  numbers  and 
follow  the  same  laws.  The  composition  of 
ratios  includes  numerical  multiplication  as  a 
particular  case,  and  combines  in  the  same  way 
with  addition  and  subtraction.  So  close  and 
far-reaching  is  this  analogy  that  the  processes 
and  results  of  the  two  sciences  are  expressed  in 
the  same  language,  verbal  and  symbolical,  while 
no  confusion  is  produced  by  this  ambiguity  of 
meaning,  except  in  the  minds  of  those  who  try 
to  make  familiarity  with  language  do  duty  for 
knowledge  of  things. 

Just  as  in  operations  of  counting  there  is  a 
comparison  of  some  aggregate  of  discrete  things 
with  a  scale  of  numbers  carried  about  with  us 
as  a  standard,  so  in  operations  of  measuring, 
real  or  ideal,  there  is  comparison  of  some  piece 
of  a  continuous  thing  with  a  scale  of  quantities. 
We  may  best  understand  this  scale  by  the 
example  of  time.  To  indicate  exactly  the  time 
elapsed  from  the  beginning  of  the  century  to 
some  particular  instant  of  to-day,  it  is  necessary 
and  sufficient  to  name  the  date  and  point  to 
the  hands  of  a  clock  which  was  going  right  and 
was  stopped  at  that  instant.  This  is  equivalent 
to  saying  that  the  whole  quantity  of  time  con- 
sists, first,  of  a  certain  number  of  hours,  specified 
by  comparison  with  the  scale  of  numbers 


PHILOSOPHY  OF  THE  PURE  SCIENCES       407 

already  constructed,  and,  secondly,  of  a  certain 
part  of  an  hour,  which  being  a  continuous 
quantity  can  only  be  adequately  specified  by 
another  continuous  quantity  representing  it  on 
some  definite  scale.  In  the  present  case  this  is 
conveniently  taken  to  be  the  arc  of  a  circle 
described  by  the  point  of  the  minute-hand. 
On  the  scale  in  which  that  whole  circumference 
represents  an  hour,  this  arc  represents  the 
portion  of  an  hour  which  remains  to  be  added. 
With  the  help  of  the  scale  of  numbers,  then, 
any  assigned  continuous  quantity  will  serve  as  a 
standard  by  which  the  whole  scale  of  quantities 
may  be  represented.  And  when  we  assert  that 
any  theorem,  e.g.  the  binomial  theorem,  is  true 
of  all  quantities  whatever,  whether  of  length,  of 
time,  of  weight,  or  of  intensity,  we  really  assert 
two  things  :  first,  this  theorem  is  true  on  the 
standard ;  secondly,  relations  of  the  measures 
of  quantities  on  the  standard  are  relations  of 
the  quantities  themselves.  The  first  is  (in 
regard  to  the  kind  of  quantity)  a  particular 
statement ;  the  second  is  involved  in  the  mean- 
ing of  the  words  "  quantity "  and  "  measure- 
ment." 

But  the  most  familiar  and  perhaps  the  most 
natural  form  of  the  scale  of  quantities  is  that  in 
which  it  is  supposed  to  be  marked  off  on  a 
straight  line,  starting  from  an  arbitrarily 
assumed  point  which  is  called  the  origin.  If 


4o8  LECTURES  AND  ESSAYS 

we  make  the  four  assumptions  of  Euclidean  or 
parabolic  geometry,  the  position  of  every  point 
in  space  may  be  specified  by  three  quantities 
marked  off  on  three  straight  lines  at  right 
angles  to  each  other,  their  common  point  of  in- 
tersection being  taken  as  origin,  and  the  direc- 
tion in  which  each  of  the  quantities  is  measured 
being  also  assigned.  Namely,  these  three 
quantities  are  the  distances  from  the  origin  to 
the  feet  of  perpendiculars  let  fall  from  the  point 
to  be  specified  on  the  three  straight  lines  re- 
spectively. In  all  space  of  three  dimensions  the 
position  of  a  point  may  be  specified  in  general  by 
a  set  of  three  quantities  ;  but  two  or  more  points 
may  belong  to  the  same  set  of  quantities,  or  two 
or  more  sets  may  specify  the  same  point ;  and 
there  may  be  exceptional  sets  specifying  not 
one  point,  but  all  the  points  on  a  curve  or  sur- 
face, and  exceptional  points  belonging  to  an 
infinite  number  of  sets  of  quantities  subject  to 
some  condition.  There  are  three  kinds  of 
space  of  three  dimensions  in  which  this  specifi- 
cation is  unique,  one  point  for  one  set  of 
quantities,  one  set  of  quantities  for  every  point, 
and  without  any  exceptional  cases.  These  three 
are  the  hypothetical  space  of  Euclid,  with  no 
curvature  ;  the  space  of  Lobatchewsky,  with 
constant  negative  curvature ;  and  the  space  I 
described  at  the  end  of  my  second  lecture, 
with  constant  positive  curvature.  In  only  one 


PHILOSOPHY  OF  THE  PURE  SCIENCES      409 

of  these,  the  space  of  Euclid,  are  the  three 
quantities  specifying  a  point  actual  distances  of 
the  point  from  three  planes.  In  this  alone 
we  have  a  simple  and  direct  representation  of 
the  scale  of  quantities.  Now,  if  we  remember 
that  the  scale  of  quantities  is  a  mental  appa- 
ratus depending  only  on  the  first  of  our  four 
assumptions  about  space,  we  may  see  in  this 
distinctive  property  of  Euclidean  space  a  prob- 
able origin  for  the  curious  opinion  that  it  has 
some  a  priori  probability  or  even  certainty,  as 
the  true  character  of  the  universe  we  inhabit, 
over  and  above  the  observation  that  within  the 
limits  of  experience  that  universe  does  approxi- 
mately conform  to  its  rules.  It  has  even  been 
maintained  that  if  our  space  has  curvature,  it 
must  be  contained  in  a  space  of  more  dimen- 
sions and  no  curvature.  I  can  think  of  no 
grounds  for  such  an  opinion  except  the  property 
of  flat  spaces  which  I  have  just  mentioned. 


END  OF  VOL.  I 


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