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LIBRARY 


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THE  MONIST 


A  QUARTERLY  MAGAZINE 


DEVOTED  TO  THE  PHILOSOPHY  OF  SCIENCE 


VOLUME  XXVII 


n  7  (i  1 7 


'/f 


CHICAGO 

THE  OPEN  COURT  PUBLISHING  COMPANY 
1917 


fc 

\ 


Nl.71 


COPYRIGHT  BY 

THE  OPEN  COURT  PUBLISHING  COMPANY 
1916-1917 


CONTENTS  OF  VOLUME  XXVII. 


ARTICLES  AND  AUTHORS. 


PACE 

Bacon,  Ernst  Lecher.     Our  Musical  Idiom   560 

Bergsonism  in  England.    By  J.  W.  Scott 179 

Bolzano,  Bernard  (1781-1848).    By  Dorothy  Maud  Wrinch  83 

Burns,  C.  Delisle.    A  Medieval  Internationalist  (Pierre  Dubois)   105 

Bussey,  Gertrude  Carman.    Mechanism  and  the  Problem  of  Freedom 295 

Cal-Dif-Fluk  Saga  (Poem).    By  J.  M.  Child 467 

Carus,  Paul.  Belief  in  God  and  Immortality,  311;  A  Chinese  Poet's  Con- 
templation of  Life  (translated  poem),  128;  Determinism  of  Free 
Will,  306;£Leibniz  and  Locke  J137;  Nirvana  (poem),  233;  Sir 
Oliver  Lodge  on  Life  After  Death,  316. 

Chatley,  Herbert.    Idealism  as  a  Force :  A  Mechanical  Analogy 151 

Child,  J.  M.    Cal-Dif-Fluk  Saga  (Poem),  467;  The  Manuscripts  of  Leib- 
niz on  his  Discovery  of  the  Differential  Calculus,  Part  II,  238,  411. 
Chinese  Poet's  Contemplation  of  Life,  A  (Su  Tung  P'o).  Tr.  by  Paul  Carus  128 

Class,  Function,  Concept,  Relation.     By  Gottlob  Frege  114 

Confucianism,  Classical.    By  Suh  Hu  157 

De  Vries,  Hugo.    The  Origin  of  the  Mutation  Theory  403 

Determinism  of  Free  Will.    By  Paul  Carus  306 

Dogma?  What  is  a.     By  Edouard  Le  Roy  481 

Dubois,  Pierre,  a  Medieval  Internationalist.    By  C.  Delisle  Burns  105 

Edmunds,  Albert  J.  The  Text  of  the  Resurrection  in  Mark  and  its  Testi- 
mony to  the  Apparitional  Theory 161 

Electronic  Theory  of  Matter,  The.    By  William  Benjamin  Smith 321 

Existents  and  Entities.    By  Philip  E.  B.  Jourdain  142 

Feingold,  Gustave  A.    The  Present  Status  of  the  Unconscious 205 

Free  Will,  Determinism  of.    By  Paul  Carus 306 

Freedom,  Mechanism  and  the  Problem  of.    By  Gertrude  Carman  Bussey. .  295 

Frege,  Gottlob.    Class,  Function,  Concept,  Relation  114 

Gerhardt,  Karl  Immanuel. ^Leibniz  in  London^ 524 

Grassmann,  Hermann  (1809'-1877).    By  A.  E.  Heath  1 

Grassmann,  The  Geometrical  Analysis  of,  and  its  Connection  with  Leibniz's 

Characteristic.    By  A.  E.  Heath  36 

Grassmann,  The  Neglect  of  the  Work  of.    By  A.  E.  Heath  22 


iv  THK    MONIST. 

PAGE 

k  l.lca*  <>i  .4n  Aiu-i  \\..rld.     By  Orland  O.  Norris  57 

I  lermann  Grassmann,  1 ;  The  Neglect  of  the  Work  of  Grass- 
niann.  22;  The  Geometrical  Analysis  of  Grassmann  and  its  Con- 
nection with  Leibniz's  Characteristic,  36. 

1  lu,  Suh.    Classical  Confucianism 157 

Idealism  as  a  Force:  A  Mechanical  Analogy.    By  Herbert  Chatley 151 

Immortality,  Primitive  and  Modern  Conceptions  of.    By  J.  II.  Leuba  608 

Internationalist,  A  Medieval.    By  C  Delisle  Burns  105 

Jourdain,  Philip  E.  B.    Existents  and  Entities,  142;  Logic  and  Psychology,  460 
Koopman,  Harry  Lyman.    Libra:  The  Eternal  Balance  of  Good  and  111 

(Poem)   455 

rLeibniz  and  LockcJ  By  Paul  Carus  137 

VLeibniz's  Characteristic  The  Geometrical  Analysis  of  Grassmann  and  its 

Connection  with.    By  A.  E.  Heath  36 

/  Leipniz  in  London.    By  Karl  Immanuel  Gerhardt  524 

\Leibniz,  The  Manuscripts  of,  on  his  Discovery  of  the  Differential  Cal- 
culus, Part  II.    By  J.  M.  Child  238,  411 

Le  Roy,  Edouard.    What  is  a  Dogma  ?  481 

Leuba,  J.  H.    The  Primitive  and  the  Modern  Conceptions  of  Personal  Im- 

.      mortality 608 

Libra:  The  Eternal  Balance  of  Good  and  111  (Poem).    By  Harry  Lyman 

Koopman 455 

Liu,  King  Shu.    The  Origin  of  Taoism  376 

Locke,  Leibniz  and.    By  Paul  Carus 137 

Logic  and  Psychology.    By  Philip  E.  B.  Jourdain 460 

Mark,  The  Text  of  the  Resurrection  in,  and  Its  Testimony  to  the  Appa- 

ritional  Theory.     By  Albert  J.  Edmunds   161 

Mechanism  and  the  Problem  of  Freedom.    By  Gertrude  Carman  Bussey. .  295 
Medical  Science  and  Practice,  The  Contributions  of  Paracelsus  to.     By 

J.  M.  Stillman 390 

Musical  Idiom,  Our.     By  Ernst  Lecher  Bacon   560 

Mutation  Theory,  The  Origin  of.    By  Hugo  De  Vries 403 

Nirvana  ( Poem) .    By  Paul  Carus  233 

Norris,  Orland  O.    Greek  Ideas  of  an  Afterworld  57 

Paracelsus,  The  Contributions  of,  to  Medical  Science  and  Practice.     By 

J.  M.  Stillman 390 

Perry,  Ralph  Barton.    Purpose  as  Systematic  Unity  352 

Philosophy  of  Science,  Notes  on  Recent  Work  in  618 

Psychology,  Logic  and.    By  Philip  E.  B.  Jourdain 460 

Purpose  as  Systematic  Unity.    By  Ralph  Barton  Perry  352 

Scarlet  Cliff,  The  (Chinese  Poem).    Tr.  by  Paul  Carus  128 

Scott,  J.  W.    Bergsonism  in  England 179 

Smith,  David  Eugene.    Notes  on  De  Morgan's  Budget  of  Paradoxes 474 

Smith,  William  Benjamin.    The  Electronic  Theory  of  Matter  321 

Stillman,  J.  M.    The  Contributions  of  Paracelsus  to  Medical  Science  and 

Practice  390 

Taoism,  The  Origin  of.    King  Shu  Liu 376 

Unconscious,  The  Present  Status  of  the.    By  Gustave  A.  Feingold  205 

Wrinch,  Dorothy  Maud     Bernard  Bolzano  (1781-1848)    .                        ...  83 


CONTENTS  OF  VOLUME  XXVII. 


BOOK  REVIEWS  AND  NOTES. 

PAGE 

American  Mathematical  Monthly 625,  628,  629 

Bollettino  di  bibliografia  e  storia  delle  sciense  matcinatichc  320 

Bulletin  of  the  American  Mathematical  Society   319,  630,  631 

Cook,  Stanley  A.    The  Study  of  Religions  480 

Dawson,  Miles  Menander.    The  Ethics  of  Confucius  157 

De  Morgan,  A.    Budget  of  Paradoxes  474 

Franceschi,  Pietro.    De  corporibus  regularibus  319 

Jourdain,  Margaret  (Tr.)     Diderot's  Early  Philosophical  Works 639 

Keyser,  Cassius  J.    The  New  Infinite  and  the  Old  Theology 479 

Le  Roy,  Edouard.    The  New  Philosophy  of  Henri  Bergson  640 

Leuba,  James  H..    The  Belief  in  God  and  Immortality 311,  608 

Lodge,  Sir  Oliver.    Raymond,  or  Life  After  Death   316 

Mind 634 

Rendiconti  delta  R.  Accademia  del  Lined  319,  621 

Revue  de  mctaphysique  et  dc  morale  320,  618,  620 

Science 623 

Scientia  624,  637,  638 

Sorel,  Georges.     Reflections  on  Violence   478 

Transactions  of  the  American  Mathematical  Society  319 


VOL.  XXVII.         JANUARY,  1917  NO.  i 

THE  MONIST 


HERMANN  GRASSMANN. 

1809-1877. 

WE  like  to  believe  that  the  final  significance  of  any 
thinker's  work  is  independent  of  his  time  and  place 
and  is  fixed  by  reference  to  some  absolute  standard.  How- 
ever that  may  be,  it  seems  quite  clear  that  his  importance 
in  his  own  age,  and  hence  his  effect  on  the  next  succeeding 
generations,  depends  to  some  extent  on  other  factors  than 
his  intrinsic  value.  And  so  in  judging  that  value  we  must 
distinguish  plainly  between  it  and  what  we  might  call  the 
relative  or  historical  importance  of  the  man's  work.  This 
latter  may  well  be  compared  to  the  potential  of  a  body  in 
electrostatics.  For  just  as  that  potential  depends  not  only 
on  the  actual  charge  on  the  body,  but  also  on  the  charges 
on  neighboring  bodies;  so  also  the  relative  importance  of 
a  man  is  not  determined  alone  by  the  content  of  his  life 
and  work,  but  is  affected  also  by  his  milieu  and  by  the 
reactions  of  that  milieu  to  it. 

This  is  the  reason  why  the  contemporary  estimate  of  a 
thinker  is  often  so  utterly  wrong.  At  the  time,  the  external 
man  and  his  work  are  more  easily  seen;  but  the  subtle 
tendencies  of  the  age  are  not  so  readily  understood,  nor 
can  the  observer  escape  the  distortion  of  vision  wrought 
by  prevailing  influences  on  himself.  So  it  comes  about 
that  he  who  is  written  down  a  failure  in  one  age  may  stand 
out  a  very  genius  in  the  next. 

These  reflections  are  quite  pertinent  to  any  inquiry  into 


2  THE  MONIST. 

the  life  and  work  of  the  author  of  the  Ausdehnungslehre — 
Hermann  Giinther  Grassmann,  the  distinguished  mathe- 
matician whose  own  generation  passed  him  by.  Although 
he  reached  eminence  in  other  branches  of  human  activity, 
we  speak  of  him  as  a  mathematician  because  that  was  cer- 
tainly the  subject  he  loved  most  and  in  which  his  influence 
will  be  most  felt  in  the  future.  Of  him,  on  the  occasion 
of  his  centenary  (1909),  F.  Engel1  could  say:  "To-day  he 
is  known  by  name  to  mathematicians,  but  few  have  read  his 
writings.  Even  where  his  ideas  and  methods  have  been  dif- 
fused in  mathematical  physics  people  learn  them  second- 
hand, sometimes  not  even  under  his  name."  So  in  Grass- 
mann we  have  a  straightforward  example  of  a  man  be- 
tween whose  work  and  whose  influence  on  his  own  and 
immediately  succeeding  generations  we  must  sharply  dis- 
tinguish if  we  are  to  avoid  underrating  his  significance. 

He  was  born  on  April  15,  1809,  m  Stettin.2  His  father, 
Justus  Gunther  Grassmann,  was  a  teacher  in  the  Gym- 
nasium there,  and  was  himself  a  good  mathematician  and 
physicist.8  His  school  days  passed  without  his  showing 
any  inclination  or  aptitude  in  special  studies.  He  had 
however  great  skill  in  and  fondness  for  music,  and  received 
a  good  foundation  in  piano  and  counterpoint  from  the 
famous  composer  Loewe.  The  latter  was  appointed  teacher 
in  the  Stettin  Gymnasium  in  1820  and  lived  for  the  first 
year  in  the  house  of  the  Grassmanns,  where  he  found  very 
congenial  society  in  Hermann  and  his  brothers  and  sisters, 
all  of  whom  were  musical.  With  them  Loewe  often  used 
to  try  over  his  new  quartettes. 

Of  Grassman's  inner  development  during  these  out- 

1  F.  Engel,  Speech  on  "Grassmann  in  Berlin,"  to  the  Berliner  Mathema- 
tische  Gesellschaft   (1909).     To  this  I  owe  most  of  the  information  about 
Grassmann's  early  life  given  in  what  follows. 

2  The  same  date  as  Euler. 

8  He  invented  an  air-pump  cock  which  was  given  his  name,  and  also  con- 
structed a  useful  index  notation  of  crystals. 


HERMANN    GRASSMANN.  3 

wardly  calm  and  uneventful  years  we  can  form  a  clear 
picture  from  his  own  writings.  For  in  1831  he  wrote  an 
account  of  his  life  in  Latin  in  connection  with  the  examina- 
tion for  his  teacher's  certificate;  and  later,  in  1834,  he 
handed  in  an  autobiography  to  the  Konsistorium  in  Stet- 
tin when  he  was  passing  his  first  theological  examina- 
tion. He  refers  to  those  earlier  years  as  a  period  of  slum- 
ber, his  life  being  filled  for  the  most  part  with  idle  reveries 
in  which  he  himself  occupied  the  central  place.  He  says 
that  he  seemed  incapable  of  mental  application,  and  men- 
tions especially  his  weakness  of  memory.  He  relates  that 
his  father  used  to  say  he  would  be  contented  if  his  son 
Hermann  would  be  a  gardener  or  artisan  of  some  kind, 
provided  he  took  up  work  he  was  fitted  for  and  that  he 
pursued  it  with  honor  and  advantage  to  his  fellow  men. 
As  he  usually  spent  his  holidays  in  the  country  among 
relatives,  and  nearly  always  in  the  families  of  clergymen, 
he  conceived  the  desire  to  prepare  himself  for  the  ministry. 
But  he  soon  came,  partly  from  the  ridicule  of  his  compan- 
ions and  partly  from  the  warnings  of  his  parents,  to  doubt 
his  capacity.  He  says  however  that,  after  his  course  of 
instruction  for  confirmation,  a  light  came  into  his  dreams. 
Suddenly  he  determined  to  exercise  all  his  intellectual 
powers  and  to  overcome  as  far  as  possible  the  phlegmatic 
character  of  his  temperament.  And  this  resolution  he 
carried  out  with  resistless  energy. 

F.  Engel4  sums  up  these  early  years  in  the  following 
words :  "He  does  not  belong  to  those  early  ripening  geniuses 
who,  even  in  childhood's  years,  know  whither  their  gifts 
will  lead  them,  and  turn  without  doubt  or  hesitation  to  that 
branch  of  knowledge  to  which  they  are  called.  He  was 
exceptionally  gifted  on  too  many  sides  for  that.  But  even 
these  many-sided  gifts  by  no  means  showed  themselves  at 
the  beginning;  and  that  they  developed  themselves  richly 


4  THE  MONIST. 

later  came  by  no  means  without  effort,  but  was  the  direct 
result  of  many  years  of  concentrated  work  which  he  did 
in  order  to  develop  his  character  and  to  solidify  his  moral 
outlook  and  grasp  of  life." 

In  the  August  of  1827  Grassmann  and  his  elder  brother 
Gustav  entered  the  University  of  Berlin  with  the  intention 
of  studying  theology.  Two  days  after  their  arrival  Her- 
mann wrote  a  droll  letter  to  his  mother  vividly  describing 
how  they  had  settled  in.  He  tells  how  they  had  to  climb 
seventy-two  steps  to  their  attic  dwelling  at  53  Dorotheen- 
strasse  at  the  corner  of  Friedrichstrasse.  They  had  only 
room  for  their  beds  and  two  chairs,  but  he  comments  hu- 
morously on  their  extra  fine  look-out  over  the  gardens  and 
houses  of  the  city,  and  adds  that  though  the  rooms  were 
small  they  could  be  the  more  easily  heated.  His  landlady 
will  be  recognized  by  students  all  the  world  over  in  his  pen 
picture,  "If  she  does  talk  too  much,  she  is  very  pleasant 
and  industrious."  Particularly  amusing  is  the  manner  in 
which  he  tells  how  they  had  spent  practically  all  their 
money  in  two  days.  He  enumerates  all  the  possible  and 
impossible  things  on  which  they  had  not  spent  the  money, 
and  finally  confesses  that  their  sudden  impecuniosity  was 
due  to  the  piano  which  they  luckily  bought  for  50  Taler. 

Grassmann  admitted  later  that  when  he  first  came  to 
the  university  he  was  quite  dependent  on  the  guidance  of 
the  professors.  He  was  easily  impressed  by  the  lectures 
he  heard  and  tended  to  fit  in  his  studies  with  the  lectures  he 
chanced  upon  rather  than  to  take  those  corresponding  to  a 
course  of  study.  At  first  he  came  specially  under  the  in- 
fluence of  the  well-known  church  historian  Neander.  Grad- 
ually, however,  he  became  attracted  more  and  more  to 
Schleiermacher  to  whom  he  acknowledges  great  indebted- 
ness. He  wrote:  "Early  in  my  second  year  I  attended 
Schleiermacher's  lectures,  which  of  course  I  did  not  under- 
stand ;  but  his  sermons  began  to  exercise  an  influence  upon 


HERMANN   GRASSMANN.  5 

me.  However  it  was  not  until  my  third,  and  last,  year  that 
Schleiermacher  entirely  engaged  my  thought,  and  although 
at  that  time  I  was  more  occupied  with  philology,  yet  I  then 
for  the  first  time  recognized  how  one  could  learn  something 
from  him  for  every  branch  of  knowledge,  because  he  aimed 
less  at  giving  positive  information  than  at  making  us  ca- 
pable of  attacking  each  investigation  in  the  right  way  and 
of  carrying  it  on  independently."  From  this  we  can  see 
how  Grassmann  was  coming  to  feel  the  joy  of  original 
creative  work. 

Though  he  had  studied  theology  with  his  heart  in  his 
subject,  he  had  by  this  time  reached  the  decision  to  lay  it 
aside.  He  says  that  he  had  noticed  that  clergymen  who 
lived  in  country  parishes,  shut  off  from  intercourse  with 
scholars,  lost  grasp  of  their  studies,  however  enthusiastic 
they  had  previously  been,  and  ceased  to  pursue  any  investi- 
gation on  their  own  account.  To  escape  such  a  fate  he 
decided  to  prepare  himself  as  broadly  as  possible.  For  this 
reason  he  began  the  study  of  philology,  but  he  continued 
it  from  sheer  love  of  the  subject.  He  had  also  made  the 
discovery  by  this  time  that  academic  lectures  are  only  of 
profit  if  taken  in  moderation ;  so  he  confined  himself  to  two 
courses  under  Professor  Boeckh,  on  the  history  of  Greek 
literature  and  on  Greek  antiquities  respectively.  But  he 
planned  out  a  tremendous  course  of  study,  intending  to 
begin  with  Greek  grammar,  then  to  read  the  Attic  authors 
chiefly  the  historians,  with  the  study  of  whom  he  would 
combine  Greek  history  and  antiquities — next  the  trage- 
dians with  mythology  and  poetic  forms,  and  afterward 
Homer  and  Herodotus.  Meanwhile  he  would  seek  variety 
by  reading  Roman  authors.  Finally,  as  he  intended  to 
follow  his  linguistic  studies  with  mathematics,  he  meant 
to  save  Plato  and  Demosthenes  until  he  began  that  study. 

This  exhaustive  program  he  was  not  able  to  complete 
in  Berlin.  When  he  had  reached  the  Attic  authors  he  was 


6  THE  MONIST. 

taken  ill  in  consequence  of  over-work.  He  describes  his 
illness  as  neither  severe  nor  dangerous,  but  it  compelled 
him  to  slow  down  and  to  introduce  more  variety  in  order 
to  avoid  mental  strain. 

In  this  way  he  was  led  to  the  study  of  the  sciences,  but 
he  showed  his  growing  independence  by  working  free  of 
the  schools.  He  did  not  attend  a  single  mathematical  lec- 
ture while  a  student  in  Berlin. 

We  may  now  see  how  wide  his  range  of  interests  was 
throughout  his  university  career.  He  seems  to  have  been 
striving  for  as  broad  a  foundation  as  possible,  while  at 
the  same  time  he  was  building  up  a  truly  scientific  attitude 
of  mind  which  would  enable  him  successfully  to  attack  any 
subject  he  might  turn  his  attention  to.  It  is  as  though,  as 
Engel  says,  he  knew  from  the  first  that  it  would  be  neces- 
sary in  his  life  to  have  more  than  one  iron  in  the  fire. 

In  the  autumn  of  1830  he  returned  to  Stettin,  and  late 
in  the  following  year  took  an  examination  for  a  teacher's 
certificate  before  the  Scientific  Examination  Commission 
in  Berlin.  It  was  at  this  examination  that  he  handed  in 
the  Latin  autobiography  we  have  previously  referred  to, 
and  concerning  which  Kopke,  rector  of  the  monastary 
school  of  the  Grey  Friars  in  Berlin,  comments  "Specimen 
turn  propter  rerum  ubertatem  turn  propter  stili  venustatem 
et  elegantiam  laude  dignum."  He  was  given  permission 
to  teach  philology,  history,  mathematics,  German  and  re- 
ligious knowledge  in  lower  and  middle  classes;  but  the 
commission  at  the  same  time  expressed  their  expectation 
that  he  might  easily  perfect  himself  for  teaching  ancient 
languages  and  mathematics  in  all  classes.  This  may  have 
stimulated  Grassmann  to  further  mathematical  studies, 
though  he  had  already  thrown  himself  with  energy  into 
them  under  the  influence  of  his  father,  whose  text-books 
he  would  naturally  use. 


HERMANN   GRASSMANN.  7 

He  became  assistant  teacher  (Hilfslehrer)  in  the  Stet- 
tin gymnasium,  and  in  1832  began  to  lay  the  foundations 
of  his  great  work,  the  "Theory  of  Extension"  (Ausdeh- 
nungslehre).  He  began  by  working  at  the  geometrical 
addition  of  straight  lines,  or  what  we  now  call  vector  addi- 
tion. From  this  he  was  led  to  the  notion  of  the  geometrical 
product  of  straight  lines.  The  direct  influence  of  his  father 
can  best  be  shown  by  his  own  words  :5  "But  I  had  not  the 
slightest  idea  into  what  a  rich  and  fruitful  province  I  had 
here  arrived;  rather  did  this  result6  appear  to  me  to  be  little 
worthy  of  notice  until  I  combined  it  with  a  closely  related 
idea.  Namely,  by  following  it  up  with  the  same  idea  of  the 
product  in  geometry  as  my  father  had  held,7  it  became 
evident  to  me  that  not  only  the  rectangle  but  also  the 
parallelogram  in  general  may  be  considered  as  the  product 
of  two  adjoining  sides." 

He  goes  on  to  add  that  he  was  surprised  to  find  that 
he  had  thus  reached  a  product  which  changed  in  sign  if 
its  factors  were  interchanged.  And  this,  together  with 
the  fact  that  he  was  drawn  into  other  spheres  of  work — 
one  of  which  was  the  passing  of  the  first  theological  exam- 
ination at  Stettin — caused  this  seed-idea  to  remain  dormant 
for  some  considerable  time. 

In  October,  1834,  Grassmann  returned  to  Berlin,  this 
time  as  mathematics  master  in  a  trade  school.  Soon  after- 
ward he  applied  for  a  better  position  than  the  one  he  held 
and  his  principal  gave  the  following  characterization  of 
him:  "Mr.  Grassmann  is  a  young  man  not  lacking  in  at- 
tainments. It  is  also  apparent  that  he  has  given  particular 
attention  to  the  elements  of  mathematics,  and  thinks  with 
especial  clearness  along  that  line,  but  he  seems  to  have  had 

6  Preface  to  the  first  edition  of  the  1844  Ausdehnungslehre. 

6  The  notion  of  writing  AB  -f-  BC  =  AC  whether  the  three  points  A,  B,  C 
are  in  the  same  straight  line  or  not. 

7  Cf.  J.  G.  Grassmann,  Raumlehre,  Part  II,  p.  164,  and  his  Trigonometrie, 
p.  10. 


g  THE  MONIST. 

little  intercourse  with  people  and  is  therefore  backward 
in  the  usual  forms  of  social  life,  shy,  easily  embarrassed 
and  then  very  awkward.  In  the  classroom  all  this  vanishes 
when  he  does  not  know  that  he  is  observed.  He  then 
moves  with  ease,  control,  and  certainty.  In  my  presence, 
in  spite  of  the  fact  that  I  have  done  all  I  could  to  give  him 
confidence,  he  has  not  been  able  to  become  fully  master  of 
his  embarrassment,  which  caused  him  much  concern.  My 
judgment  of  him  is  therefore  as  yet  uncertain,  and  I  cannot 
say  whether  he  will  be  able  suitably  to  fill  the  present 
vacancy." 

As  a  matter  of  fact  the  vacancy  was  not  an  easy  one  to 
fill,  since  it  had  previously  been  held  by  no  less  a  person 
than  Jacob  Steiner,  the  geometrician,  who  had  been  ap- 
pointed to  the  university  but  retained  some  of  the  higher 
classes  in  geometry.  Grassmann  obtained  the  appoint- 
ment ;  and  as  Steiner  had  bound  himself  to  initiate  his  suc- 
cessor as  far  as  possible  into  his  own  method  of  geometrical 
instruction,  one  would  have  expected  interesting  develop- 
ments from  the  contact  between  the  two  men.  There  ap- 
pears however  to  have  been  very  little  intimacy  between 
them.  There  was  a  difference  of  thirteen  years  in  their 
ages,  and  a  wide  contrast  in  temperament — the  one  self- 
reliant  but  thoroughly  one-sided,  the  other  diffident  and 
many-sided.  To  these  differences  in  personal  character- 
istics Carl  Musebeck8  is  inclined  to  attribute  their  small 
effect  on  each  other.  Victor  Schlegel's9  view  was  that  it 
was  caused  by  the  great  difference  in  the  methods  em- 
ployed by  the  two  mathematicians.  Whatever  may  have 

8  Carl  Musebeck,  article  on  Hermann  Grassmann,  No.  3,  Jahrgang  6  of  the 
Mathematisch-Natururissenschaftliche  Blatter,  p.  1,  note. 

9  It  is  curious  to  note  that  V.  Schlegel,  who,  as  we  shall  see,  was  one  of  the 
first  appreciators  of  Grassmann's  work,  long  afterward  used  the  methods  of 
Grassmann's  "Geometrical  Analysis"  to  attack  the  problem  of  the  minimum 
sum  of  the  distances  of  a  point  from  given  points  (Bull.  Amer.  Math.  Soc., 
Vol.  I,  1894,  p.  33)  and  reached  a  general  result  which  reduces  to  Steiner's 
form  of  solution  as  a  special  case ;  thus  illustrating  the  power  of  the  method. 


HERMANN   GRASSMANN.  9 

been  the  cause  it  is  at  any  rate  clear  that  Steiner's  method 
of  handling  geometry  had  no  influence  whatever  upon 
Grassmann's  manner  of  thinking. 

Several  things  combined  to  make  Grassmann's  stay  in 
Berlin  short.  He  was  greatly  distressed  by  the  loss  of  his 
youngest  sister,  who  was  scarcely  four  years  old,  and  this 
increased  his  inclination  to  religious  brooding — to  which 
he  was  the  more  inclined  as  he  lacked  suitable  companion- 
ship. His  eyesight  also  gave  him  some  trouble,  so  that 
after  a  year  and  a  quarter  he  gladly  returned  to  Stettin  on 
January  I,  1836,  and  became  teacher  in  the  Ottoschule. 

He  had,  however,  pleasant  memories  of  these  months 
in  Berlin,  as  we  can  see  from  a  letter  written  to  his  brother 
Robert,  in  which  after  speaking  with  pleasure  of  his  return 
to  Stettin  he  acknowledges  the  freedom  and  mental  stimu- 
lation afforded  by  Berlin.  At  first  glance  this  move  from 
the  capital  seems  a  pity,  since  recognition  of  his  talents 
might  have  come  to  him  if  he  had  stayed  on.  But  we  must 
remember  to  set  against  this,  that  he  was  very  high-strung 
and  energetic  in  mind  and  could  be  easily  over-stimulated 
— an  effect  helped  by  the  quiet  life  he  lived — and  also  that 
a  calmer  atmosphere  was  more  suitable  to  the  long  and 
careful  development  of  his  very  original  way  of  thought. 

While  still  at  the  Ottoschule  Grassmann  entered  for 
and  passed  the  second  theological  examination  in  Stettin 
in  July,  1839.  We  may  note  here  that  he  was  deeply  at- 
tached to  the  study  of  positive  theology  throughout  his  life. 
After  passing  his  theological  examinations  he  became  sec- 
retary and  then  president  of  the  "Pomeranian  Central  So- 
ciety for  the  Evangelization  of  China."  And  it  is  note- 
worthy in  this  respect  that  his  last  work  was  on  "The 
Falling  Away  from  Belief." 

A  few  months  before  he  submitted  his  essay  for  this 
last  theological  test,  he  was  examined  by  the  Berlin  Scien- 


IO  THE  MONIST. 

tific  Examination  Commission  in  mathematics  and  physics. 
It  was  in  connection  with  this  that  an  event  fraught  with 
great  consequences  to  his  lifework  happened  to  Grassmann ; 
for  he  was  set  the  task,  by  Professor  Conrad  of  the  Joa- 
chimsthal  Gymnasium,  of  developing  the  theory  of  tides. 
It  is  uncertain  whether  the  subject  was  chosen  by  Conrad 
on  his  own  initiative  or  was  suggested  by  Grassmann  him- 
self. In  any  case  it  was  precisely  the  practical  need  which 
was  best  calculated  to  spur  him  on  to  the  development  of 
his  dormant  mathematical  ideas.  Later  on  he  spoke10  of  the 
necessity,  in  expounding  the  claims  of  a  new  mathematical 
discipline,  of  showing  its  application.  And  it  seems  clear 
that,  faced  with  the  difficulties  and  complications  of  La- 
place's tidal  theory,  he  was  led  at  once  to  the  idea  of  trans- 
forming analytical  mechanics  by  the  introduction  of  his 
own  rudimentary  analytical  notions.  He  found  to  his  de- 
light that  the  new  analysis  proved  a  powerful  simplifying 
tool  when  applied  to  the  equations  of  Lagrange's  Mecanique 
analytique.  This  initial  success  encouraged  him  to  extend 
his  method  and  to  clothe  many  other  conceptions  such  as 
exponentials,  the  angle,  and  the  trigonometrical  functions, 
in  the  form  of  that  analysis.  He  was  then  able  to  simplify 
and  render  symmetrical  the  intricate  formulas  of  the  tidal 
theory.  Furthermore  he  found  that  the  elimination  of  ar- 
bitrary coordinates  so  effected  left  the  ideas,  their  develop- 
ment, and  their  interrelations  much  less  obscured  by  ana- 
lytical machinery. 

The  thesis  Grassmann  sent  to  Berlin  in  April  1840  was 
of  an  unusual  size ;"  and,  in  the  opinion  of  Engel,12  "judged 
by  the  number  of  new  thoughts  and  methods  contained  in 
it,  there  is  only  one  other  to  be  compared  with  it — the  thesis 
which  Weierstrass  submitted  a  year  later  to  the  Commis- 

10  In  the  Preface  to  the  first  edition  of  the  Ausdehnungslehre  of  1844. 

11  It  fills  190  pages  of  royal  octavo  in  the  third  volume  of  his  Werke. 

12  F.  Engel,  he.  cit. 


HERMANN    GRASSMANN.  II 

sion  at  Munster."  The  two  works  were,  however,  ac- 
corded very  different  receptions;  and  it  is  evident  that 
Professor  Conrad  had  no  idea  of  the  remarkable  work  he 
had  called  into  being.  His  report  runs:  "The  test  treats 
the  theory  of  the  tides  with  thoroughness  and  strength 
throughout;  and  he  has  chosen,  not  unhappily,  a  peculiar 
method  which  departs  in  many  particulars  from  the  theory 
of  Laplace."  It  remains  an  evil  omen  for  the  fate  of 
Grassmann's  later  work  that  his  examination  thesis  should 
thus  have  failed  to  find  recognition.  It  must  be  added 
that  Conrad  could  scarcely  have  read  the  work  and  still 
less  have  been  able  to  estimate  it  at  its  true  value.  For 
he  received  it  on  May  26  and  returned  it  five  days  later  at 
the  oral  examination — in  which  Grassmann  fared  better, 
being  granted  full  recognition  of  his  mathematical  ability. 
Grassmann  probably  realized  that  this  thesis  on  tidal 
theory  was  but  a  first  fruit  of  his  methods  and  that  those 
methods  themselves  were  much  more  general  and  capable 
of  immense  development.  This  work  he  threw  himself 
into  with  characteristic  energy  in  the  next  few  years.  He 
left  the  Ottoschule  at  Michaelmas,  1842,  and  spent  six 
months  teaching  at  the  Stettin  Gymnasium;  after  which 
he  entered  the  Friedrich-Wilhelm-Schule  which  had  been 
founded  a  few  years  before,  and  of  which  his  eldest  son 
Justus  Grassmann  is  now  the  principal. 

By  1842  Grassmann  had  completed  the  main  outlines 
of  his  new  analytical  method.  He  tried  to  make  the  ideas 
known  to  his  own  circle  by  lectures,  in  which  he  showed 
the  power  of  the  new  "science  of  extended  magnitudes" 
by  further  application  to  mechanics  and  crystallography. 
Desiring  to  expound  his  method  by  reference  to  well-known 
results  he  was  led  to  the  barycentric  calculus  of  Mobius 
and  to  Poncelet.  The  first  of  these  illustrations  was  the 
"Theorie  der  Zentralen"  (Crelle's  Journal,  Vol.  XXIV, 


12  THE  MONIST. 

1842)  in  which,  without  using  his  own  analysis,  he  made 
a  general  statement  in  which  not  only  all  Poncelet's  results 
but  also  further  important  general  properties  of  curves 
and  surfaces  are  contained  as  special  cases.  Such  wide 
generalization  is  characteristic  of  his  method.  In  1844  h*3 
Ausdehnungslehre  was  published,  being  designed  as  the 
first  part  of  the  complete  work.  This  part,  which  he  pro- 
posed to  follow  up  with  a  second  later,  he  called"  Die  lineale 
Ausdehnungslehre,  a  new  branch  of  mathematics." 

The  fate  of  this  book  was  a  tragic  one.  It  remained 
unread  and  unsold  until  the  publisher  had  to  get  rid  of  the 
whole  edition  as  waste  paper.  Not  even  a  review  was 
granted  to  it;  and  what  criticism  there  was  had  so  little 
basis  of  understanding  that  it  led  to  no  deeper  study  of  the 
work.  Gauss  wrote  of  it,  in  1844,  that  its  tendencies  partly 
went  in  the  same  direction  in  which  he  himself  for  almost 
half  a  century  had  wandered;  but  there  seemed  to  him  to 
be  only  a  partial  and  distant  resemblance  in  the  tendency. 
He  thought  it  would  be  necessary  to  familiarize  oneself 
with  the  special  terminology  to  get  at  the  real  kernel  of 
the  book.  Grunert  declared  that  he  had  not  completely 
succeeded  in  forming  a  definite  and  clear  opinion  about  the 
work.  Mobius,  whom  Grassmann  had  asked  for  a  review 
in  some  critical  journal  because  he  stood  nearest  to  the 
ideas  in  the  book,  answered  that  this  mental  relationship 
only  existed  in  regard  to  mathematics,  not  with  reference 
to  philosophy;  and  that  he  considered  himself  incapable  of 
estimating  and  appreciating  the  philosophical  element  of 
the  excellent  work — which  lies  at  the  base  of  all  mathe- 
matics. But  he  added  that  he  recognized  that,  next  to  the 
great  simplification  of  method,  the  principal  gain  consisted 
in  the  fact  that  by  a  more  general  comprehension  of  funda- 
mental mathematical  operations  the  difficulties  of  many 
analytical  concepts  are  removed. 


HERMANN   GRASSMANN.  13 

Without  entering  in  detail  into  a  discussion  of  the 
causes  of  this  neglect  of  Grassmann's  work13  we  may  note 
that  its  great  generality,  its  philosophical  form,  and  its 
original  and  technical  symbolism  were  contributing  factors 
which  also  make  it  very  difficult  to  give  any  account  of  the 
work  for  the  general  reader.14  But  the  importance  of  the 
ideas  hidden  away  in  this  forbidding  volume  may  be  gath- 
ered from  the  words  written  of  it  by  Carl  Miisebeck  many 
years  later:  "Earlier  than  Riemann,  Grassmann  evolved 
manifolds  of  n  dimensions  in  mathematical  analysis.  In 
a  lighter  and  less  constrained  manner  Grassmann  arrives 
by  his  combinatory  multiplication  at  the  fundamental  prin- 
ciples of  determinant-theory,  and  the  elementary  solution 
of  various  problems  of  elimination.  In  him  one  finds  indi- 
cated both  Bellavitis's  Equipollences  and  Hamilton's  Qua- 
ternions." And  yet  the  only  recognition  given  by  mathe- 
maticians to  the  ideas  of  Grassmann  was  the  award  to  him 
by  the  Jablonowski  Society  at  Leipsic  for  a  prize  essay15  on 
the  "Geometrical  Calculus  of  Leibniz"  in  1846. 

It  must  not  be  supposed,  however,  that  Grassmann  sat 
quietly  down  to  neglect.  He  brought  out  the  importance 
and  applicability  of  his  investigation  by  numerous  valuable 
articles  in  Crelle's  Journal,  and  later  in  Mathematische 
Annalen  and  the  Nachrichten  of  the  Royal  Society  of  Sci- 
ence of  Gottingen.  Furthermore,  in  1845  ne  published  in 
Grunert's  Archiv,  Vol.  VI,  a  detailed  abstract16  of  the  Aus- 
dehnungslehre,  intended  for  mathematicians.  Thirty  years 
later  Grassmann  spoke  to  Delbriick17  with  youthful  ardor 

18  See  the  article  below  on  "The  Neglect  of  the  work  of  H.  Grassmann." 

14  An  attempt  was  made  to  do  this  by  Justus  Grassmann  in  an  address 
delivered  at  the  opening  of  his  school  year  on  April  16,  1909,  when  the  cen- 
tenary of  his  father  was  being  celebrated. 

15  Geometrische  Analyse,  published  1847.    This  treatise  is  to  some  extent 
a  substitute  for  the  second  part  of  the  Ausdehnungslehre  of  1844,  anticipated 
in  the  preface  to  that  work  but  never  written. 

16  Reprinted  in  the  Werke,  Vol.  I,  Part  I,  p.  297. 

17  B.  Delbriick,  "Hermann  Grassmann,"  Supplement  to  the  Allgemeine  Zei- 
tung,  Oct.  18,  1877. 


14  THE  MONIST. 

of  this  period  as  one  of  happy  restlessness  and  joy  in  dis- 
covery. Such  joy  in  original  work  and  faith  in  the  power 
of  his  mathematical  methods  he  always  retained  in  spite 
of  a  succession  of  disappointments  which  would  have 
quenched  a  less  ardent  spirit. 

It  is  an  extraordinary  thing  that  it  was  not  only  in  his 
mathematical  work  that  he  failed  to  find  recognition,  but 
also  in  his  contributions  to  physics.  In  1845  ne  published 
in  Poggendorff's  Annalen  a  statement  of  the  mutual  inter- 
action of  two  electric  stream  lines  which  was  re-discovered 
thirty-one  years  later  by  Clausius.  In  a  school  syllabus 
in  1854  Grassmann  stated  that  the  vowels  of  the  human 
voice  owe  their  character  to  the  presence  of  certain  partial 
tones  of  the  mouth  cavity,  a  view  of  the  nature  of  vowel 
sounds  which  is  usually  ascribed  to  Willis  and  Helmholtz. 
Of  his  other  purely  physical  work  we  may  mention  his 
notes  on  the  mixing  of  colors  and  his  design  of  a  very 
simple  but  practical  heliostat.18  Still  he  continued  to  hope 
that  the  value  of  his  work  would  be  appreciated.  He  had 
himself  foreseen19  that  the  dislike  of  mathematicians  for  a 
philosophical  form  might  deter  them  from  considering  his 
work,  and  the  comments  of  Mobius  and  Grunert  on  this 
had  shown  his  fears  to  be  well  founded.  So  he  yielded  to 
the  often  expressed  wish  of  Mobius  that  he  should  rewrite 
the  Ausdehnungslehre.  in  a  form  more  attractive  to  mathe- 
maticians. In  the  new  work,  published  in  1862,  he  chose 
a  more  deductive  method — one  moreover  which  is  not  alto- 
gether suited  to  the  subject  matter,  but  it  did  succeed  in 
bringing  forward  more  clearly  the  original  operations  and 
characteristics  of  the  Ausdehnungslehre.  All  was  in  vain. 
Neither  genius  nor  indomitable  energy  could  contend 
against  so  unresponsive  an  environment. 

We  must  remember  that  Grassmann's  continued  output 

18  A  model  was  constructed  by  the  Stettin  Physical  Society. 

19  Preface  to  the  first  edition  of  the  1844  Ausdehnungslehre. 


HERMANN    GRASSMANN.  1 5 

of  virile  original  work  was  done  in  the  scanty  leisure  of 
an  energetic  schoolmaster.  He  had  been  nominated  head- 
teacher  at  the  Friedrich-Wilhelm-Schule  in  1847,  and  five 
years  later  he  was  appointed  successor  to  his  father  at  the 
gymnasium.  There  he  remained  for  a  quarter  of  a  cen- 
tury. He  had  hoped  that  his  mathematical  writings  would 
win  for  him  some  position  in  which  he  would  have  more 
leisure  for  research  and  be  in  closer  contact  with  other 
scientific  workers.  But  it  must  not  be  supposed  for  an 
instant  that  this  lessened  his  intense  interest  in  the  work 
at  hand.  He  wrote  articles  on  educational  subjects  as 
well  as  a  number  of  text-books  for  school  use.  Of  these 
his  Arithmetik,  written  in  collaboration  with  his  brother 
Robert  showed  a  strictness  in  its  proofs  which  made  it  a 
good  introduction  to  the  theory  of  numbers.  His  Trigono- 
metrie  has  a  richness  of  content  in  small  space  and  an 
originality  of  plan  not  often  then  found  in  elementary  hand- 
books. 

Miisebeck  has  questioned  some  of  Grassmann's  pupils 
on  his  methods  of  teaching.  They  appear  in  the  main  to 
agree  with  Wandel,  who  says  in  his  "Studies  and  Char- 
acters from  Ancient  and  Modern  Pomerania"  that  he  was 
a  lovable  and  painstaking  master  whose  kindly  instruction 
was  sometimes  too  difficult  for  them.  The  lively  interest 
he  took  in  the  independence  of  those  he  taught  is  shown  by 
the  fact  that,  according  to  Schlegel,  he  formed  a  society 
out  of  every  three  scholars  in  his  chemistry  class,  the 
members  of  which  had  to  demonstrate  and  lecture  to  the 
others  on  some  substance  and  its  combinations.  The  pleas- 
ant footing  he  established  between  himself  and  his  classes 
may  be  judged  from  the  fact  that  they  were  willing  to  co 
operate  in  classwork  with  him  when  in  later  years  he  had 
to  be  taken  to  school  in  a  wheeled  chair.  Whenever  any  of 
his  old  pupils  speak  of  him  they  do  so  with  the  greatest 
admiration  and  respect. 


l6  THE  MONIST. 

It  is  difficult  in  thus  giving  an  account  of  Grassmann's 
educational  and  scientific  activity  to  avoid  at  the  same 
time  conveying  the  impression  of  a  mere  enthusiastic  ped- 
ant. It  does  not  seem  that  there  could  have  been  time  for 
anything  else.  And  yet  such  a  view  would  be  widely  re- 
moved from  the  truth.  For  in  the  midst  of  all  these  exact- 
ing duties  he  had  many  social  and  general  interests.  In 
1848  he  took  an  active  part  in  politics,  expressing  anti- 
revolutionary  sympathies;  he  attempted  to  introduce  a 
German  plant-terminology  into  botany;  and  his  early  de- 
veloped love  for  music  found  expression  in  organizing  an 
orchestra  of  his  scholars  and  in  collecting  numerous  folk- 
songs, which  he  set  for  three  voices,  to  be  sung  in  his 
family. 

We  have  been  led,  by  the  necessity  of  obtaining  some 
idea  of  the  actual  conditions  under  which  Grassmann 
worked,  to  speak  of  his  later  life.  We  must  now  return  to  the 
time  when  he  first  began  to  realize  how  slight  a  recognition 
was  to  be  accorded  to  his  mathematical  writings — that  is 
to  say  about  the  year  1852.  Great  as  was  his  inner  sure- 
ness  of  the  value  of  the  work,  yet  his  was  not  the  type  of 
mind  to  be  satisfied  with  a  partial  success.  And  so  he  took 
the  astonishing  (and  almost  unprecedented)20  step  of  turn- 
ing his  attention  to  another  field  of  knowledge  altogether 
and  quickly  winning  the  recognition  of  experts.  The  plia- 
bility of  his  genius  enabled  him  to  force  his  way  into  a  new 
subject,  philology,  and  to  produce  results  of  outstanding 
merit  in  it. 

B.  Delbriick21  gives  an  interesting  account  of  how 
Grassmann  turned  to  philology.  The  rules  of  the  tradi- 
tional school  grammar  with  its  mass  of  exceptions  must 
have  been  painful  to  his  mathematical  understanding,  and 

20  The  equally  neglected  English  genius  Thomas  Young  combined  mathe- 
matical and  philological  ability. 

21  B.  Delbruck,  loc.  cit. 


HERMANN   GRASSMANN.  I? 

so  he  first  planned  a  grammar  and  reading  book  in  which 
scientific  laws  replaced  the  old  rule-of-thumb  methods 
wherever  possible.  It  is  natural  therefore  that  he  should 
next  turn  his  attention  to  that  sphere  of  language  in  which 
such  laws  are  most  easily  recognizable,  namely  phonetics. 
His  first  attempt  in  the  realm  of  comparative  philology 
was  on  this  subject.  It  was  an  article,  published  in  1859, 
on  the  influence  of  v  and  j  on  neighboring  consonants,  and 
on  certain  phenomena  in  connection  with  aspiration.  Del- 
briick  expresses  the  opinion  that  his  work  in  this  field  is 
not  distinguished  either  for  breadth  of  scholarship,  since 
he  worked  with  few  books,  or  for  etymological  depth. 
"But,"  he  says,  "it  is  the  clearness  of  reflection  which  pene- 
trates into  all  corners  of  the  subject,  the  persistence  with 
which  the  material  has  been  so  long  accumulated  until  it 
became  possible  to  reach  the  simplest  formulation  of  the 
governing  law,  and  the  untiring  nature  of  the  mathematical 
abstraction  which  in  these  undertakings  so  clearly  comes 
to  light." 

Grassmann  must  have  quickly  recognized  how  valuable 
in  all  researches  into  comparative  philology  a  deep  acquain- 
tance with  the  oldest  Indian  languages  would  be,  and  he 
determined  with  his  usual  persistency  to  make  himself  at 
home  in  the  hymns  of  the  Vedas.  These  Sanskrit  studies 
led  him  to  the  production  of  works  which  rendered  his  name 
famous.  In  1861  he  had  only  the  first  volume  of  the  up- 
right text  and  scarcely  half  of  the  Bohtlingk-Roth  diction- 
ary. Yet  with  these  means  he  succeeded  in  mastering  the 
extraordinary  difficulties  of  the  texts,  and  began  his  dic- 
tionary and  translation  of  the  Rig- Veda.  He  arranged  his 
dictionary  in  an  original  manner  so  as  to  be  able  to  give 
the  meaning  of  each  form  according  to  the  place  in  which 
it  occurred.  Although  Delbriick  credits  the  first  volume 
of  the  dictionary  with  etymological  value  for  its  grammat- 
ical subdivision  of  the  roots,  yet  he  regards  the  arrange- 


l8  THE  MONIST. 

ment  just  mentioned  as  unphilological.  It  aims  less  at 
giving  definite  historical  and  philological  information  than 
at  making  successive  attempts  at  explanation.  As,  how- 
ever, the  work  progressed,  aided  by  the  stream  of  material 
reaching  the  author  from  the  growing  Roth  dictionary 
and  elsewhere,  it  became  more  philological.  Still  the 
method  pursued  was  the  same,  and  Grassmann  completed 
the  translation  side  by  side  with  the  dictionary.  For  long 
these  works  formed  a  useful  tool  in  attacking  the  difficul- 
ties of  the  Vedas.  The  recognition  of  experts  was  worthily 
expressed  by  Rudolph  Roth,  on  whose  word  the  University 
of  Tubingen  conferred  upon  Grassmann  the  honorary  de- 
gree of  Doctor  of  Philosophy.  He  spoke  of  him  as  a  man 
qui  acutissima  vedicorum  carminum  interpretation  nomen 
suum  reddidit  illustrissimum. 

During  this  period  of  his  life  when  he  was  winning 
fame  in  another  sphere  of  work,  Grassmann's  mathemat- 
ical writings  were  gradually  obtaining  the  recognition 
which  was  their  due.  Toward  the  end  of  the  sixties  con- 
siderable attention  was  paid  by  mathematicians  to  higher 
algebra,  and  the  quickening  of  thought  along  those  lines 
made  recognition  much  more  likely.  Hermann  Hankel  in 
his  Theorie  der  complexen  Zahlensysteme  of  1867  was  the 
first  to  call  attention  to  Grassmann's  work.  Clebsch22  also 
shortly  afterward  accorded  him  a  full  measure  of  admira- 
tion. Grassmann23  believed  that  Clebsch  would  have  fer- 
tilized the  theory  of  extension  with  far-reaching  new  ideas 
of  his  own  if  death  had  not  cut  short  his  promising  career. 
Some  of  the  younger  teachers  at  the  Stettin  Gym- 
nasium had  become  pupils  of  Grassmann ;  and  one  of  these, 
the  mathematician  Victor  Schlegel,  in  his  System  der 

22  Clebsch,  Zum  Ged'dchtniss  an  Julius  Pliicker,  1872. 

28  See  preface  to  second  edition  of  the  Ausdehnungslehre  of  1844,  pub- 
lished in  1878. 


HERMANN    GRASSMANN.  19 

Raumlehre  (ist  part  1872,  2d  part  1875)  made  his  works 
more  accessible  by  a  clear  exposition  and  application  of 
them.  The  best  kind  of  approval  from  authorities  came  to 
him  in  their  use  of  his  methods  in  various  fields ;  and  Grass- 
mann  himself,  after  a  long  interval,  again  took  up  his 
mathematical  labors.  Of  the  many  articles  from  his  pen, 
we  may  mention  especially  that  on  the  application  of  his 
work  to  mechanics,24  because  it  was  in  this  domain  that  he 
considered  the  theory  of  extension  to  be  particularly  suc- 
cessful. He  expressed  the  desire  that  it  might  be  granted 
to  him  to  write  a  treatise  on  mechanics  based  on  his  prin- 
ciples. This  was  denied  him.  He  lived,  however,  to  see 
a  second  edition  of  his  ill-fated  Ausdehnungslehre  of  1844 
called  for ;  and  died,  while  it  was  passing  through  the  press, 
on  September  26,  1877,  in  his  sixty-ninth  year.  To  the 
last,  in  spite  of  great  bodily  suffering,  he  retained  his  vigor 
and  enthusiasm.  Five  essays  published  in  the  year  of  his 
death  testify  to  this. 

It  is  a  pleasant  thing  to  think  that  he  received  such 
rich  recognition  before  he  died;  though  it  must  always 
remain  a  source  of  regret  that  he  never  succeeded  in  ob- 
taining the  position  he  hoped  for,  which  would  have  enabled 
his  powers  to  be  more  fully  developed  and  his  influence 
more  widely  expressed.  And  yet,  there  can  be  no  cause 
for  sorrow  if  we  think  of  the  fortitude  of  this  strong  soul, 
and  remember  the  firm  conviction  expressed  in  the  closing 
words  of  the  introduction  to  the  Ausdehnungslehre  of  1862, 
that  his  mathematical  ideas  would  some  day  arise  again, 
though  perhaps  in  a  new  form,  and  become  part  of  living 
thought.  To  some  extent  that  conviction  has  proved  a 
justifiable  one.  The  publication  of  his  Collected  Works 
was  suggested  by  Professor  Klein  of  Gottingen.  After  ob- 

24  "Die  Mechanik  nach  den  Principien  der  Ausdehnungslehre,"  Math. 
Annalen,  Vol.  XII,  1877. 


2O  THE  MONIST. 

taining  the  consent  of  Grassmann's  relatives  he  laid  the 
matter  before  the  Royal  Saxon  Academy  of  Sciences  in 
October,  1892.  A  committee  was  formed  and  F.  Engel 
made  chief  editor.  The  first  part  of  the  first  volume  ap- 
peared in  1894. 

Since  then  there  have  been  many  works  on  the  calculus 
of  extension,  but  it  can  scarcely  be  held  that  they  have 
done  more  than  make  a  beginning  of  the  development  of  the 
suggestions  in  Grassmann's  work.  What  has  been  done 
has  been  mainly  in  the  domains  of  spatial  theory  and  higher 
algebra ;  mechanics  remains  still  burdened  with  traditional 
coordinate  systems.  This  is  the  more  remarkable  since  the 
principle  of  relativity,  with  its  demand  for  a  generalized  dy- 
namics of  which  ordinary  dynamics  is  a  special  case,  offers 
such  a  promising  field  of  application. 

There  is  usually,  in  the  sphere  of  thought,  a  rational 
explanation  of  apparently  irrational  facts.  A  minute  in- 
fluence translated  into  action  by  the  mass  of  thinking  men 
may  give  rise  to  the  spirit  of  their  age ;  and  thus  its  effects, 
and  the  negative  effects  may  be  just  as  great  as  the  positive, 
carried  forward  in  ever-increasing  circles  to  distant  gen- 
erations. So  it  has  been  with  whatever  lies  at  the  base  of 
the  neglect  of  Hermann  Grassmann.  There  has  been  be- 
queathed to  us  something  like  an  unreasoning  distaste  for 
his  and  similar  analytical  methods,  from  which  has  arisen 
the  need  for  a  definite  effort  to  break  the  spell  of  the  past. 
The  formation  of  an  "International  Association  for  Pro- 
moting the  Study  of  Quaternions  and  Allied  Systems  of 
Mathematics"  took  its  origin  from  such  a  need.25  It  may 
therefore  be  that  a  just  estimate  both  of  the  value  and  limi- 
tations of  Grassmann's  work  will  only  come  by  the  appli- 
cation of  a  critical  method  of  wider  scope  than  those  of  his 

28  P.  Molenbrock  and  Shunkichi  Kimura,  letter  to  Nature,  Oct.  3,  1895. 


HERMANN   GRASSMANN.  21 

own  period.  Indications  are  indeed  not  wanting  that  in 
the  modern  theory  of  transformation-groups26  lies  the  cri- 
terion for  a  final  judgment. 

A.  E.  HEATH." 
BEDALES,  PETERSFIELD,  ENGLAND. 

26  Lie  and  Engel,  Theorie  der  Transformationsgruppen,  Vol.  II,  p.  748 ; 
M.  Abraham  and  P.  Langevin,  "Notions  geometriques  fondamentales,"  Encyc. 
des  sciences  mathematiques,  Tome  IV,  Vol.  5,  p.  2. 

27  I  wish  to  thank  Miss  Vinvela  Cummin  and  Mr.  R.  E.  Roper  for  help 
in  the  translation  of  materials  for  this  sketch. 


THE  NEGLECT  OF  THE  WORK  OF  H.  GRASS- 
MANN. 

IT  must  not  be  supposed  that  the  neglect  of  Hermann 
Grassmann's  mathematical  work  by  his  contemporaries 
is  merely  an  incident  of  his  biography.  Its  consideration 
involves  a  much  larger  question,  because  Grassmann's  fate 
was  shared  by  other  mathematicians  of  the  period  in  whose 
work  stress  was  laid  on  form  rather  than  content.  The 
distinction  between  the  two  may  be  illustrated  by  reference 
to  the  mathematical  treatment  of  quantity.  As  soon  as 
analysis  had  generalized  that  idea  so  as  to  include  complex 
quantities,  a  mathematics  based  on  formal  definitions  and 
of  a  general  character  could  be  developed  to  include  them. 
The  meaning  of  the  propositions  of  such  a  calculus  need 
not  enter  into  this  study.  The  propositions  would  consti- 
tute a  formal  deductive  series  which  could  be  developed 
without  any  reference  to  content.  That  Grassmann  was 
a  pioneer  in  the  movement  which  made  magnitude  sub- 
ordinate and  posterior  to  a  science  of  form  was  recognized 
by  Hankel,1  who  says,  "It  was  Grassmann  who  took  up  this 
idea  for  the  first  time  in  a  truly  philosophical  spirit  and 
treated  it  from  a  comprehensive  point  of  view."  In  the 
Introduction  (A)  to  the  Ausdehnungslehre  of  1844  Grass- 
mann puts  the  matter  thus :  "The  chief  division  of  all  sci- 
ences is  that  into  real  and  formal.  The  former  sciences 

1  Theorie  der  complexen  Zahlensysteme,  p.  16. 


THE  NEGLECT  OF  THE  WORK  OF  H.  GRASSMANN.        2$ 

image  in  thought  the  existent  as  independent  of  thinking, 
and  their  truth  consists  in  the  agreement  of  the  thought 
with  the  existent;  the  latter  sciences  on  the  contrary  have 
for  their  subject-matter  that  which  has  been  determined 
by  thought  itself,  and  their  truth  is  shown  in  the  mutual 
agreement  between  processes  of  thought."  He  goes  on  to 
consider  mathematics  and  formal  logic  as  branches  of  a 
general  science  of  form,  and  seeks  to  dissociate  this  science 
from  such  real  sciences  as  the  geometry  of  actual  space, 
although  it  must  form  the  basis  on  which  all  such  are  built. 
That  the  neglect  accorded  to  Grassmann  had  nothing 
to  do  with  any  accident  of  birth  or  position  is  shown  by 
the  fact  that  Leibniz,  whose  name  was  famous  in  both 
mathematical  and  philosophical  circles,  shared  the  same 
fate  in  regard  to  his  Dissertatio  de  Arte  Combinatoria 
and  later  writings  of  the  same  kind,  in  which  he  sought 
to  set  up  a  formal  symbolical  calculus  with  similar  aims. 
Of  Grassmann's  contemporaries  who  worked  in  the  same 
field,  we  need  mention  only  George  Boole  (1815-1864) 
who  failed  to  obtain  anything  like  a  due  recognition  of  his 
genius;  and  Sir.  W.  R.  Hamilton  whose  early  papers  on 
quaternions  were  regarded  as  mere  curiosities.  Even  when 
the  applications  of  these  generalized  formal  methods  to  the 
founding  of  a  calculus  of  directed  quantities  of  immediate 
value  to  physics  had  been  made,  we  find  the  important 
work  of  Willard  Gibbs  waiting  for  years  before  it  became 
known  and  made  full  use  of.  If,  then,  we  are  to  explain 
the  neglect  of  Grassmann's  work  we  shall  have  to  analyze 
the  causes  of  the  apathy  and  mistrust  with  which  all  such 
work  has  been  received. 

The  view  held  by  Carl  Miisebeck  is  that  in  the  almost 
exclusively  philosophical  form  of  representation,  which 
however  was  grounded  in  the  whole  system,  we  have  to 
seek  the  reason  why  the  contemporaries  of  Grassmann 


24  THE  MONIST. 

drew  back  in  terror  from  deeper  study  of  his  early  work. 
He  says2:  "Such  a  height  of  mathematical  abstraction  in 
which,  with  the  help  of  a  new  calculus,  laws  are  inferred 
in  abstract  regions  about  the  mutual  dependence  of  abstract 
constructions  in  which  not  even  the  character  of  the  spatial 
is  maintained,  although  at  the  conclusion  of  almost  every 
section  it  is  shown  how  the  new  method  could  be  used  with 
advantage,  was  never  before  known."  That  this  has  been 
a  very  important  factor  cannot  be  doubted.  Dislike  of  the 
philosophical  form  of  his  work  was  expressed  to  Grass- 
mann  by  the  few  mathematicians  who  noticed  his  first  Aus- 
dehnungslehre.  He  himself  says  in  the  preface  to  the 
second  edition  of  this  book  that  he  expected  the  work  to 
find  fullest  recognition  from  the  more  philosophically  in- 
clined reader.  It  is  only  necessary  to  refer  to  the  appli- 
cation and  extension  of  his  ideas  which  have  come  from 
A.  N.  Whitehead3  in  England  and  from  G.  Peano4  and  C. 
Burali-Forti5  in  Italy  to  show  how  well-founded  this  fore- 
cast was.  But  the  analysis  cannot  rest  there.  We  must 
inquire  further  how  this  dislike  arose. 

J.  T.  Merz6  in  his  chapter  on  "The  Development  of 
Mathematical  Thought  in  the  iQth  Century,"  inclines  to 
the  view  that  a  definite  distaste  for  a  philosophical  form 
had  set  in  among  German  mathematicians  as  a  part  of  the 
reaction  against  the  exaggerations  of  the  metaphysical  uni- 
fication of  knowledge  in  the  schools  of  Schelling  and  Hegel. 
But  mathematicians  in  modern  times  have,  on  the  whole, 
been  singularly  unaffected  by  philosophical  movements. 
Furthermore  the  calculus  of  extension  and  allied  systems 
have  not  fully  come  into  their  own  even  in  our  own  day, 

2  In  his  memoir  of  Hermann  Grassmann,  Stettin,  1877. 

3  Universal  Algebra,  Cambridge,  1898. 

4  Calcolo  Geometrico  secondo  I'Ausdehnungslehre  di  H.  Grassmann,  Turin, 
1888. 

e  Introduction  a  la  geometric  differ entielle,  suivant  la  methode  de  H.  Grass- 
mann, Paris,  1897. 

6  History  of  European  Thought  in  the  loth  Century,  Vol.  I,  p.  243. 


THE  NEGLECT  OF  THE  WORK  OF  H.  GRASSMANN.        25 

when  wide  syntheses  are  eagerly  sought.  It  seems  to  the 
present  writer  that  it  is  in  the  attitude  of  the  plain  anti- 
metaphysical  mathematician  that  we  must  seek  for  the 
explanation  of  the  want  of  understanding  which  leads  to 
mistrust  of  philosophical  form.  An  immense  amount  of 
prejudice  barred  the  way  to  the  full  development  of  a  gen- 
eral science  of  form — prejudice  due  to  non-realization  of 
the  purely  formal  claims  of  such  a  calculus.7  And  if  we 
could  get  at  the  bottom  of  this  not  altogether  unreasoning 
mistrust  it  might  be  possible  to  clear  away  some  of  the 
hindrances  to  a  proper  understanding  of  the  fundamental 
importance  of  Grassmann's  work. 

To  do  this  we  must  push  our  analysis  a  step  further. 
What  steady  cause  can  have  been  operating  over  such  a 
long  period  which  could  so  affect  the  attitude  of  the  indi- 
vidual as  to  create  what  amounts  almost  to  a  general 
blindness  to  the  importance  of  a  whole  body  of  contribu- 
tions to  thought  ?  I  believe  that  the  root  of  the  matter  lies 
in  wrong  principles  of  instruction.  It  may  be  that  this  at 
first  sight  appears  too  small  an  influence  to  have  such  con- 
sequences; but  so  did  the  minute  geological  influences  of 
the  uniformitarians  to  those  who  sought  for  explanations 
in  more  dramatic  cataclysms.  It  is  as  unscientific  to  neglect 
the  unobtrusive  but  persistent  influences  of  educational 
methods  on  pure  thought  as  it  would  be  to  treat  of  the 
social  conditions  of  a  people  without  taking  into  account 
their  mind-development. 

We  will  only  give  one  well-recognized  example  of  the 
importance  of  methods  of  exposition  on  mathematical  his- 
tory. Merz  places  Gauss  at  the  head  of  the  critical  move- 
ment which  began  the  nineteenth  century.  He  adds,8  how- 
ever, that  it  was  not  to  him  primarily  that  the  great  change 

7  Cf.  the  article  below  on  "The  Geometrical  Analysis  of  Grassmann  and  its 
Connection  with  Leibniz's  Characteristic,"  §  2. 

8  Op.  cit.f  Vol.  II,  p.  636. 


26  THE  MONIST. 

which  came  over  mathematics  was  due,  but  to  Cauchy. 
Gauss,  while  issuing  finished  and  perfect  though  some- 
times irritatingly  unintelligible  tracts,  hated  lecturing;  in 
contrast  to  this  Cauchy  gained  the  merit,  through  his  en- 
thusiasm and  patience  as  a  teacher,  of  creating  a  new 
school  of  thought — and  earned  the  gratitude  of  the  greatest 
intellects,  such  as  Abel,  for  having  pointed  out  the  right 
road  of  progress.  But  it  is  not  so  much  upon  the  manner 
of  exposition  of  original  mathematicians  themselves  that 
stress  must  be  laid.  It  has  without  doubt  often  happened 
that  writers  of  great  analytical  insight  have  failed  to  see 
that  it  is  no  more  a  descent  to  a  common  level  to  seek  out 
and  use  the  best  methods  of  enforcing  consideration  of 
their  work,  than  it  is  to  use  a  printing-press  instead  of  a 
town  crier  the  more  effectively  to  reach  their  audience. 
Grassmann  himself,  however,  did  all  that  was  humanly 
possible  in  this  way,  although  Jahnke  is  of  the  opinion 
that  he  was  inclined  to  the  belief  that  even  first  instruction 
should  be  rigorous;  and  kept  back  applications  until  too 
late.  It  is  rather  that  teaching  methods  in  general  dur- 
ing the  nineteenth  century  have  always  lagged  too  far  be- 
hind discovery.  And  so  they  have  left  the  students  of 
one  generation,  who  are  the  potential  original  workers 
of  the  next,  with  minds  unreceptive  to  newer  and  more 
delicate  methods.  It  might  be  urged  that  this  would 
affect  equally  all  branches  of  mathematics,  but  I  think  it 
can  be  shown  that  it  is  on  the  reception  of  such  funda- 
mental analytical  methods  as  Grassmann's  that  its  evil  in- 
fluence more  particularly  falls. 

It  is  quite  obvious  that  the  subject  must  be  limited  if 
we  are  to  deal  in  detail  with  the  suggested  effects  of  in- 
adequate educational  methods.  So  I  shall  confine  myself 
in  what  follows  to  the  consideration  of  the  difficulties  which 
beset  the  path  of  the  teacher  who  has  to  explain  the  ordi- 


THE  NEGLECT  OF  THE  WORK  OF  H.  GRASSMANN.        2? 

nary  concepts  of  mechanics ;  and  attempt  to  show  how  fail- 
ure to  realize  the  nature  of  those  difficulties  tends  to  pro- 
duce an  unreceptive  attitude  to  modern  analysis.  I  have 
chosen  this  subject  for  two  reasons.  Firstly,  it  seems  to 
me  that  if  the  concepts  of  mechanics  were  properly  treated 
they  would  finally  appear  to  the  pupil  as  useful  construc- 
tions instead  of  as  the  dogmatically  asserted  existents  they 
are  still  commonly  held  to  be;  and  so  the  formal  science 
underlying  the  real  science  of  mechanics  would  naturally 
arise  for  him  as  the  final  result  of  analysis,  and  not  as  the 
unreal  fabric  of  a  philosopher's  dream.  And  secondly,  it 
is  the  domain  to  which  the  various  "extensive  algebras" 
have  peculiar  applicability,  as  Grassmann  himself  felt 
strongly.  It  is  highly  significant  therefore  that  it  is  pre- 
cisely Grassmann's  suggestive  applications  to  mechanics 
whose  neglect  is  the  most  noticeable.  That  this  is  so  is. 
on  my  view,  because  sounder  and  more  philosophical  no- 
tions of  geometrical  as  opposed  to  mechanical  concepts 
were  already  coming  into  exchange  in  Grassmann's  own 
day  so  that  geometrical  applications  were  thereby  rendered 
more  understandable. 

At  the  very  outset  of  our  discussion  we  are  faced  with 
the  difficulty  that  so  much  difference  of  opinion  exists  be- 
tween teachers  of  mechanics  that  many  have  been  forced 
into  the  conclusion  that,  since  the  enthusiast  with  an  un- 
philosophical  method  of  his  own  can  yet  reap  good  results, 
method  is  unimportant.  This,  of  course,  is  only  partially 
true.  If  it  were  wholly  true  it  would  mean  an  end  to  all 
possibility  of  coordination — an  end,  in  fact,  to  the  claims 
of  education  to  be  a  science.  To  grant  that  education  is 
an  art  is  not  to  forego  all  its  claims  to  be  a  science.  For 
we  must  regard  all  art  as  applied  science  "unless  we  are 
willing,  with  the  multitude,  to  consider  art  as  guessing 


28  THE  MONIST. 

and  aiming  well."1  Beneath  the  apparent  chaos  of  opinion 
on  the  teaching  of  mechanics  there  is  however  some  order 
if  one  can  avoid  certain  sources  of  confusion  which  have 
led  to  superficial  differences  of  opinion  where  nothing 
deeper  exists. 

One  source  of  confusion  is  the  absence  of  a  clear  idea 
of  the  difference  in  educational  theory  between  an  imper- 
sonal principle  and  the  more  personal  element — the  method 
of  applying  the  principle.  This  distinction  is  insisted  on 
by  Mr.  E.  G.  A.  Holmes,10  and  seems  a  real  one.  If  once 
we  realize  it  we  can  see  how  it  is  possible  for  there  to  be 
fairly  well  accepted  scientific  principles  of  teaching  at  the 
same  time  as  a  wide  divergence  of  method  in  use  by  dif- 
ferent teachers  under  differing  conditions.  And  indeed  if 
one  looks  carefully  into  much  of  the  polemical  writing  on 
mechanics  teaching  it  is  seen  to  be  caused  less  by  funda- 
mental differences  of  principle  than  by  differences  of  method. 
It  is  still  more  necessary  to  clear  away  a  second  source  of 
unsatisfactory  discussion.  A  superficial  glance  through  the 
mass  of  controversial  writing  on  science  teaching  in  recent 
years  would  lead  one  to  suppose  there  was  a  sharp  division 
of  principle  between  those  who  believe  in  a  logically  ordered 
course  with  emphasis  on  what  one  may  call  the  instruc- 
tional method,  and  those  who  prefer  a  looser,  more  empir- 
ical, treatment  usually  embodying  heuristic  methods.  It 
would  be  possible,  however,  to  reconcile  many  of  the  com- 
batants if  they  could  be  persuaded  to  see  that  so  direct  an 
opposition  is  far  too  simple  a  statement  of  the  problem,  and 
that  each  may  be  partial  statements  of  the  real  solution. 
And  this  becomes  possible,  I  think,  if  once  the  disputants 
grant  the  importance  of  the  biogenetic  or  embryonic  prin- 
ciple as  applied  to  education — the  principle,  that  is  to  say, 

8  Reference  to  Plato,  Philebus :  G.  Boole,  The  Mathematical  Analysis  of 
Logic,  note  p.  7. 

10  E.  G.  A.  Holmes,  The  Montessori  System  of  Education,  English  Board 
of  Education  Pamphlet,  No.  24,  p.  3. 


THE  NEGLECT  OF  THE  WORK  OF  H.  GRASSMANN.        2Q 

that  the  development  of  the  individual  is  a  recapitulation 
of  the  development  of  the  race.  It  seems  strange  that  it 
should  be  necessary  at  this  stage  to  call  attention  to  a 
principle  so  well  known11  and  so  much  applied,  and  yet 
one  often  has  the  spectacle  of  a  successful  teacher  of  higher 
classes  urging  the  claims  of  logical  order  against  an  equally 
successful  empiricist  whose  experience  has  been  with 
younger  pupils.  The  truth  is,  of  course,  that  no  one 
method  is  applicable  to  all  ages.  If  the  biogenetic  law 
holds,  then  the  natural  principle  would  be  to  use,  in  general, 
modes  of  teaching  a  subject  similar  at  each  stage  to  those 
by  which  the  race  has  gathered  its  knowledge  of  that  sub- 
ject. In  mechanics  this  would  mean  that  a  more  rigidly 
logical  course  would  follow  empirical  experiments  and  the 
handling  of  simple  machines. 

We  will  now  pass  on  to  our  main  investigation  of  the 
factors  which  must  be  taken  into  account  in  avoiding  the 
creation  of  an  atmosphere  uncongenial  to  a  final  abstract 
analysis.  In  doing  so  I  will  indicate  what  appear  to 
be  the  general  principles  by  which  one  must  work  in 
giving  to  beginners  living  ideas  of  the  entities  of  mechan- 
ics, and  failure  to  comply  with  which  leads  to  the  produc- 
tion of  passively  instructed,  rather  than  of  irritable  and 
responsive,  organisms.  The  concepts  of  mechanics  are 
produced  from  the  raw  material  of  experience  by  the  proc- 
ess of  abstraction,  and  a  beginner  must  therefore  pass 
through  an  experimental  stage  before  he  is  introduced  to 
the  logically  defined  concepts  themselves.  In  fact  he  must 
first  use  and  handle  rough  ideas  and  thence  be  led  to  build 
up  the  more  rigidly  exact  definitions  of  them  for  himself. 
It  follows  from  this  that  any  information  we  can  glean 

11  It  is  a  very  remarkable  thing  that  De  Morgan  in  his  Study  and  Diffi- 
culties of  Mathematics,  first  published  in  1831,  or  28  years  before  the  Origin 
of  Species,  should  have  stated  this  principle  so  concisely  in  the  words  (p.  186) 
referring  to  discussions  of  first  principles :  "the  progress  of  nations  has  ex- 
hibited throughout  a  strong  resemblance  to  that  of  individuals." 


3O  THE  MONIST. 

about  the  actual  historical  process  by  which  man  came  to 
form  and  use  concepts  may  be  of  vital  importance  to  a 
teacher.  In  mechanics  particularly,  where  the  concepts 
are  less  obvious  than  in  geometry  (the  first  ideas  of  force, 
mass,  acceleration  and  energy,  regarded  however  not  as 
constructions  but  as  real  entities,  were  only  developed  to 
any  clearness  after  Galileo — that  is  at  quite  a  late  stage 
in  man's  history)  any  fogginess  about  their  nature  and 
use  means  endless  confusion;  and  that  accounts  for  most 
of  the  difficulties  commonly  experienced. 

It  was  Locke  who  first  plainly  showed  how  concepts 
arise  from  the  material  of  immediate  perception.  If  we 
think  of  the  flux  and  confusion  of  our  perceptions — the 
colors,  sounds,  smells,  sensations  of  touch,  at  any  instant 
we  find  our  attention  drawn  to  some  more  insistent  parts 
of  that  flux.  When  these  continually  recur  we  use  nouns, 
adjectives  and  verbs  to  identify  them.  Such  is  the  begin- 
ning of  the  formation  of  concepts.  These  are  regrouped  to 
form  other  concepts.  Thus  a  wide  experience  of  animals 
would  lead  us  to  group  them  and  to  speak,  for  example, 
of  a  class  "dog."  Once  classed  we  can  treat  all  instances 
as  having  the  general  properties  of  the  class.  The  prac- 
tical advantages  are  obvious.  "The  intellectual  life  of  man 
consists  almost  wholly  in  his  substitution  of  a  conceptual 
order  for  the  perceptual  order  in  which  his  experience 
originally  comes,"  says  William  James.12  Once  concepts 
are  formed  they  enable  us  to  handle  our  immediate  ex- 
perience with  greater  ease.  And  by  building  up  more  and 
more  complex  concepts  and  tracing  the  connections  be- 
tween them  we  create  our  mathematics  and  our  sciences. 

Even  animals  may  form  rough  concepts.18  A  dog  by 
experience  comes  to  know  the  difference  between  "man" 

12  Some  Problems  of  Philosophy,  p.  51. 

18  This  treatment  of  the  origination  of  concepts  is  founded  largely  on  that 
of  E.  Mach  in  his  chapter  on  Concepts  in  the  volume  Erkenntnis  und  Irrtum. 


THE  NEGLECT  OF  THE  WORK  OF  H.  GRASSMANN.        3! 

and  other  animals.  Furthermore  if  he  met  a  dummy  man 
he  would  soon  find  out  that  the  reactions  he  ordinarily 
associated  with  "man"  failed  to  be  reproduced,  and  so 
would  reject  that  experience  for  his  man-class.  In  a 
similar  way  man  must  have  formed  concepts  becoming 
more  and  more  complicated  but  more  firm  in  outline  as 
his  experience  became  richer.  But  it  is  to  be  noticed  that 
the  growth  of  concepts  in  a  body  of  experience  depends 
on  the  number  and  interest  of  our  observations  in  the 
region  concerned.  For  this  reason  interest  in,  and  con- 
sequent familiarization  with,  simple  machines  and  mechan- 
ical toys  may  well  be  the  child's  best  introduction  to  me- 
chanics. Model  monoplanes,  an  old  petrol  engine  from  a 
motor  cycle,  pumps,  a  screw,  levers,  a  jack,  Hero's  turbine 
model — all  these  can  easily  be  got  at;  few  young  children 
will  show  no  interest,  while  many  of  them  will  possess 
already  in  these  days  of  mechanical  toys  a  considerable 
knowledge  of  manipulation.  Simple  explanations  of  the 
working  of  such  apparatus  are  absorbed  with  astonishing 
readiness.  In  larger  schools  where  there  is  an  engineering 
workshop  this  method  of  introducing  young  boys  to  me- 
chanics by  way  of  machinery  has  been  tried  with  con- 
siderable success.  Knowledge  gets  picked  up  as  it  were 
"by  contact."  The  concepts  which  arise  at  this  stage  are 
necessarily  crude — general  ideas  of  force,  speed,  work  and 
friction;  this  latter  is,  of  course,  one  of  the  first  things 
to  notice — not  the  last  to  be  dealt  with  as  is  usually  the 
case.  Simple  as  these  considerations  are,  they  are  not  yet 
fully  appreciated.  The  London  Mathematical  Association's 
Report  on  the  Teaching  of  Elementary  Mechanics  sug- 
gested some  time  ago  that  the  phrase  "Mechanical  Advan- 
tage" be  replaced  by  "Force-Ratio."  For  beginners  neither 
of  these  is  intelligible ;  but  they  very  soon  know  "how  much 
stronger"  a  machine  makes  you.  And  that  conception  is 
quite  good  enough  for  them  to  use. 


32  THE  MONIST. 

In  introducing  simple  mechanical  concepts  to  beginners, 
therefore,  the  principle  to  use  is  that  the  concepts  must 
arise  naturally  from  experience  and  not  be  handed  out  as 
definitions.  Dictated  definitions  not  founded  on  sufficient 
knowledge  of  facts  are  flimsy  constructions  ready  to  fall 
at  the  first  breath  of  difficulty.  They  do  not  perform  that 
primary  function  of  concepts  of  helping  one  to  classify  and 
handle  facts,  because  the  facts  to  be  handled  are  not  in  the 
mind  when  the  concept  is  formulated.  "How  much  stronger 
a  machine  makes  you"  is  a  phrase  which  reminds  the 
hearer  at  once  of  the  assistance  it  gives  him  in  grouping 
machines  and  using  them  intelligently  for  different  pur- 
poses. A  note-book  definition  of  "mechanical  advantage" 
is  likely  to  present  another  arithmetical  puzzle  instead 
of  serving  to  remind  the  learner  of  the  solution  of  old 
ones.  The  principle  here  advocated  was  well  expressed 
in  the  discussion  on  mechanics  teaching  at  the  British 
Association  in  1905  by  the  president  of  the  section,  Pro- 
fessor Forsyth.  He  said,  "What  you  want  to  do  in  the 
first  instance  is  to  accustom  the  boys  to  the  ordinary  rela- 
tions of  bodies  and  of  their  properties,  and  afterward  you 
can  attempt  to  give  some  definitions  which  will  be  more  or 
less  accurate;  but  do  not  begin  with  the  definitions,  begin 
with  the  things  themselves."  And  the  philosophical  basis 
for  the  principle  is,  that  the  significance  of  concepts  is 
always  learned  from  their  relations  to  perceptual  particu- 
lars, their  utility  depending  on  the  power  they  give  us  of 
coordinating  perceptual  facts.  From  this  it  follows  further 
that  concepts  and  names  should  never  be  introduced  where 
there  is  no  direct  and  immediate  gain  in  so  doing.  Such 
terms  as  "centrifugal"  and  "centripetal"  forces,  and  the 
endless  discussion  to  which  they  lead,  are  thus  beside  the 
mark.  "Force  toward,  or  away  from,  the  center"  does 
all  that  is  necessary  without  introducing  new  words  of 
really  less  precision, 


THE  NEGLECT  OF  THE  WORK  OF  H.  GRASSMANN.        33 

It  should  be  noted  that  some  of  the  crude  concepts 
arrived  at  in  the  early  stages  are  really,  when  one  comes 
to  analyze  them,  very  complex,  and  Ostwald's  warning14 
against  the  error  of  supposing  that  the  less  simple  concepts 
have  always  been  reached  by  compounding  simple  ones 
has  application  here.  As  he  says,  complex  concepts  often 
in  origin  have  existed  first.  We  can  now  see  more  clearly 
why  the  teacher  of  mechanics  so  often  complains  of  the 
difficulty  of  giving  the  average  child  a  satisfactory  notion 
of  force.15  The  difficulty  is  largely  due  to  the  teacher  who 
knows  the  concept  to  be  complicated,  and  seeks  to  define  it 
in  terms  of  mass-acceleration — thus  involving  two  more 
concepts,  one  of  which  (mass)  is  at  least  as  difficult  to 
understand  as  force.  A  rough  idea  of  force,  considered 
simply  as  a  "push"  or  a  "pull,"  can  be  assimilated  at  a  very 
early  stage ;  that  of  mass-acceleration  must  come  very  much 
later. 

The  bearing  of  this  preliminary  stage  in  the  formation 
of  concepts  on  our  main  thesis  may  now  be  traced.  It  is 
quite  evident  that  the  individual  has  very  limited  powers 
of  absorbing  the  logically  ordered  account  of  a  science  in 
which  stress  is  laid  on  abstract  notions  before  such  notions 
have  grown  up  naturally  by  use.  Now  this  difficult  step 
for  the  beginner  from  the  perceptual  to  the  conceptual  is 
very  similar  to  that  which  leads  from  ordinary  mechanics 
to  such  a  treatment  of  the  subject  as  that  of  Grassmann. 
Both  lead  into  regions  of  greater  abstraction.  In  the  latter 
case  we  can  get  rid  of  concepts  in  so  far  as  they  relate  to 
the  existent,  and  reach  a  statement  of  mechanical  principles 
in  terms  of  a  generalized  form-theory.  We  may  illustrate, 
roughly,  the  meaning  of  this  by  the  following  analogue. 
At  different  stages  in  the  history  of  physics  various  the- 

14  Ostwald,  Natural  Philosophy,  p.  20. 

15  Cf.  C.  Godfrey,  Brit.  Association  Report  on  Mechnics  Teaching,  p.  41. 


34  THE  MONIST. 

ories  of  light  have  been  held.  The  concepts  used  in  these 
theories  (corpuscle,  elastic-solid  ether,  electro-magnetic 
medium)  have  possessed  widely  different  "qualities";  but 
the  equations  expressing  the  relation  between  the  concep- 
tual elements  have  throughout  possessed  similarity  of  form. 
A  science  of  form  would  hence  lay  emphasis  on  the  in- 
variant relations,  refine  away  the  particular  concepts,  and 
leave  a  much  more  abstract  and  generalized  science. 

But  if  racial  development  is  in  the  main  similar  to  the 
progress  of  the  individual  this  will  explain  the  great  diffi- 
culty experienced  by  whole  generations  of  mathematicians 
in  understanding  work  of  the  type  of  Grassmann's. 

Furthermore,  it  is  at  this  point  that  defective  scientific 
training  looms  into  importance.  For  unless  great  care 
has  been  taken  in  avoiding  the  too  early  definition  of  con- 
cepts, a  rigid  view  of  them  is  promulgated.  The  older 
dogmatic  and  orderly  methods  of  teaching  tended  inevi- 
tably to  this.  The  consequence  was  that  when  the  time 
came  for  polishing  and  development  of  the  concepts  ob- 
tained, and  for  the  deliberate  building  up  of  more  com- 
plex ones — it  was  found  that  the  capacity  for  subtle  gen- 
eralized views  had  been  destroyed.  A  mind  forced  into 
passivity  and  filled  with  inert  knowledge  cannot  suddenly 
be  brought  to  discard  it  in  response  to  the  stimulus  of  a 
tentative  generalization.  To  take  a  simple  example,  the 
idea  of  a  new  kind  of  addition,  applicable  to  vectors,  shocks 
and  confuses  a  pupil  who  has  been  dogmatically  instructed 
in  algebra  as  though  it  were  a  sacred  rite.  As  with  the 
child  under  such  a  system,  so  with  the  generation  of  which 
he  forms  a  part.  Jahnke  states  that  many  mathematicians 
were  put  off  by  meeting  in  Grassmann's  work  a  product 
which  equals  zero  without  either  factor  doing  so.  Formal 
logical  development  often  leads  to  conclusions  which  are 
not  capable  of  any  mental  image.16  Such  abstractions  are 

"  Cf.  F.  Klein,  The  Evanston  Colloquium,  Lecture  6. 


THE  NEGLECT  OF  THE  WORK  OF  H.  GRASSMANN.        35 

out  of  reach  of  those  who  have  never  been  freed  from  the 
confines  of  the  existent  world. 

Cajori17  in  a  notice  in  1874  of  the  publication  called 
The  Analyst,  Des  Moines,  Iowa,  said  that  it  bore  evidence 
of  an  approaching  departure  from  antiquated  views  and 
methods,  of  a  tendency  among  teachers  to  look  into  the 
history  and  philosophy  of  mathematics.  My  thesis  is  that 
such  a  movement,  which  certainly  has  not  yet  been  realized, 
would  remove  the  main  cause  of  the  neglect  of  Hermann 
Grassmann's  work,  which  even  in  these  days  is  often 
granted  the  kind  of  recognition  accorded  to  certain  literary 
classics,  which  are  famous  but  never  read.  Perhaps  it  is 
an  earnest  of  the  future  that  the  copy  of  The  Analyst  re- 
ferred to  by  Cajori  contained  a  brief  account18  of  the  essen- 
tial features  of  Grassmann's  Ausdehnungslehre. 

A.  E.  HEATH. 

BEDALES,  PETERSFIELD,  ENGLAND. 

17  Teaching  and  History  of  Mathematics  in  the  United  States. 

18  Translated  by  W.  W.  Beman  of  the  University  of  Michigan. 


THE  GEOMETRICAL  ANALYSIS  OF  GRASSMANN 
AND  ITS  CONNECTION  WITH  LEIB- 
NIZ'S CHARACTERISTIC. 

§  I- 

BY  a  curious  turn  of  fate  Grassmann  wrote,  in  the 
introduction  to  his  "Geometrical  Analysis,"  concern- 
ing Leibniz's  early  work  on  the  same  subject,  words  which 
were  to  apply  with  prophetic  force  to  his  own  Ausdeh- 
nungslehre.  "When  the  special  power  of  a  genius ....  is  so 
revealed  that  he  is  able  to  grasp  and  extend  the  ideas 
toward  which  the  development  of  his  time  is  directed,  and 
so  appears  representative  of  his  period,  then  that  power 
shows  itself  still  more  remarkable  when  it  can  seize  ideas 
in  those  realms  of  thought  in  advance  of  their  day  and 
forecast  for  hundreds  of  years  the  line  of  their  develop- 
ment. While  ideas  of  the  first  kind  are  often  developed 
simultaneously  by  the  outstanding  spirits  of  the  age,  as 
when  both  Leibniz  and  Newton  founded  the  differential 
calculus — a  certain  stage  of  fruition  being  reached — ideas 
of  the  latter  kind  appeared  as  the  special  characteristic 
of  the  individual,  the  innermost  revelations  of  his  mind 
into  which  only  a  few  elect  contemporaries  could  enter 
and  have  a  foreshadowing  of  the  developments  which  were 
to  spread  from  them  in  the  future.  While  the  first  received 
great  applause  and  aroused  movement  in  their  own  day, 


THE  GEOMETRICAL  ANALYSIS  OF  GRASSMANN.  37 

because  they  represent  the  summit  of  the  epoch,  the  others 
for  the  most  part  fall  without  effect  in  the  contemporary 
period  since  they  are  only  understood  by  a  few,  and  then 
only  partially.  Often  afterward  does  such  thought  be- 
come the  seed  of  a  rich  harvest.  That  this  great  idea  of 
Leibniz — namely,  the  idea  of  a  true  geometrical  analysis — 
belongs  to  this  preparatory  and,  as  it  were,  prophetic  class 
cannot  be  doubted  for  a  moment.  It  has  also  shared  the 
fate  of  such.  Indeed  by  a  special  ill  favor  of  circumstances 
it  has  remained  hidden  far  beyond  the  time  when  it  might 
have  had  a  powerful  influence.  For  even  before  it  was 
brought  out  of  its  hiding  place  by  Uylenbroek,  paths 
toward  a  similar  analysis  had  been  made  in  other  ways." 

At  the  time  when  these  words  were  written  Grassmann 
could  have  had  no  idea  of  the  disappointment  which  was 
to  come  to  him  in  the  neglect  of  his  own  work.  The  first 
edition  of  the  Ausdehnungslehre,  or  theory  of  extended 
magnitudes,  had  been  published  in  1844  and  had  received 
no  attention  from  mathematicians  with  the  exception  of  a 
few  individuals.  Grassmann,  however,  believed  that  rec- 
ognition was  only  a  matter  of  time  and  sought  to  bring 
out  the  importance  and  applicability  of  the  new  analysis. 
For  the  year  1845  (but  extended  to  1846  to  coincide  with 
the  two  hundredth  anniversary  of  Leibniz's  birth)  the 
Jablonowski  Society  of  Leipsic  set  a  prize  essay  demanding 
the  restatement  or  further  development  of  the  geometrical 
calculus  discovered  by  Leibniz  or  the  setting  up  of  a  simi- 
lar calculus;  and  the  award  was  made  to  Grassmann  for 
the  essay,  printed  in  1847,  from  which  I  made  the  above 
quotation.  This  was  the  first  and  the  only  acknowledg- 
ment of  the  value  of  his  work  which  he  received  from 
mathematicians  until  long  after  many  of  the  ideas  he 
formulated  had  been  reached  and  applied  by  other  methods 
and  other  thinkers. 

I  have  laid  stress  on  the  similarity  of  treatment  meted 


38  THE  MONIST. 

out  to  the  fundamentally  important  work  of  the  two  men 
because  I  believe  that  in  some  elements  of  its  explanation 
lies  the  clue  to  unravel  the  difficulties  of  their  subject 
matter  and  connection  with  each  other.  The  more  general 
aims  of  both  Leibniz  and  Grassmann  were  the  same — the 
setting  up  of  a  convenient  calculus  or  art  of  manipulating 
signs  by  fixed  rules,  and  of  deducing  therefrom  true  propo- 
sitions for  the  things  represented  by  the  signs,  for  use  as 
a  generalized  mathematics.  In  each  case  their  geometrical 
calculus  was  a  particular  application  to  geometry  of  a 
wider  calculus  for  which  each  desired  more  than  mere 
applicability  to  mathematics. 

In  a  letter  to  Arnauld,  dated  January  14,  1688,  Leibniz 
writes1:  "Some  day,  if  I  find  leisure,  I  hope  to  write  out 
my  meditations  upon  the  general  characteristic  or  method 
of  universal  calculus,  which  should  be  of  service  in  the 
other  sciences  as  well  as  in  mathematics.  I  have  defini- 
tions, axioms,  and  very  remarkable  theorems  and  prob- 
lems in  regard  to  coincidence,  determination,  similitude, 
relation  in  general,  power  or  cause,  and  substance,  and 
everywhere  I  advance  with  symbols  in  a  precise  and  strict 
manner  as  in  algebra.  I  have  made  some  applications  of 
it  in  jurisprudence."  Similarly  Grassmann2  says:  "By  a 
general  science  of  symbols  (Formenlehre)  we  understand 
that  body  of  truths  which  apply  alike  to  every  branch  of 
mathematics,  and  which  presuppose  only  the  universal  con- 
cepts of  similarity  and  difference,  connection  and  disjunc- 
tion." The  symbols  are  made  so  general  as  to  be  applicable 
to  both  logic3  and  mathematics,  although  in  the  Ausdeh- 

1  George  R.   Montgomery   (trans.),  Leibniz:  Discourse  on  Metaphysics, 
Correspondence  with  Arnauld,  and  Monadology,  p.  241. 

2  Ausdehnungslehre  of  1844,  p.  2. 

8  The  application  of  such  a  general  science  of  symbols  to  formal  logic 
was  made  by  both  H.  Grassmann  and  his  brother  Robert. 


THE  GEOMETRICAL  ANALYSIS  OF  GRASSMANN.  39 

nungslehre  they  are  only  applied  to  the  domain  of  mathe- 
matics.4 

It  is  clear  that  both  Leibniz  and  Grassmann,  but  espe- 
cially the  former,  claimed  great  scope  for  their  calculus, 
a  fact  which  tended  to  make  their  writings  generalized 
and  difficult  to  understand.  In  the  preface  to  his  Universal 
Algebra  (1898)  Professor  Whitehead  expresses  his  belief 
that  lack  of  unity  in  presentation  (which  of  course  would 
be  the  tendency  in  dealing  with  a  method  applicable  to 
many  fields)  discourages  attention  to  such  a  subject.  But 
that  is  not  all.  A  new  mathematical  method,  to  make 
itself  known,  has  to  appeal  in  the  main  to  mathematicians 
and  not  to  philosophers.  So  that  a  wide  and  philosophical 
treatment  is  apt  to  be  discounted  by  the  ordinary  man  who 
thinks  logic  can  be  made  to  prove  anything. 

§2. 

Before  we  condemn  this  attitude  we  must  first  of  all 
inquire  as  to  what  exactly  the  common  man  means  by  the 
dangers  of  logic.  What  he  really  fears  is  not  logic  but 
fallacy.  Without  realizing  it  he  distrusts  a  mechanical 
dexterity  in  reasoning  because  the  attainment  of  truth  de- 
pends not  only  on  a  facility  in  manipulating  logical  proc- 
esses but  also  on  the  sifting  of  first  principles.  When 
Leibniz  claims  for  his  char  act  eristic  a  universalis  or  "uni- 
versal mathematics,"  the  germ  of  which  he  produced  in 
his  De  arte  combinatoria  published  when  he  was  twenty, 
that  " .  . .  .  there  would  be  no  more  need  of  disputation 
between  two  philosophers  than  between  two  accountants. 
For  it  would  suffice  to  take  their  pencils  in  their  hands,  to 
sit  down  to  their  slates,  and  to  say  to  each  other  (with  a 

*In  the  Ausdehnungslehre ,  however,  are  expressions  directly  applicable 
to  logic,  e.  g.,  there  is  the  generalized  expression  for  the  result  of  division 
C+  O/B  where  O/B  is  an  indefinite  form  (p.  213) — an  anticipation  of  Boole's 
use  of  p/O  to  symbolize  perfect  indefiniteness,  as  pointed  out  by  Venn  in  his 
Symbolic  Logic,  note  p.  268  (2d  ed.). 


40  THE  MONIST. 

friend  as  witness,  if  they  liked) :  Let  us  calculate" — he  is 
running  counter  to  the  plain  man's  knowledge  that  there 
are  two  parts  of  a  logical  process,  the  first  the  choosing  of 
an  assumption,  the  second  the  arguing  upon  it. 

Now  Leibniz  realized  of  course  that  premises  are  re-- 
quired first,  but  he  thought  they  could  be  obtained  very  sim- 
ply. By  analyzing  any  notion  until  it  was  simple  he  thought 
that  all  axioms  or  assumptions  followed  as  identical  propo- 
sitions. Thus  he  was  led,  by  his  view  of  ideas,  to  believe 
that  even  the  axioms  of  Euclid  could  be  proved.  So  in  his 
New  Essays,  "I  would  have  people  seek  even  the  demon- 
stration of  the  axioms  of  Euclid ....  And  when  I  am  asked 
the  means  of  knowing  and  examining  innate  principles,  I 
reply.  . .  .we  must  try  to  reduce  them  to  first  principles, 
i.  e.,  to  axioms  which  are  identical,  or  immediate  by  means 
of  definitions  which  are  nothing  but  a  distinct  exposition 
of  ideas."  This  is  connected  with  his  view  that  all  our 
ideas  are  composed  of  a  very  small  number  of  simple  ideas, 
which  together  form  an  alphabet  of  human  thoughts.  But, 
as  Couturat  remarks,5  there  are  many  more  simple  ideas 
than  Leibniz  believed;  and  furthermore  there  is  no  great 
philosophical  interest  in  such.  "An  idea  which  can  be 
defined  or  a  proposition  which  can  be  proved,  is  only  of 
subordinate  philosophical  interest."6  It  is  precisely  the 
business  of  philosophy  to  deal  with  the  primitive,  intuitive 
assumptions  on  which  any  calculus  must  be  based. 

So  the  plain  man  is  to  some  extent  justified  in  his  mis- 
trust of  the  uncritical  application  of  a  calculus. 

§3. 

It  is  very  necessary,  however,  to  see  exactly  what  is,  and 
what  is  not,  here  granted  to  the  plain  man.  It  is  true  that 
in  using  a  calculus  we  must  be  careful  not  to  over-empha- 

B  L.  Couturat,  La  logique  de  Leibniz,  p.  431. 
8  B.  Russell,  The  Philosophy  of  Leibniz,  p.  431. 


THE  GEOMETRICAL  ANALYSIS  OF  GRASSMANN.  4! 

size  the  results  at  the  risk  of  forgetting  the  premises  from 
which  they  have  been  obtained.  But  that  being  admitted, 
thus  making  the  final  development  of  the  universal  char- 
acteristic a  matter  not  of  philosophy  but  of  a  sort  of  gener- 
alized mathematics  of  which  formal  logic7  and  geometry 
are  special  cases,  it  does  not  follow  that  there  must  be 
limits  to  the  applicability  of  the  calculus  in  these  spheres. 
Yet  that  is  what  the  modern  representative  of  our  plain 
man  asserts.  His  criticism  of  a  logical  calculus  has  put  on 
a  more  philosophical  form,  but  remains  essentially  the 
same.  Henri  Poincare  may  justly,  I  think,  be  taken  as 
such  a  representative.  For  he  says,  "I  appeal  only  to 
unprejudiced  people  of  common  sense.  . .  .they  [the  logis- 
ticians]  have  shown  that  mathematics  is  entirely  reducible 
to  logic,  and  that  intuition  plays  no  part  in  it  whatever."8 
This  belief  led  Poincare  to  the  view  that,  since  he  knew 
from  his  own  experience  as  a  mathematician  of  great  in- 
sight the  important  part  intuition  plays  in  mathematical 
discovery,  therefore  the  nature  of  mathematics  cannot  be 
logical. 

This  reasoning  is  founded  on  a  very  common  fallacy 
which  I  will  call  the  genetic  error — the  error,  namely, 
which  lies  in  the  assumption  that  the  origin  of  a  thing  in 
some  way  determines  its  nature.9  If  this  assumption  is 
made  it  follows  that  since  intuition  plays  a  part  in  dis- 


7  Leibniz  himself  foresaw  this  development  carried  out  by  Boole,  Peano, 
Frege,  Whitehead  and  Russell  and  their  school  of  symbolic  logicians.    In  fact 
he  made  discoveries  in  this  field  but  did  not  publish  them  because  they  contra- 
dicted certain  points  in  the  traditional  doctrine  of  the  syllogism.     In  some 
points  he  even  advanced  beyond  Boole  (See  Couturat,  op.  cit.,  p.  386). 

8  Science  et  Methode,  p.  155;  cf.  also  C.  J.  Keyser  in  Bull.  Amer.  Math. 
Soc.,  Jan.  1907,  pp.  197,  198. 

8  This  error  has  been  very  common  in  philosophy.  It  underlies  much 
argument  against  rationalism,  denying  that  knowledge  reached  empirically 
can  be  anything  other  than  empirical.  (Cf.  Leibniz,  New  Essays,  IV,  1  §9, 
against  Locke.)  It  is  at  the  basis  of  many  criticisms  leveled  against  any 
generalization  of  number,  since  the  idea  of  number  arose  from  perceptual  ex- 
perience. It  vitiates  pragmatism,  which  inquires  into  the  causes  of  our  judging 
things  to  be  true  in  order  to  get  at  the  nature  of  truth.  (See  B.  Russell,  Philo- 
sophical Essays,  p.  110.) 


42  THE  MONIST. 

covery,  the  nature  of  mathematics  cannot  be  purely  formal, 
and  therefore  it  cannot  be  expressed  in  terms  of  symbolic 
logic.  Now  all  such  references  to  the  origins  of  mathe- 
matics are  irrelevant.  Once  the  premises  have  been  made, 
and  that  is  where  intuition  comes  in,  symbolic  logic  is 
merely  "an  instrument  for  economizing  the  exertion  of 
intelligence."1  The  mind,  being  relieved  of  unnecessary 
work  by  a  good  symbolism,  is  set  free  to  attack  more  diffi- 
cult problems;  for  as  Professor  Whitehead  says,11  "Opera- 
tions of  thought  are  like  cavalry  charges  in  a  battle — they 
are  strictly  limited  in  number."  Nor  is  that  the  only  ad- 
vantage of  this  modern  development  of  Leibniz's  universal 
mathematics.  It  "has  the  effect  of  enlarging  our  abstract 
imagination  and  providing  an  infinite  number  of  possible 
hypotheses  to  be  applied  in  the  analysis  of  any  complex 
fact."1  And  so  it  lends  itself  to  the  production  of  just 
such  novel  fundamental  hypotheses  as  are  needed  in  sub- 
jects like  the  dynamics  of  relativity. 

So  finally,  we  must  say  of  the  symbols  of  a  universal 
calculus  what  Hobbes  said  of  words,  "They  are  wise  men's 
counters,  they  do  but  reckon  by  them;  but  they  are  the 
money  of  fools."  Yet  it  must  be  recognized  that  when  it 
is  confined  to  dealing  with  mathematics  in  its  widest  sense 
(taken  to  include  formal  logic), — within  the  limits  im- 
posed on  his  own  calculus  by  Grassmann,  in  fact, — it  serves 
as  a  powerful  and  legitimate  tool. 

§4. 

This  discussion  of  the  neglect  and  mistrust  of  mathe- 
maticians for  the  generalized  calculus  of  both  Leibniz  and 
Grassmann  has,  I  hope,  shown  what  the  nature  of  such  a 

10  W.  E.  Johnson  in  Mind,  N.  S.,  Vol.  I,  pp.  3,  5.    Cf.  Stout,  "Thought  and 
Language,"  Mind,  April,  1891. 

11  An  Introduction  to  Mathematics,  Home  Univ.  Library,  p.  59.    See  also 
P.  E.  B.  Jourdain  in  The  Monist,  Jan.  1914,  p.  141. 

12  B.  Russell,  Our  Knowledge  of  the  External  World,  pp.  58,  242. 


THE  GEOMETRICAL  ANALYSIS  OF  GRASSMANN.          43 

calculus  is.  Moreover,  it  accounts  for  the  long  period 
which  elapsed  before  their  fruitful  application  of  these 
methods  of  calculation  to  special  fields  obtained  the  notice 
they  deserved. 

The  particular  application  we  are  here  concerned  with 
is  that  to  geometry.  In  a  letter  to  Huygens  of  September 
8,  1679,  Leibniz  complained  that  he  was  not  satisfied  with 
the  algebraic  methods,  and  adds:  "I  believe  that  we  must 
have  still  another  properly  linear  geometrical  analysis, 
which  directly  expresses  situm  as  algebra  expresses  mag- 
nitudinem.  And  I  believe  I  have  the  means  for  it,  and  that 
one  could  represent  figures  and  even  machines  and  move- 
ments in  symbols,  as  algebra  represents  number  or  magni- 
tude; I  am  sending  you  an  essay  which  seems  to  me 
notable."  This  essay  contained  an  account  of  his  geo- 
metrical calculus  in  which  the  relative  position  of  points 
is  denoted  by  simple  symbols  and  fixed  without  the  help  of 
the  magnitude  of  lines  and  angles.  It  differs  therefore  from 
ordinary  algebraic  analytical  geometry.  The  further  de- 
velopment of  this  calculus  was  the  subject  of  Grassmann's 
Geometrlsche  Analyse13  which  we  have  already  noted  as 
being  crowned  by  the  Jablonowski  Society.  This  was  done 
on  the  recommendation  of  Mobius,  who  found  in  Grass- 
mann's essay  a  generalization  and  extension  of  his  own 
barycentric  calculus. 

We  will  now  consider  the  geometrical  calculus  of  Leib- 
niz with  a  view  to  discovering  if  Grassmann's  develop- 
ment of  it  has  fulfilled  in  any  way  Leibniz's  hopes  of  its 
ultimate  importance. 

§5. 

The  letters  and  papers  of  Leibniz  in  which  he  deals 
with  his  project  of  a  geometrical  calculus  are  many,  and 

13  This  treatise  develops  some  of  the  subjects  which  Grassmann  had  in- 
tended for  a  second  part  of  the  1844  Ausdehnungslehre,  which  was  never 
written. 


44  THE  MONIST. 

spread  over  a  considerable  period  of  time.14  The  most  im- 
portant is  the  Char  act  eristic  a  geometrica,  a  sketch  of  the 
notion  which  he  made  for  fear  it  should  be  lost  if  he  found 
himself  unable  to  develop  it.  The  essay  enclosed  in  the 
letter  to  Huygens  in  1679  was  an  extract  from  this.  From 
these  writings  it  seems  clear  that  the  starting  point  was 
his  conviction  of  the  imperfection  of  algebra  as  the  logical 
instrument  of  geometry.  Thus,  "Algebra  itself  is  not  the 
true  characteristic  of  geometry,  but  quite  another  must  be 
found,  which  I  am  certain  would  be  more  useful  than 
algebra  for  the  use  of  geometry  in  the  mechanical  sciences. 
And  I  wonder  that  this  has  hitherto  been  remarked  by  no 
one.  For  almost  all  men  hold  algebra  to  be  the  true  math- 
ematical art  of  discovery,  and  as  long  as  they  labor  under 
this  prejudice,  they  will  never  find  the  true  characters  of 
other  sciences."  It  must  be  n'oted  that  algebra  is  here  used 
by  Leibniz  in  its  ordinary  sense,  not  as  a  general  term  for 
any  calculus. 

He  saw  that  analytical  geometry  only  expressed  geo- 
metrical facts  in  a  complicated  and  roundabout  manner. 
A  figure  such  as  the  circle  is  not  defined  by  its  internal  re- 
lations, but  by  reference  to  its  relations  to  arbitrary  coor- 
dinates. So  a  set  of  magnitudes  foreign  to  the  figure  are 
introduced  and  obscure  the  purely  geometrical  relation- 
ships. Further,  to  reduce  relations  of  position  to  relations 
of  size  presupposes  Thales's  theorem  about  similar  tri- 
angles and  the  theorem  of  Pythagoras.15  In  other  words 
analytical  geometry  is  thus  made  dependent  on  synthetic. 
The  analysis  not  being  pushed  far  enough,  it  has  not  the 
logical  perfection  which  belongs  to  a  purely  rational  sci- 
ence.16 He  realized  the  want  of  rigor  and  generality  of 

14  An  interesting  bibliography  of  them  together  with  an  account  of  the 
main  ideas  which  inspired  and  directed  his  search  for  a  geometrical  charac- 
teristic is  given  in  Couturat,  La  logique  de  Leibniz,  1901. 

16  Characteristica  geometrica,  §  5. 

18  Cf.  his  letter  to  Bodenhausen. 


THE  GEOMETRICAL  ANALYSIS  OF  GRASSMANN.  45 

intuitive  methods,  but  dreamed  of  a  method  which  would 
be  completely  analytical  and  rational  while  still  possessing 
all  the  advantages  of  a  synthetic  method. 

In  this  his  aim  was  similar  to  that  which  he  had  in 
mind  for  his  universal  characteristic,  which  was  to  be  a 
logical  calculus  replacing  concepts  by  combinations  of 
signs,  and  which  furthermore  was  not  merely  to  furnish 
demonstrations  of  propositions  but  to  be  the  means  of  dis- 
covering new  ones.  So,  in  like  manner,  his  geometrical 
calculus  was  to  combine  analysis  with  guidance  of  the  in- 
tuition.17 A  fusion  of  analysis  and  synthesis  being  made, 
the  divorce  between  calculation  and  construction  would 
disappear.  "This  new  characteristic.  . .  .will  not  fail  to 
give  at  the  same  time  the  solution,  construction,  and  geo- 
metrical demonstration,  the  whole  in  a  natural  manner  and 
by  an  analysis."1  It  is  clear  that  the  final  goal  was  a 
science  of  form  of  very  wide  application.19  This  aim  we 
must  distinguish  carefully  from  the  manner  in  which  he 
attempted  to  realize  it. 

As  Grassmann  points  out  in  the  introduction  to  his 
"Geometrical  Analysis,"  this  distinction  between  the  dis- 
tant goal  and  his  attempt  toward  a  new  characteristic 
which  he  connects  with  it  to  render  the  thought  more  real- 
izable, is  recognized  fully  by  Leibniz.  Although  the  char- 
acteristic he  provided  will  be  seen  to  be  only  a  small  first 
step  toward  the  goal  he  had  set  himself,  yet  he  had  esti- 
mated the  essential  advantages  of  a  final  geometrical  anal- 
ysis to  an  extraordinary  completeness.  Grassmann  says: 
"Just  this  eminent  talent  of  Leibniz  of  being  able  to  fore- 
see in  presentiment  a  whole  series  of  developments  without 
being  able  to  work  it  out  and  without  dismembering  and 

17  Leibniz  conjectured  that  the  ancients  had  some  natural  and  spontaneous 
analysis  of  this  kind  resting  on  the  abstract  relations  of  figures,  which  under- 
lay and  helped  their  synthetic  methods.     (De  analyst  situs.) 

18  Letter  to  Huygens. 

19  Ibid.    "I  believe  that  one  could  handle  mechanics  by  these  means  almost 
like  geometry." 


46  THE  MONIST. 

dissecting  it,  yet  to  make  it  present  to  himself  with  pro- 
phetic mind  and  to  recognize  the  importance  of  its  conse- 
quences— it  is  just  this  talent  which  led  him  to  such  great 
discoveries  in  almost  all  domains  of  knowledge." 

§6. 

Leibniz  founds  his  fundamental  definitions  on  con- 
gruence, which  means  the  possibility  of  coincidence.  He 
represents  points  whose  positions  are  known  by  the  first 
letters  of  the  alphabet,  and  those  which  are  unknown  or 
variable  by  the  last  letters.  Any  two  combinations  of  cor- 
responding points  are  said  to  be  congruent  if  both  can  be 
brought  to  coincide  without  the  mutual  position  of  the 
points  being  changed  in  either  of  the  two  combinations; 
so  that  every  point  of  one  combination  covers  a  correspond- 
ing point  in  the  other.  Congruence  (geometrical  equality) 
is  a  union  of  two  relations — similarity  and  equality  (quan- 
titative equivalence). 

All  points  are  equal  and  similar,  so  all  points  are  con- 
gruent.20 Hence  if  we  use  =  for  congruence,  the  ex- 
pression a  -  x,  where  a  is  fixed  and  x  is  variable,  is  a  defi- 
nition of  space. 

It  must  be  noticed  that  in  defining  figures  by  congru- 
ence the  axiom  of  congruence  or  free  mobility21  must  be 
postulated.  If  we  do  this,  ax  =  be  represents  a  sphere  of 
center  a  and  radius  be. 

Also,  ax  =  bx  represents  a  plane  which  bisects  ab  per- 
pendicularly. 

The  above  can  be  taken  as  the  definition  of  the  sphere 
and  the  plane  respectively.  Again  ax  =  bx  =cx  gives  the 
locus  of  the  center  of  all  spheres  which  pass  through  a,  b, 
c;  and  so  it  is  a  straight  line. 

If  ax^ac  and    bx  =  bc 

they  together  give  the  common  trace  of  two  spheres. 

20  Characteristica  geometrica. 

91  See  B.  Russell,  Foundations  of  Geometry,  Cambridge,  1897. 


THE  GEOMETRICAL  ANALYSIS  OF  GRASSMANN.  47 

Combined  they  are  written  abx  =  abc.  This  therefore 
represents  the  locus  of  points  whose  distances  from  the 
points  a,  b  are  the  same  as  the  distances  of  c  from  a,  b. 
That  is,  it  is  a  circle. 

The  economical  nature  of  the  symbolism  is  shown  by 
the  fact  that  if  we  take  this  as  a  definition  of  the  circle,  it 
does  not  imply  the  idea  of  the  straight  line  or  the  plane; 
nor  does  it  require  (as  the  circle  defined  by  an  algebraic 
equation)  that  the  center  of  the  circle  must  be  known. 

As  an  example  of  a  proof  consider  the  proposition 
that  the  intersection  of  two  planes  is  a  straight  line. 
Let      ay  =  by  be  one  plane 
and     ay  =  cy  be  the  other. 

Then  ay  =  by  =  cy,  and  this  we  saw  above  to  be 
the  form  of  the  congruences  representing  a  straight  line. 

In  these  examples  is  a  faint  foreshadowing  of  the  side 
by  side  development  of  construction,  proof  and  analysis. 
And  since  all  kinds  of  spatial  relationships  can  be  devel- 
oped from  the  line  and  the  sphere,  the  method  is  capable 
of  wide  extension. 

§7. 

There  are  several  obvious  defects  in  it,  however.  .These 
appear  at  once  if  we  attempt  by  means  of  it  to  solve  the 
fundamental  problem  in  geometry  of  finding  the  expres- 
sion for  a  straight  line  passing  through  two  given  points. 
Leibniz  had  previously  attacked  the  problem  only  to  find 
himself  involved  in  difficulties.22 

Grassmann's  treatment  is  as  follows :  We  saw  above  the 
expression  for  a  straight  line  was 

ax  =  bx  -  ex. 

If  we  now  take  three  auxiliary  points,  a! ' ,  b' ',  c',  which 
are  not  in  a  straight  line,  and  write 

**  Couturat  gives  a  clear  account  of  this,  op.  cit.,  pp.  420-427. 


THE  MONIST. 

'x  =  b'x  =  c'x 
a'a  =  b'a  =  c'a 
a'b  =  b'b  =  c'b, 

then  together  these  congruences  represent  the  required 
straight  line  through  a,  b,  as  the  locus  of  x. 
Combining  the  last  two  we  get 

$    a'x=   b'x^c'x 
\  aba'  =  abb'  =  abc'. 

This  then  expresses  that  the  auxiliary  points  lie  on  the 
circle  the  plane  of  which  is  cut  at  its  center  by  the  line  ab 
at  right  angles. 

If  this  expression  is  to  have  the  necessary  simplicity, 
it  must  be  possible  to  eliminate  the  arbitrary  auxiliary 
points  which  have  nothing  to  do  with  the  nature  of  the 
problem,  and  to  combine  the  group  of  formulas  into  one. 
That  being  impossible,  the  characteristic  has  failed  to 
serve  its  purpose. 

Indeed  the  failure  of  the  method  followed  at  once  from 
the  choice  of  congruence  as  the  fundamental  relation.  For, 
as  we  have  seen,  this  complex  relation  contains  a  quanti- 
tative element,  and  so  prevents  any  freeing  of  geometry 
from  considerations  of  magnitude.  In  fact,  as  the  above 
expression  for  the  line  through  ab  shows,  we  are  still  left 
with  arbitrary  coordinates.  Further,  in  this  system  there 
is  also  ambiguity,  as  Couturat  has  shown.23  In  other  words 
the  analysis  had  not  gone  far  enough.  If  what  remained 
of  magnitude  had  been  eliminated — not  merely  by  taking 
the  relation  of  similarity,  for  Leibniz  had  himself  shown 
that  to  imply  metrical  relations24 — but  by  reducing  figures 
to  their  projective  properties  and  relations,  at  least  a  real 
geometry  of  position25  would  have  followed.  But  such  a 
projective  geometry,  while  satisfying  Leibniz's  desire  to 

28  Ibid.,  p.  428,  note  2. 

24  "Elementa  Nova  Matheseos  Universalis." 

28  Developed  by  Staudt,  Geometric  der  Lage,  1847. 


THE  GEOMETRICAL  ANALYSIS  OF  GRASSMANN.  49 

eliminate  algebraic  methods  from  geometry,  would  not 
have  been  a  geometrical  calculus  with  points  as  elements. 
Nor  could  it  have  had  the  wide  application  which  he  sought 
for  in  his  calculus ;  for  if  it  was  to  be  applicable  to  mechan- 
ics and  physics,  it  must  at  some  point  have  been  susceptible 
of  metrical  development. 

Now,  throughout  our  discussion  we  have  seen  that 
Leibniz  was  seeking  for  a  characteristic  particularly  ap- 
plicable to  geometry  but  akin  to  his  universal  character- 
istic. At  the  end  of  the  letter  to  Huygens  he  says:  "I 
believe  it  is  possible  to  extend  the  characteristic  to  things 
which  are  not  subject  to  imagination."  In  other  words 
he  was  seeking  a  formal  calculus,  an  abstract  mathematics 
lying  at  the  base  of  geometry  and  applicable  not  only  to  it 
but  also  to  logic.  Now  Grassmann  had  already  developed 
such  a  science  of  form  in  his  Ausdehnungslehre  of  1844. 
So  when  the  Jablonowski  Society  announced  the  subject 
of  their  prize  essay  he  took  the  opportunity  of  expounding 
his  science  of  extensive  magnitudes,  not  as  he  had  orig- 
inally derived  it,  but  starting  from  Leibniz's  characteristic. 

§8. 

When  he  had  proved  the  insufficiency  of  the  relation 
of  congruence  as  Leibniz  had  left  it,  he  tried  to  give  it  a 
form  in  which  substitution  would  be  possible.  What  are 
congruent  to  the  same  thing  are  congruent  to  each  other, 
but  that  does  not  mean  that  we  can  in  a  general  way  place 
instead  of  a  given  term  in  a  congruent  expression  one 
congruent  to  it.  So  substitution  is  not  possible.  This  can 
be  seen  at  once.  All  points  are  congruent.  Therefore, 
if  one  could  substitute  the  congruent,  one  could  place  abc 
congruent  to  every  combination  of  three  points — which  is 
absurd. 

Grassmann  rightly  regarded  the  fact  that  substitution 
was  not  possible  as  a  serious  defect  in  the  calculus.  So  he 


50  THE  MONIST. 

inquired  what  equations  would  hold  between  the  points 
a,  b,  c,  d,  e,  f,  if  abc  =  def. 

There  must  be  some  function  f  such  that,  when  the 
above  holds, 

f(a,b,c)=f(dfe,f). 

So  he  was  led  to  the  general  linear  relation  of  collinearity. 
Now  in  the  Ausdehnungslehre  Grassmann  had  reached 
the  fruitful  idea  of  a  true  geometrical  multiplication  which 
has  the  peculiarity  that  if  any  two  factors  of  the  product 
are  interchanged  the  sign  of  the  product  is  changed,  that  is, 

AB  =  — BA. 

This  combinatory  multiplication  enabled  him  now  to  give 
an  intrinsic  definition  of  geometrical  figures  in  terms  of 
points,  and  so  to  accomplish  what  Leibniz  had  failed  to  do. 
Thus  the  product  ab  determines  the  straight  line  between 
the  points  a,  b ;  the  product  of  three  points  determines  the 
plane,  and  so  on.  But  since  the  product  is  non-commuta- 
tive these  figures  when  so  defined  have  a  sense  represented 
by  the  signs  +  or  — .  Furthermore,  he  conceived  the 
notion  of  using  these  products  to  express  not  only  relations 
of  position  but  also  of  magnitude.  So  that  the  same  anal- 
ysis which  gave  a  geometry  of  position  also  gave,  side  by 
side  and  without  confusion,  a  metrical  geometry.  In  making 
this  step  he  had  to  define  (§3,  Geometrlsche  Analyse)  what 
he  meant  by  a  point  magnitude.  Each  element  (point, 
line,  plane)  has  two  aspects — its  position  in  space,  and  its 
intensity.  In  the  case  of  the  point,  this  latter  was  repre- 
sented by  a  positive  or  negative  "mass." 

By  now  defining  a  line  magnitude  as  the  combination 
ab  of  the  point  magnitudes  a,  b —  the  direction  of  which  is 
through  a  and  b,  and  the  intensity  of  which  can  be  defined ; 
and  also  defining  the  point  magnitude  as  the  combination 
AB  of  two  line  magnitudes,  the  position  of  which  is  the 
intersecting  point  of  A  and  B  and  the  mass  value  of  which 


THE  GEOMETRICAL  ANALYSIS  OF  GRASSMANN.  $1 

can  be  made  the  subject  of  a  definition — then  by  an  as- 
sumption which  makes  ab  =  O  and  AB  =  O  represent 
coincident  lines  and  points,  it  is  possible  to  write  in  the 
form  of  an  equation  every  linear  dependence. 

Thus  (ab)  (cd)e  =  O  denotes  that  e  is  the  intersecting 
point  of  ab  and  cd. 

So  the  principle  of  collineation  can  be  expressed,  though 
cumbrously  without  further  adaptation,  by  such  combina- 
tion equations. 

In  this  way  equality  is  made  to  include  the  two  relations 
of  identity  of  position  and  equality  of  intensity.  So  pro- 
jective  and  metrical  relations  can  be  expressed  in  one  form, 
and  considered  either  separately  or  together. 

§9. 

It  is  impossible  to  follow  Grassmann's  development26 
further  without  setting  up  a  technical  symbolism,  but  it 
may  easily  be  shown  how  brilliantly  Leibniz's  hopes  of  an 
analysis  specially  applicable  to  mechanics  have  been  ful- 
filled. 

In  terms  of  this  calculus  the  sum  of  n  points  is  their 
mean  point.  If  intensities  are  considered,  the  metrical 
relation  follows.  Thus  if  the  intensities  represent  masses 
at  the  points  the  sum  gives  the  center  of  gravity  of  the 
system — a  point  whose  intensity  will  be  the  sum  of  the 
other  intensities.  If  the  intensities  represent  parallel  forces 
acting  at  the  point  the  sum  gives  the  point  of  application 
of  the  resultant.  The  barycentric  calculus  of  Mobius  is 
thus  included  in  this  more  general  analysis. 

Furthermore,  the  line  magnitude  of  Grassmann  ex- 
presses a  force  with  exactitude.  Composition  of  forces 
thus  becomes  the  addition  of  line  magnitudes.  The  general 
equations  of  dynamics  can  also  be  represented  (§  n,  Geo- 

26  Needless  to  say  the  above  is  a  mere  sketch  of  the  beginning  of  Grass- 
mann's "Analysis."  In  particular  no  mention  is  made  of  his  distinction  be- 
tween inner  and  outer  products. 


52  THE  MONIST. 

metrische  Analyse)  by  means  of  this  calculus,  as  soon  as 
certain  modes  of  treating  infinitesimals  have  been  evolved. 

Moreover  the  possibility  of  attaching  a  metrical  co- 
efficient to  each  point  in  space  opens  at  once  many  fields 
of  application  in  physics. 

We  must  notice  in  addition  that  the  "Geometrical  Anal- 
ysis" does  not  treat  of  the  quotients  of  non-parallel 
stretches,  a  subject  which  leads  to  a  calculus  for  dealing 
with  powers,  roots,  logarithms  and  angles. 

Grassmann  can  claim  justly  therefore,  as  he  does,  in 
the  concluding  remarks  to  this  work,  that  his  mode  of 
treatment,  if  transferred  to  physics  in  general,  would  sim- 
plify the  mathematical  treatment  in  a  splendid  manner. 
He  himself  has  shown  the  great  advantages  of  the  calculus 
in  many  fields.  In  the  essay  we  have  several  times  referred 
to,  Leibniz  wrote,  "If  it  [the  characteristic]  were  set  up 
in  the  manner  I  conceive,  one  could  construct  in  symbols, 
which  would  only  be  the  letters  of  the  alphabet,  the  de- 
scription of  any  machine.  . .  .  One  could  by  these  means 
make  exact  descriptions  of  natural  objects." 

As  an  example  of  such  descriptive  power  Grassmann 
mentions  his  application  of  the  calculus  to  crystallography 
(cf.  Ausdehnungslehre  of  1844,  §  171). 

§  10. 

Apart  from  the  adaptability  of  the  geometrical  cal- 
culus to  different  provinces,  there  are  other  good  reasons 
for  believing  that  it  realizes  the  ideal  toward  which  Leib- 
niz looked  forward.27  Grassmann's  claim  put  forward  in 
his  concluding  remarks  will,  I  think,  be  granted  by  any 
one  willing  to  master  the  symbolism  sufficiently  to  under- 

27  Letter  to  Huygens :  "Algebra  is  nothing  but  the  characteristic  of  in- 
determinate numbers,  or  of  magnitude.  But  it  does  not  express  exactly  situa- 
tion, angles  and  movement....  But  this  new  characteristic. ..  .cannot  fail  to 
give  at  the  same  time  the  solution,  the  construction  and  the  geometrical  proof, 
the  whole  in  a  natural  manner  and  by  analysis.  That  is  by  determined  ways." 


THE  GEOMETRICAL  ANALYSIS  OF  GRASSMANN.  53 

stand  any  of  his  theorems.  "As  in  the  analysis  demon- 
strated here  every  equation  is  only  the  expression,  clothed 
in  the  form  of  the  analysis,  of  a  geometrical  relation,  and 
this  relation  expresses  itself  clearly  in  the  equation  without 
being  obscured  by  arbitrary  magnitudes — as  for  example 
the  coordinates  of  the  usual  analysis — and  therefore  can  be 
read  off  from  it  without  further  trouble;  and  as  further 
every  form  of  such  equation  is  only  the  expression  of  a 
corresponding  construction,  then  it  follows  that  as  a  matter 
of  fact,  by  means  of  the  analysis  here  given,  the  solution 
of  a  geometrical  problem  results  at  the  same  time  as  the 
construction  and  the  proof.  As  further  nothing  arbitrary 
....  need  be  introduced,  the  kind  of  solution  must  always 
be  according  to  the  nature  of  the  problem;  and  as  it  is  in 
the  form  of  analysis,  therefore  a  necessary  one  in  which 
there  can  be  no  question  of  any  seeking  round  for  methods 
of  solution."  In  other  words  the  fusion  of  synthetic  and 
analytic  methods  which  Leibniz  hoped  for  is  fully  accom- 
plished. 

It  must  be  noted  that  in  one  respect  Grassmann  has 
not  only  realized  the  prophetic  vision  of  Leibniz  but  also 
cleared  away  the  inconsistency  which  vitiates  his  attempt 
at  making  his  dream  come  true.  For  Leibniz,  seeing  that 
the  fundamental  analysis  of  geometry  must  rest  on  non- 
metrical  relations,  yet  desired  its  final  application  to  me- 
chanics and  natural  science,  in  which  metrical  relations 
are  all  important.  So  he  was  led  to  a  half-hearted  attempt 
at  non-metrical  analysis  by  means  of  a  relation — congru- 
ence— which,  while  showing  the  way  to  a  geometry  not 
based  on  algebra,  yet  failed  itself  to  travel  far  in  that  direc- 
tion. The  special  merit  of  Grassmann  has  been  to  found 
a  geometrical  analysis  free  of  magnitude  and  yet  so  to 
develop  it  that  metrical  considerations  may  be  introduced 
without  disturbing  the  form  of  that  analysis.  Projective 
geometry,  therefore,  only  partly  fulfils  Leibniz's  hopes; 


54  THE  MONIST. 

their  complete  realization  is  found  in  Grassmann's  theory 
of  extension. 

§». 

We  began  our  discussion  of  the  relation  between  Grass- 
mann's calculus  and  the  characteristic  of  Leibniz  by  an 
analysis  of  the  manner  in  which  their  work  has  been  re- 
ceived by  the  average  mathematician.  It  seems  to  me  that 
we  can  profitably  return  to  these  historical  considerations 
for  a  moment,  and  look  at  them  from  another  view-point. 

There  is  some  reason,  as  I  have  tried  to  show  else- 
where,28 for  citing  lack  of  historical  perspective  on  the  part 
of  mathematicians  as  the  cause  of  the  unsympathetic  atti- 
tude commonly  taken  up  in  regard  to  work  of  philosophical 
breadth;  and  that  if  more  regard  were  paid  to  historical 
development  in  mathematical  education  wider  and  more 
penetrating  vision  would  result.  The  position  taken  up  is 
well  expressed  by  Branford29 :  "The  path  of  most  effective 
development  of  knowledge  and  power  in  the  individual 
coincides,  in  broad  outline,  with  the  path  historically  tra- 
versed by  the  race  in  developing  that  particular  kind  of 
knowledge  and  power."  At  the  same  time,  however,  we 
must  realize  that,  if  we  alter  our  attitude  to  this  slightly, 
and  regard  it  not  from  the  point  of  view  of  the  education- 
alist but  from  that  of  the  original  worker  himself,  obsession 
with  origins  seems  inevitably  to  lead  to  what  I  have  called 
above  the  genetic  error.  The  effective  point  of  departure 
in  attaining  knowledge  of  geometry  may  be  from  such 
empirical  and  utilitarian  experiments  as  form  its  historical 
origin.  But  that  must  not  be  allowed  to  create  an  at- 
mosphere hostile  to  any  recognition  of  the  a  priori  and 
formal  nature  of  that  science. 

Furthermore  the  historical  method  may  lead  to  a  cer- 
tain ex  cathedra  manner,  a  reliance  on  authority  and  tra- 

28  "The  Neglect  of  the  Work  of  H.  Grassmann." 

29  B.  Branford,  A  Study  of  Mathematical  Education,  1908. 


THE  GEOMETRICAL  ANALYSIS  OF  GRASSMANN.  55 

dition.  It  is  this  factor  which  especially  concerns  us  in 
our  attempt  to  see  the  work  of  Leibniz  and  Grassmann 
in  true  relation  to  each  other  and  to  mathematical  thought. 
For  Couturat  points  out30  that  what  probably  hindered 
Leibniz's  development  of  his  geometrical  calculus  and  ren- 
dered abortive  his  attempt  at  its  realization  was  the  author- 
ity of  Euclid.  He  says,  "Why,  amidst  all  the  relations 
which  Leibniz  catalogued,  did  he  give  preference  to  the 
relation  of  congruence  and  neglect  the  relations  of  simi- 
larity, inclusion,  situation,  which  serve  to-day  as  the  bases 
of  quite  new  sciences31  which  he  foresaw  and  would  have 
been  able  to  found  ?  It  is  evidently  because  tradition,  rep- 
resented and  embodied  by  the  Elements  of  Euclid,  limited 
geometry  to  the  study  of  the  metrical  properties  of  space. 
Now  the  tradition  is  not  explicable  by  any  reason  of  theo- 
retical order  (considering  that  metrical  relations  are  more 
complex  and  less  general  than  projective  relations)  but 
solely  by  reason  of  historical  and  practical  order." 

I  have  already  in  the  previous  section  shown  that  an- 
other explanation  may  be  held  of  this  clinging  to  a  metrical 
relation  by  Leibniz.  However  that  may  be,  the  authority 
of  the  Euclidean  tradition  may  have  had  some  influence 
on  his  work  in  geometry,  as  the  Aristotelian  tradition  had 
in  his  foreshadowing  of  a  logical  characteristic.32  In  fact 
we  shall  not  be  laying  over-emphasis  on  the  tendencies  of 
an  exaggerated  reliance  on  historical  method  if  we  say  that 
its  final  result  is  the  attitude  of  the  young  critic  in  Shaw's 
play88  who  says,  in  effect,  "Give  me  the  name  of  the  author 
and  I'll  tell  you  if  it's  a  good  play."  If  that  critic  held  a 
university  chair  of  historical  criticism  he  would  doubtless 
be  able  to  .find  valid  arguments  for  his  position — for  how 

80  La  logique  de  Leibniz,  pp.  438-440.   Russell  however  attributes  Leibniz's 
failure  to  his  holding  the  relational  theory  of  space,  Mind,  1903,  p.  190. 

81  Theory  of  aggregates,  modern  Analysis  situs,  projective  geometry,  etc. 

82  See  note  7  above. 

88  "Fanny's  First  Play." 


56  THE  MONIST. 

(he  might  ask)  can  one  judge  competently  without  a  com- 
plete set  of  data,  and  is  not  authorship  an  important  datum  ? 
It  is  irritation  at  this  standpoint  which  causes  Mr.  Bertrand 
Russell,  whom  I  have  heard  speak  very  forcibly  on  the  sub- 
ject, inveigh  against  this  hyper-historical  method.  But  the 
objection  can  be  stated  in  a  much  stronger  form.  "Erudition 
often  does  violence  to  inventive  power:  and  the  proof  is 
that  the  modern  discoverers  of  symbolic  logic,  Boole  and 
his  successors,  have  all  ignored  (and  rightly)  the  example 
and  precedent  of  Leibniz ;  it  has  even  been  remarked34  that 
they  have  almost  all  been  ignorant  of  one  another,  and  if 
this  ignorance  has  been  a  source  of  error,  it  has  been  above 
all  a  condition  of  originality."3  Now  it  does  not  appear  to 
me  that  the  essential  defect  of  such  an  extreme  anti-histor- 
ical attitude  has  been  that  it  caused  error.  Staudt  realized 
the  ambitions  of  Leibniz  in  some  degree  in  founding  his 
projective  geometry,  and  Grassmann  in  still  further  degree 
in  creating  his  theory  of  extension,  without  knowing  that 
their  historical  origins  lay  in  his  work.  No  great  harm 
comes  from  this,  although  an  original  genius  will,  as  a 
general  rule,  be  less  likely  to  be  deflected  from  his  way 
by  the  work  of  others  than  to  find  in  them  sources  of  stim- 
ulation. But  to  the  mass  of  us,  who  form  the  bulk  of  man- 
kind, narrowness  is  a  mental  blinker  which  hides  the  full 
splendor  of  the  creations  of  genius.  The  real  toll  taken 
by  historical  ignorance  is  in  neglect  of  originality,  and  the 
loss  of  power  and  influence  consequent  on  it.88 

A.  E.  HEATH. 
BEDALES,  PETERSFIELD,  ENGLAND. 

84  J.  Venn,  Symbolic  Logic,  Introd.,  pp.29,  30. 
88  Couturat,  op.  cit.,  p.  440. 

88  I  have  to  thank  Miss  Vinvela  Cummin  for  valuable  help  in  translating 
the  Geometrische  Analyse. 


GREEK  IDEAS  OF  AN  AFTERWORLD. 

A  STUDY  OF  THE  RELATION  BETWEEN  PRACTICE  AND  BELIEF. 

WE  have  a  free  and  easy  way  of  generalizing  the  after- 
world  of  Greek  religious  belief  as  an  underworld. 
This  is  indeed  the  usual  form  of  the  belief  from  Hesiod 
onward,  and  it  is  the  view  generally  disclosed  by  Homer 
both  in  the  Iliad  and  Odyssey.  Yet  the  fact  is  that  the 
most  deliberate  and  detailed  Greek  presentation  of  the 
approach  to  that  dread  world,  that  of  the  eleventh  book  of 
the  Odyssey,  does  not  at  all  represent  it  as  an  underworld 
like  the  infernal  regions  of  Vergil's  fancy,  but  as  a  far 
western  realm.  The  far-wandering  Odysseus  sails  to  the 
distant  west,  out  of  the  sea,  and  across  the  mighty  ocean 
stream  to  its  farther  shore;  he  beaches  his  black-hulled 
ship  on  a  lone  waste  beach  where  stand  the  barren  groves 
of  Persephone ;  thence  he  directs  his  steps  inland  to  a  great 
white  rock  at  the  confluence  of  the  Styx,  Pyriphlegethon 
and  Acheron ;  and  there  it  is  that  he  enters  the  purlieus  of 
the  many-peopled  house  of  dark-browed  Hades. 

The  Odyssean  realm  of  the  dead  is  reached  neither  by 
descent  into  a  cave  nor  by  passage  underneath  an  over- 
hanging ledge.  It  is  of  the  same  level  as  the  land  of  living 
men.  Its  darkness  is  apparently  due  to  its  location  beyond 
the  path  of  the  western  sun,  which,  descending  into  Ocean 
Stream,  disappears  somewhere  from  the  sight  of  mortal 
men  to  be  ushered  in  anew  by  rosy-fingered  Eos,  each 
succeeding  morn.  To  speak  of  Odysseus  as  descending  into 
an  underworld  is  to  have  but  little  regard  for  the  language 
of  Homer.  Clearly  to  discern  the  picture  that  he  actually 


58  THE  MONIST. 

presents  is  to  become  aware  of  a  striking  contrast  between 
it  and  the  afterworld  of  classic  Greek  and  Roman  belief; 
and  this  contrast  raises  the  problem  of  explaining  and  ac- 
counting for  such  different  views,  obviously  related  in  the 
same  way  to  the  same  fact, — the  fact  of  death.  An  obvious 
relation,  I  say ;  if  this  appears  to  be  but  a  bold  assumption, 
I  trust  it  will  be  justified  in  the  course  of  my  argument. 

A  study  of  early  man's  beliefs  about  an  afterworld  in- 
volves a  consideration  of  two  groups  or  series  of  facts — 
mental  facts  and  motor  facts,  or  facts  of  belief  and  facts 
of  practice, — both  associated  with  the  event  of  death.  Ap- 
parently these  two  kinds  of  facts  do  not  simply  constitute 
two  parallel  series  that  were  mutually  unrelated  in  life  and 
thought  and  that  may  therefore  be  studied  and  understood 
apart  from  each  other ;  they  seem  to  have  an  intimate  and 
genetic  relationship.  This,  however,  is  not  to  say  that  they 
are  absolutely  simultaneous  in  origin,  or  that  one  may  not 
be  primary  and  the  other  secondary,  both  in  origin  and 
importance.  On  the  contrary,  in  their  genesis,  either  belief 
is  antecedent  and  causal  to  practice,  or  practice  is  antece- 
dent and  causal  to  belief. 

It  is  popularly  supposed  that  belief  originates  and  dic- 
tates practice,  or  custom,  which  is  thus  regarded  as  secon- 
dary to  belief.  Anthropologists  generally  confirm  the  sup- 
position, and  whole  systems  of  social  interpretation  and 
philosophy  have  been  built  upon  the  assumption.  Professor 
Seymour,  in  his  Greek  Life  in  the  Homeric  Age,  insists 
upon  this  relation  in  the  case  of  Greek  mortuary  practice 
and  belief,  and  cautions  the  reader  against  assuming  that 
the  Greeks  who  maintained  certain  customs  may  have  "in- 
herited also  the  beliefs  on  which  those  customs  were  orig- 
inally based."  He  brings  to  bear  upon  the  case  the  author- 
ity of  the  German  scholar  Rohde,  declaring  that  "Rohde 
gives  as  the  cause  of  the  adoption  of  cremation  by  the 
ancestors  of  the  Homeric  Greeks,  a  desire  to  rid  themselves 


GREEK  IDEAS  OF  AN  AFTERWORLD.  59 

of  the  souls  of  the  dead;  and  as  a  result  of  the  change,  the 
abandonment  of  the  old  ritual  and  sacrifices." 

According  to  Professor  Brinton,  "The  funeral  or  mor- 
tuary ceremonies,  which  are  often  so  elaborate  and  so 
punctiliously  performed  in  savage  tribes,  have  a  twofold 
purpose.  They  are  equally  for  the  benefit  of  the  individual 
and  for  that  of  the  community.  If  they  are  neglected  or 
inadequately  conducted,  the  restless  spirit  of  the  departed 
cannot  reach  the  realm  of  joyous  peace,  and  therefore  re- 
turns to  lurk  about  his  former  home  and  to  plague  the  sur- 
vivors for  their  carelessness. 

"It  was  therefore  to  lay  the  ghost,  to  avoid  the  anger  of 
the  disembodied  spirit,  that  the  living  instituted  and  per- 
formed the  burial  ceremonies;  while  it  became  to  the  in- 
terest of  the  individual  to  provide  for  it  that  those  rites 
should  be  carried  out  which  would  conduct  his  own  soul 
to  the  abode  of  the  blest." 

Here  again  practice  is  regarded  as  secondary  to  belief, 
and  is  interpreted  by  reference  to  belief.  Professor  Frazer, 
also,  the  dean  of  living  anthropologists,  insists  upon  this 
relation  between  our  two  series  of  facts,  and  cannot  admit 
or  conceive  of  the  opposite  as  being  true.  I  intend,  how- 
ever, to  take  the  other  side  of  the  question  here  involved, 
advancing  the  proposition  that  it  was  mortuary  practice 
that  constituted  the  motive  for  belief  in  an  af terworld ;  and 
especially  shall  I  endeavor  to  indicate  the  application  of 
this  formula  to  the  genetic  interpretation  of  Greek  ideas 
of  an  afterworld. 

The  Hellenic  peoples  of  whom  we  have  knowledge  uni- 
versally believed  in  an  afterworld,  whither  the  souls  of 
mortals  departed  at  death  and  where  they  had  a  continued 
existence.  But  they  entertained  not  merely  the  two  con- 
flicting beliefs  already  mentioned;  they  held  in  developed 
form  at  least  four  quite  different  beliefs  regarding  the 
destination  and  abode  of  the  souls  of  the  dead.  According 


6O  THE  MONIST. 

to  one  of  these  beliefs  the  souls  of  dead  men  ascended  to 
Olympus,  as  did  that  of  Heracles  in  story ;  according  to  an- 
other they  descended  into  an  underworld;  in  the  eleventh 
book  of  the  Odyssey  Homer  places  them  in  a  continental 
region  beyond  the  western  verge  of  Ocean  Stream;  and 
Pindar  places  the  souls  of  great  heroes  in  "Islands  of  the 
Blest"  in  the  far  Western  Ocean. 

It  may  be  well  at  this  point  to  note  some  apparently 
fundamental  resemblances  between  these  last  two  beliefs. 
Pindar  places  the  souls  of  sinful  mortals  in  an  underworld, 
subject  to  sentences  reluctantly  imposed  upon  them.  Hesiod 
declares  that  the  men  of  the  Golden,  Silver  and  Bronze 
ages  were  hidden  away  in  earth ;  and  it  is  but  natural,  be- 
cause of  the  different  types  of  life  imputed  to  them,  that 
he  should  fancy  different  conditions  for  them  after  death. 
But  the  souls  of  his  age  of  heroes,  he  says,  were  given  a 
life  and  an  abode  apart  from  men,  and  established  at  the 
ends  of  the  earth  in  "Islands  of  the  Blest  by  deep-eddying 
Ocean."  He  does  not  state  the  direction  of  these  wondrous 
islands,  but  undoubtedly  their  direction,  like  that  of  the 
Pindaric  Islands  of  the  Blest  and  that  of  the  Odyssean 
realm,  was  already  so  fixed  in  the  tradition  of  his  day  that 
there  was  no  need  of  indicating  it.  It  would  appear,  then, 
that  in  essential  characteristics  the  continental  Odyssean 
realm  and  the  Islands  of  the  Blest  are  alike  in  being 
conceived  as  western,  and  differ  only  in  geographical  form 
and  extent.  From  this  it  would  further  appear  that 
the  notions  of  these  two  similar  abodes  of  the  dead  are 
variants  derived  from  a  single  source.  But  if  these  two 
notions  did  grow  out  of  a  single  origin  there  was  certainly 
a  reason  for  the  divergence,  which  it  should  be  part  of  our 
task  to  discover.  And  yet,  on  the  other  hand,  it  may  be 
unnecessary  or  even  incorrect  to  assign  their  origin  to  the 
same  people,  even  though  we  may  feel  compelled  to  assume 


GREEK  IDEAS  OF  AN  AFTERWORLD.          6l 

that  the  significant  common  element  of  direction  must  in- 
here in  a  common  element  of  antecedent  cause. 

Whence  came  these  three  or  four  differing  beliefs  ?  That 
is  to  say,  upon  what  difference  of  psychological  ground  do 
they  severally  stand  ?  No  one  man  could  at  one  time  enter- 
tain so  many  and  so  contradictory  beliefs  upon  one  subject; 
neither  could  one  homogeneous  people,  as,  for  example,  a 
single  city  state  of  the  Mycenean  civilization,  or  even  the 
Minoan  civilization  of  Crete  as  a  whole.  Wherefore  we 
should  probably  look  for  this  difference  of  belief  either  in 
the  several  racial  stocks  amalgamated  to  form  the  historic 
Greek  people,  or  in  part  to  their  respective  traditional  be- 
liefs and  in  part  to  alien  streams  of  influence.  But  in  either 
case  it  will  be  pertinent  to  inquire  how  different  races  and 
racial  stocks  should  have  come  thus  to  act  and  believe  so 
differently  in  the  face  of  the  same  fact,  death.  To  trace  a 
belief  or  practice  from  one  people  back  to  another  should 
never  be  taken  as  an  explanation;  this  done,  the  question 
of  real  origin  and  motive  still  remains,  as  insistent  as  ever. 
Neither  should  identity  of  belief  or  practice  be  taken  as 
necessary  evidence  of  racial  relationships,  or  of  racial  con- 
tacts; nor  difference  of  belief  or  practice  as  evidence  of 
difference  of  race.  There  are  others  besides  the  human 
factor  that  enter  into  the  origin  and  development  of  prac- 
tice, as  we  shall  presently  see. 

With  regard  to  mortuary  practice,  the  Greek  world 
furnishes  only  two  types  of  historically  attested  facts.  The 
Homeric  Achseans  cremated  their  dead,  and  the  practice 
survived  far  beyond  the  Homeric,  and  even  the  Periclean 
age.  The  Mycenean  civilization  laid  its  dead  beneath  the 
surface  of  the  earth,  and  this  practice  gradually  superseded 
cremation,  even  among  the  descendants  of  the  Achaeans. 
Thus  the  Greeks  of  historical  times  had  two  strongly  con- 
trasted modes  of  disposing  of  their  dead,  corresponding  to 
two  of  the  contrasted  beliefs  we  have  mentioned.  For  there 


62  THE  MONIST. 

is  undoubtedly  a  genetic  relation  between  cremation  and 
belief  in  a  heavenly  abode  of  souls,  and  between  inhuma- 
tion and  belief  in  an  underworld.  But  which  is  cause  and 
which  effect?  And  how  did  the  causal  series  itself  orig- 
inate? And  how  could  the  belief  in  a  western  abode  of 
souls  be  related  to  either  of  these,  either  as  antecedent  or 
as  consequent? 

These  two  series  of  facts  in  Hellenic  life  give  rise  to 
three  problems  of  immediate  significance;  to  say  nothing 
of  others  more  remote,  as  for  example,  how  man  came  to 
believe  that  he  had  a  soul  at  all,  how  nearly  the  belief  coin- 
cides with  actuality,  the  origin  of  religious  fears,  etc.  The 
three  special  problems  thus  isolated  for  present  considera- 
tion are: 

1.  What  is  the  genetic  relation  and  order  of  precedence 
between  practice  and  belief, — between  cremation  and  belief 
in  a  heavenly  abode  of  souls,  and  between  inhumation  and 
belief  in  an  underworld? 

2.  In  case  either  belief  or  practice  is  found  to  be  antece- 
dent to  the  other,  how  then  did  this  antecedent  series  take 
its  rise? 

3.  Whence  and  how  came  the  belief  in  a  far  western 
abode  of  souls,  and  why  the  apparently  twofold  differen- 
tiation of  this  belief,  which  we  have  noted? 

In  the  interest  of  brevity  I  may  appear  to  be  cutting 
the  Gordian  knot  rather  than  untying  it;  but  I  feel  sure 
that  the  drift  of  my  argument  will  be  caught,  and  that  its 
essential  truth  must  make  a  strong  appeal  for  assent. 

In  the  first  place,  let  us  consider  this  intimate  and  in- 
herent correspondence  between  mortuary  practice  and  be- 
lief about  the  dead,  under  conditions  where  we  can  see  more 
plainly  the  part  played  by  geographical  environment,  and 
where  at  the  same  time  we  can  be  sure  of  the  soil  on  which 
our  two  series  of  facts  originated;  for  we  know  not  yet 
where  the  practice  of  inhumation  originated  among  the 


GREEK  IDEAS  OF  AN  AFTERWORLD.  63 

Myceneans  and  Minoans,  nor  where  cremation  first  de- 
veloped among  the  Achaeans. 

The  ancient  Egyptians  and  the  Incas  of  Peru  preserved 
their  dead  by  mummification,  and  both  believed  in  a  bodily 
resurrection  of  the  dead.  We  are  reasonably  sure  that  the 
land  where  each  of  these  peoples  developed  was  likewise 
the  soil  upon  which  their  respective  traditions  in  this  mat- 
ter originated;  we  shall  be  still  more  sure  of  this  local 
origin  as  we  proceed.  Did  the  belief  or  the  practice  pre- 
cede? 

Now  no  matter  what  we  may  imagine  them  to  have 
thought  about  soul  and  body  and  their  mutual  relations 
before  the  practice  began,  the  Egyptians  and  Peruvians 
could  not  have  cremated  their  dead ;  both  Egypt  and  Peru 
lacked  that  abundant  supply  of  fuel  which  would  be  neces- 
sary for  this  practice  among  a  numerous  people.  Neither 
could  either  people  long  have  inhumed  its  dead  in  the  fertile 
valley  land  of  its  abode.  These  restricted  valleys  early 
became  the  seat  of  such  dense  populations  that  productive 
land  could  not  be  permanently  set  aside  for  burial  pur- 
poses ;  nor  could  land  under  cultivation  be  wantonly  tram- 
pled over  for  this  common  social  purpose,  even  though  six 
feet  of  earth  were  sufficient  for  the  individual  grave.  Of 
necessity,  therefore,  the  adjacent  desert  ridges  were  em- 
ployed for  the  purpose,  and  the  earliest  mode  of  burial  there 
was  inhumation.  But  the  dry  climate  and  the  nitrous  char- 
acter of  the  upland  soil,  both  in  Egypt  and  Peru,  tended 
naturally  to  preserve  the  bodies  of  the  dead.  The  action 
of  wind  and  wild  animals,  however,  tended  often  to  exhume 
them,  at  the  same  time  disclosing  a  high  degree  of  preser- 
vation. In  order  to  protect  their  dead,  especially  to  prevent 
the  work  of  their  hands  from  being  made  of  none  effect, 
the  Egyptians,  in  particular,  came  to  build  rock  tombs. 
But  this  required  much  labor  and  expense.  Yet  it  was 
cheaper  to  build  one  tomb  large  enough  for  many  burials, 


64  THE  MONIST. 

for  whole  families,  even  through  successive  generations, 
than  to  build  many  individual  tombs.  Hence,  by  mutual 
suggestion  and  social  rivalry  through  long  stretches  of 
time,  the  mighty  Pyramids  of  Egypt  came  to  be  developed. 

But  under  these  conditions  a  tomb  must  be  entered  from 
time  to  time  for  new  burials;  and  in  spite  of  their  high 
degree  of  preservation  by  natural  means,  the  bodies  of  the 
dead  within  gave  rise  to  noisome  odors.  Hence  arose  the 
practice  of  embalming  with  aromatic  spices,  to  counteract 
or  obscure  the  evil  odors  of  decomposition.  What  but  this 
fact  of  unpleasant  odors  could  first  have  suggested  the  use  of 
expensive  spices  in  embalming?  With  the  prominent  Egyp- 
tian nose  was  undoubtedly  associated  a  keen  sense  of  smell. 
Wrappings  of  linen  served  in  the  first  instance  to  retain 
the  spices.  The  embalming  tended  to  more  perfect  preser- 
vation of  the  flesh,  and  this  result  also  helped  to  accomplish 
the  primary  object  of  the  practice,  which  was  the  laying 
of  unpleasant  odors.  Upon  this  combination  of  facts  arose 
a  profession  of  embalmers,  who  developed  a  more  and 
more  elaborate  technique.  When  death  and  funerals  had 
thus  become  an  economic  burden  upon  the  living,  for  which 
no  obvious  or  adequate  return  was  received,  the  question  of 
meaning  inevitably  arose  and  persistently  pressed  for  a 
satisfactory  answer.  It  is  exceedingly  difficult  for  man  to 
admit  that  he  is  spending  sacred  energies  in  vain  or  pur- 
poseless quests,  and  thus  making  a  fool  of  himself;  and  so 
the  practice,  entailing  so  large  an  expense,  insistently  re- 
quired a  sanction,  and  a  tremendous  one  at  that. 

Now  the  care  lavished  upon  the  dead  body,  by  tending 
to  preserve  it  for  an  indefinite  length  of  time,  embodied 
within  it  an  inherent  and  obvious  suggestion  of  the  primary 
sanction  that  actually  came  to  be  formulated.  For  by  this 
time  embalming  had  come  to  take  place  before  the  process 
of  decomposition  had  set  in ;  and  the  original  cause  of  the 
practice  was  no  longer  making  its  appearance,  even  though 


GREEK  IDEAS  OF  AN  AFTERWORLD.  65 

from  allied  experience  the  agents  may  well  have  been 
aware  of  what  would  soon  happen  without  embalming  and 
burial.  So  now,  instead  of  really  knowing  that  they  are 
trying  to  forestall  or  allay  the  noisome  odors  of  decompo- 
sition, they  detect  but  one  purpose  in  the  practice,  the 
preservation  of  the  body.  But  why  should  the  body  of  the 
dead  be  preserved  ?  With  this  query  arose  the  first  sugges- 
tion of  a  mystical  or  transcendental  idea  in  association 
with  the  practice,  and  the  first  attempt  to  formulate  an 
ultra-pragmatic  or  other-world  sanction  for  it.  This  sanc- 
tion was  formulated  as  an  explanation.  It  was  from  the 
first  employed  for  this  purpose,  and  as  all  thinking  indi- 
viduals were  implicated  in  the  practice  no  one  was  in  a 
position  to  question  or  challenge  it. 

It  might  be  urged  on  this  latter  ground  that  the  ques- 
tion of  purpose  or  value  could  never  have  arisen;  but  we 
must  not  overlook  the  fact  of  foreign  contacts — especially 
among  the  Egyptians — wherein  contrasted  practices  would 
raise  the  question  from  without,  if  not  from  within.  Be- 
sides this,  they  had  always  the  poor  with  them,  who, 
from  contrast  with  their  own  meager  efforts  in  the  same 
field,  would  be  forced  to  think  about  values.  And  above 
all,  there  was  always  growing  up  among  them  the  supreme 
pragmatist, — the  eager,  curious  child. 

Thus  this  question  of  values,  like  the  ghost  of  Banquo, 
was  ever  likely  to  confront  the  living,  and  only  a  powerful 
sanction  would  serve  to  lay  it.  The  priesthood  and  the 
professional  embalmers,  in  particular,  had  constant  need 
of  the  sanction,  as  a  means  of  justifying  their  existence. 
Thus  it  is  that  this  sanction  arose,  and  that  it  has  been 
passed  on  and  received  as  an  explanation  even  by  the 
wisest,  even  unto  the  present  day.  And  that  is  in  brief  the 
story  of  the  Egyptian  and  Peruvian  practice  of  mummi- 
fication, and  of  their  belief  in  a  bodily  resurrection.  It  all 
comes  back  in  the  last  analysis  to  the  fact  of  decomposition 


66  THE  MONIST. 

and  the  despised  sense  of  smell,  which  would  move  men  to 
acts  of  aversion  and  riddance. 

But,  one  may  ask,  is  it  not  after  all  just  possible  that 
this  practice  arose  out  of  an  antecedent  idea  of  souls  and 
the  notion  that  the  body  must  be  preserved  against  a  future 
resurrection  and  a  reincarnation  of  the  soul?  Rather  is 
it  not  far  more  reasonable  to  see  that  the  belief  arose  out 
of  the  practice,  as  a  sanction  for  the  care  and  expense  in- 
volved in  it?  On  the  first  alternative  we  must  certainly 
congratulate  the  Egyptians,  and  the  Peruvians  too,  on 
having  found  a  geographical  location  so  congenial  to  their 
belief.  What  would  they  have  practiced,  or  how  could  this 
belief  have  survived,  had  they  lived  in  the  valley  of  the 
Congo  or  Amazon,  or  even  in  Greece  ?  Or  how  could  they 
have  come  to  believe  in  a  heavenly  abode  of  souls,  when 
they  did  not  cremate  ?  And  if  the  belief  in  a  bodily  resur- 
rection came  before  the  practice  of  mummification,  then 
how  did  this  notion  and  belief  arise? 

Now  let  us  take  a  look  at  barren,  hungry,  frost-bitten 
Tibet.  What  burial  practices  and  what  cognate  beliefs 
about  the  dead  have  from  the  first  been  inherent  in  the 
natural  environment  of  man  presented  by  the  Himalayan 
highland?  Let  us  picture  to  ourselves  a  people  making 
here  its  arduous  ascent  from  lowest  savagery  to  barbar- 
ism. As  they  come  to  have  a  settled  place  of  abode,  how 
shall  they  secure  for  themselves  riddance  from  the  discom- 
forting odors  of  decomposition  that  follow  in  the  train  of 
death?  Suppose  that  they  have  attained  to  such  a  degree 
of  economic  efficiency  as  to  have  left  behind  the  practice 
of  cannibalism,  and  that  they  are  as  yet  without  any  meta- 
physical or  transcendental  ideas  and  beliefs;  how  then 
shall  they  dispose  of  their  dead  ?  Or  what  shall  they  believe 
about  their  dead,  if  they  have  as  yet  paid  no  attention  to 
them  save  by  the  simplest  modes  of  seposition  and  aban- 
donment ? 


GREEK  IDEAS  OF  AN  AFTERWORLD.  67 

Here  in  Tibet  is  a  people  that  could  not  cremate  its 
dead ;  for  here,  too,  fuel  is  scarce.  Neither  could  it  inhume 
its  dead ;  for  during  a  considerable  portion  of  the  year  the 
deeply  frozen  ground  is  proof  against  even  the  tools  of  civil- 
ized man.  Here  preservation  of  the  dead  by  natural  means, 
that  of  freezing,  may  be  assured  for  a  season ;  but  should 
this  be  relied  upon  temporarily,  final  burial  by  one  means 
or  another  would  become  imperative  with  the  advent  of 
spring.  Shall  the  Tibetans  preserve  the  bodies  of  their 
dead  through  the  long  winter,  to  the  end  that  they  may 
give  them  some  sort  of  approved  burial  in  the  spring? 
What  could  originally  have  suggested  to  them  the  notion  of 
an  approved  form  of  burial,  and  of  the  preservation  of  their 
dead  against  the  time  when  this  should  become  possible? 
The  primary  function  of  burial  by  whatever  means  is  avoid- 
ance or  riddance  of  certain  after  effects  of  death ;  and  with 
an  abundance  of  carnivorous  animal  life  scouring  the  coun- 
try for  the  means  of  subsistence,  how  could  the  immediate, 
practical  function  of  burial  be  more  readily  or  more  easily 
secured  than  by  calling  in  the  aid  of  dogs  and  vultures  that 
infest  the  land  ?  Now  this  is  exactly  what  the  Tibetans  do, 
even  to-day.  And  from  their  own  hard  struggle  for  ex- 
istence they  furthermore  feel  it  an  act  of  charity  thus  to 
minister  to  these  scavengers  of  their  land.  There  is  no 
other  people  on  earth  with  whom  charity  is  so  highly  es- 
teemed as  a  virtue,  and  so  universally  encouraged.  Under 
the  hard  conditions  of  life,  charity,  generosity,  is  a  neces- 
sary practice  among  their  own  kind.  And  furthermore, 
the  leisure-class  priesthood,  which  is  very  numerous,  in 
its  own  self-interest  has  need  of  encouraging  this  funda- 
mentally necessary  virtue ;  and  finally,  this  virtue  is  invoked 
as  a  sanction  for  the  feeding  of  their  dead  to  the  beasts  of 
the  field  and  the  birds  of  the  air.  Without  some  notion  of 
other  ways  of  securing  this  same  object,  they  could  feel 
no  need  of  this  or  of  any  other  sanction. 


68  THE  MONIST. 

In  the  case  of  a  very  few  individuals  of  the  highest  rank 
cremation  is  allowed  as  a  special  honor,  and  naturally  this 
privilege  is  mostly  restricted  to  the  religious  hierarchy.  It 
is  evidently  not  the  native  Tibetan  practice,  but  was  plainly 
introduced  into  Tibet  by  the  Buddhists  of  India,  with  whom 
it  was  native.  But  the  great  majority  of  the  Tibetan  dead 
go  to  feed  the  hungry  dogs  and  vultures,  which  are  highly 
esteemed  for  this  purpose;  and  this,  despite  the  fact  that 
Tibet  has  for  a  dozen  centuries  been  subjected  to  Buddhist 
influence,  which  would  naturally  favor  cremation,  its  own 
native  mode  of  burial,  if  this  were  economically  possible. 
Here  in  Tibet  the  native  mode  of  burial  is  directly  apposite 
to  geographical  conditions,  even  as  it  was  in  Egypt  and 
Peru;  and  the  beliefs  by  which  it  is  explained  are  merely 
so  many  sanctions,  or  justifications,  which  have  been  de- 
veloped out  of  the  practice  itself. 

But  what  are  the  Tibetan  beliefs  about  the  dead  ?  When 
once  they  have  acquired  the  notion  of  a  soul  that  survives 
the  event  of  death,  whether  originally  or  by  adoption  from 
other  peoples,  we  should  expect  them  to  hold  a  belief  in 
some  sort  of  transmigration.  From  seeing  the  bodies  of 
the  dead  devoured  by  animals,  they  would  seem  naturally 
to  think  that  souls  also  passed  into  the  bodies  of  these  liv- 
ing sepulchers.  This  is  exactly  what  they  believe.  We 
should  furthermore  expect  them  to  have  a  preference  for 
transmigration  into  the  winged  vulture  that  sails  so  easily 
through  the  air,  to  taking  up  their  abode  within  the  body 
of  a  lazy,  grunting  pig  or  snarling  dog.  Here  too  our  sur- 
mises are  correct.  In  the  course  of  centuries,  as  the  rela- 
tion between  practice  and  belief  has  become  obscured,  their 
beliefs  have  been  elaborated  and  graduated,  so  that  even 
non-carnivorous  animals  are  included  in  certain  cycles  of 
transmigration.  But  in  this  fact  of  feeding  their  dead  to 
animals  is  certainly  to  be  found  the  original  germ  and  sug- 
gestion of  their  belief  in  transmigration.  Tibetan  religious 


GREEK  IDEAS  OF  AN  AFTERWORLD.  69 

ideas  and  beliefs  are  not  so  definitely  conceived  nor  so  sys- 
tematically organized  as  are  those  of  some  other  peoples, 
because  their  authors  have  never  devoted  so  much  personal 
care  and  energy  to  the  disposal  of  their  dead.  They  have 
not  felt  so  strong  a  necessity  for  justifying  their  practice 
as  have  the  Egyptians  and  some  other  peoples. 

Again,  let  us  consider  the  case  of  India,  where  Brah- 
manism  and  Buddhism  have  their  origin  and  home.  The 
Indians,  like  the  Tibetans,  hold  a  belief  in  transmigration, 
and  of  course  for  that  same  fundamental  reason.  That  a 
mighty,  far-scattered  people  like  the  Indians  exhibits  a 
characteristic  belief  or  practice  does  not  mean  that  all  in- 
dividuals of  the  group  hold  it  in  common.  It  would  be  too 
much  to  expect  such  a  people,  or  any  people  at  all,  to  be 
really  homogeneous  in  belief  and  practice  from  the  early 
stage  when  human  burial  began  among  their  forebears 
until  the  present  time.  Thousands  of  families  in  India  to- 
day are  too  poor  to  afford  the  most  characteristic  tradi- 
tional form  of  burial  for  their  dead,  and  throw  them  into 
rivers,  or  otherwise  dispose  of  them.  In  Indo-China  those 
to  poor  tu  afford  cremation  commonly  carry  out  their  dead 
to  be  eaten  by  the  beasts  of  the  jungle.  On  the  Ganges, 
"When  the  pyre  is  built  the  nearest  relative  of  the  deceased 
goes  to  the  temple  and  haggles  with  the  keeper  of  the 
sacred  fire  over  the  price  of  a  spark ;  and  having  paid  what 
is  required  he  brings  the  fire  down  in  smouldering  straw 
and  lights  the  pile.  If  the  family  can  afford  to  buy  enough 
wood,  the  body  is  completely  consumed;  in  any  case  the 
ashes  or  whatever  is  left  on  the  exhaustion  of  the  fire  is 
thrown  into  the  sacred  river ; .  . .  .  and  any  failure  on  the 
part  of  the  fire  to  do  its  full  duty  is  made  good  by  the  fish 
and  the  crocodiles."1  Thus  it  is  easy  to  see  how  in  bygone 
days  the  Indian,  at  least  in  the  lower  social  strata,  became 
possessed  of  a  belief  in  transmigration,  and  how,  through 

1  Pratt,  India  and  its  Faiths.    New  York,  1915,  p.  44. 


7O  THE  MONIST. 

ignorance  of  its  primary  source  and  relationships,  carried 
it  over  into  relationships  bearing  little  or  no  connection 
with  its  parent  practice,  as  in  his  abstinence  from  eating 
flesh. 

And  yet  India,  with  its  wide  extent  and  countless  popu- 
lation, has  more  constant  elements  of  religious  and  philo- 
sophical belief  than  would  at  first  seem  possible, — a  result 
of  mutual  contacts  and  social  cooperation  through  long 
stretches  of  time.  "The  central  point  of  Hindu  thought  is 
the  soul.  It  is  from  the  soul  or  self  that  all  the  reasoning 
of  the  Hindu  starts  and  to  it  that  all  his  arguments  finally 
return."2  Probably  the  most  widely  known  characteristic 
of  Indian  religious  philosophy  is  the  doctrine  of  the  im- 
manence and  absoluteness  of  the  supreme  soul  Brahman, 
with  its  correlate  doctrine  of  the  oneness  of  the  individual 
self  with  the  All, — the  merging  of  the  objective,  phenom- 
enal world  into  the  universal  absolute,  which  is  Brahman. 
Yet  it  is  plain  that  this  interest  in  the  objective  world 
begins  with  the  individual  human  self.  "This  unity  of  the 
soul  with  God  is  at  the  foundation  not  only  of  Hindu  meta- 
physics, but  of  Hindu  ethics  as  well.  The  great  aim  of  life 
is  the  full  realization  of  that  God-consciousness,  the  sig- 
nificance of  which  forms  the  central  point  of  Hindu 
thought.  Before  this  can  be  fully  attained,  the  soul  must 
be  liberated  from  the  mass  of  particular  interest  and  petty 
wishes  and  self-born  illusions  which  weigh  it  down  and 
hide  from  it  the  beatific  vision.  Hence  liberation  and  reali- 
sation may  be  called  the  twin  ideals  of  Hinduism,  and  it  is 
these  that  determine  all  its  ethical  theory."3 

The  doctrine  of  "liberation"  and  "realization,"  the  doc- 
trine of  Nirvana,  the  yoga-systems,  and  other  character- 
istic Indian  notions  would  be  meaningless  and  impossible 
without  the  basic  body  of  "religious  intuitions"  that  make 

2  Pratt,  p.  91. 
8  Pratt,  p.  92. 


GREEK  IDEAS  OF  AN  AFTERWORLD.          7! 

up  the  Brahmanistic  doctrine  of  the  Upanishads.  But  an 
"intuition"  has  in  primordial  genesis  some  sensuous  basis, 
direct  or  indirect;  and  so,  instead  of  seeking  for  the  idea 
or  philosophy  back  of  the  practices  associated  with  these 
and  other  beliefs,  we  should  undoubtedly  seek  in  practice 
for  the  sensuous  elements  of  suggestion  that  formed  the 
basis  of  the  beliefs,  and  then  seek  in  turn  for  the  sensuous 
motive  of  the  practice  itself. 

We  may  admit  that  the  Indians  are  a  peculiar  people; 
yet,  when  we  pin  ourselves  down  to  minute  details,  we  note 
that  the  testimony  of  their  senses,  the  ultimate  constituent 
of  all  intellectual  forms,  is  the  same  as  our  own.  Their 
intellectual  peculiarity  consists  not  in  their  physical  or  psy- 
chological selves,  but  in  the  differences  of  their  objective 
environment,  part  of  which  they  themselves  make,  and  in 
the  various  ways  in  which  the  sensuous  details  of  expe- 
rience with  it  have  been  combined  through  generations  of 
spontaneous  social  collaboration. 

If,  then,  we  consider  these  doctrines  of  the  "infinite 
ocean  of  the  absolute  Brahman";  of  the  essential  oneness 
of  the  one  with  the  All;  of  the  soul's  struggle  for  liberation, 
to  realize  and  complete  this  oneness  in  "Nirvana,  or  re- 
absorption  into  the  eternal  light" :  as  we  contemplate  these 
doctrines,  seeking  to  discover  their  source  in  sensuous  ex- 
perience at  a  time  antedating  the  rise  of  science  with  its 
theories  of  atoms  and  corpuscles,  can  we  not  almost  see 
before  our  eyes  the  primitive  populace  of  India  cremating 
its  dead  and  beholding  the  body  ascending  in  the  form  of 
flame  and  smoke,  thus  becoming  absorbed  in  the  ocean  of 
air,  which  to  them,  at  that  time,  seems  infinite  ? 

We  examine  Indian  burial  practices,  both  present  and 
past,  and  we  find  that  from  time  immemorial  cremation 
has  been  a  characteristic  Indian  mode  of  burial.  When 
men  actually  beheld  the  body  of  a  deceased  friend  dissolve 
and  mingle  with  the  elements,  they  were  bound  to  have 


72  THE  MONIST. 

different  thoughts  about  the  destiny  of  the  individual  than 
if  it  were  laid  away  in  earth  to  decompose  by  degrees  for 
an  unknown  length  of  time,  or  if  it  were  altogether  pre- 
served by  embalming  against  decomposition.  And  from 
seeing  the  individual  thus  pass  so  visibly  from  a  corporate 
existence  into  thin  air,  they  would  also  be  moved  more 
strongly  to  contemplate  the  other  end  of  individual  exist- 
ence, the  whence  as  well  as  the  whither.  There  could  be  no 
doubt  that  the  deceased  had  attained  to  freedom  from  the 
bonds  and  ills  of  terrestrial  existence ;  and  the  living,  from 
their  own  desires  to  live  beyond  the  usual  limits  of  life, 
would  be  brought  face  to  face  with  the  question  whether 
they  should  ever  live  again  and  how  their  scattered  selves 
could  realize  another  conscious  existence.  To  hold  before 
them  the  notion  of  another  life  as  something  to  be  desired 
was  to  believe  in  it;  and  from  this  point  it  was  an  easy 
matter  to  identify  the  conditions  of  existence  before  birth 
and  after  death,  whence  Brahman  becomes  the  source,  the 
end,  and  the  essential  constituent  of  individual  existence. 

Add  to  all  this  the  practice  of  feeding  to  animals  either 
the  entire  body  or  the  remains  of  partial  cremation,  already 
noted — the  differences  of  practice  being  characteristically 
in  agreement  with  differences  of  social  rank — and  we  have 
the  proper  sensuous  background  in  practice  for  the  doctrine 
of  transmigration,  which  we  find  embodied  in  the  doctrine 
of  Karma  and  fused  with  the  doctrine  of  Nirvana. 

Geographical  conditions  undoubtedly  favored  crema- 
tion in  India  in  the  days  when  fuel  was  abundant  and 
easily  secured.  But  with  a  numerous  population  making 
large  demands  upon  the  wood-supply  through  scores  of 
generations,  the  practice  has  become  more  and  more  ex- 
pensive, and  the  demand  for  sufficient  sanction  has  become 
more  imperative.  Thus  in  the  course  of  many  centuries 
the  beliefs  genetically  inhering  in  these  practices  have  be- 
come much  elaborated ;  and,  by  the  development  of  an  elab- 


GREEK  IDEAS  OF  AN  AFTERWORLD.  73 

orate  logic  and  metaphysic,  they  have  in  turn  modified  the 
practice  itself.  It  is  in  this  way  that  the  religious  institu- 
tion has  justified  its  ways  and  made  itself  indispensable 
to  men. 

Here  again  we  may  claim  without  fear  of  successful 
contradiction  that  burial  practice  arose  as  a  purely  prac- 
tical matter  and  by  its  form  dictated  the  form  that  belief 
about  souls  must  take,  when  once  the  notion  of  soul  itself 
arose  out  of  the  practice.  The  sense  of  smell  together  with 
the  simple,  practical  knowledge  of  the  purifying  agency 
of  fire  suggested  and  motivated  the  practice;  here  it  is 
that  we  find  the  sensuous  motive  behind  the  practice,  which 
in  turn  motivated  the  belief.  Primarily,  the  belief  is  a 
supposed  explanation  of  the  practice,  invented  when  the 
practice  had  become  so  highly  elaborated  as  to  conceal  its 
real  cause  and  thus  to  demand  justification.  Men  do  not 
feel  the  need  of  explaining  or  justifying  the  obviously  prac- 
tical. 

But  the  explanation  given  of  this  and  other  kinds  of 
practice  is  not  an  explanatiaon  of  the  covert  act;  rather  is 
it  intended  to  explain  or  justify  the  care  and  energy  de- 
voted to  it  or  required  by  it  in  the  name  of  social  form. 
The  overt  act  merely  affords  suggestions  toward  the  ex- 
planation that  is  evolved.  It  is  only  after  a  long  lapse  of 
time  during  which  a  practice  has  by  social  concurrence 
become  highly  elaborated  that  a  justification  is  required. 
Men  acting  in  unison,  with  a  common  sense  or  emotional 
interest,  will  do  extravagant  things  not  dreamed  of  in  indi- 
vidual life.  But,  having  participated  in  such  an  act,  un- 
sophisticated man  can  easily  find  a  justification  for  his  act, 
suggested  by  the  act  itself.  It  seems  to  be  a  characteristic 
of  universal  human  nature,  in  the  absence  of  a  true,  an- 
tecedent cause  for  specified  conduct,  to  seek  about  for  some 
consequent  justification ;  and  the  race  seems  equally  prone 


74  THE  MONIST. 

to  accept  such  a  justification  as  a  statement  of  antecedent 
cause. 

And  now  we  may  return  to  the  case  of  Greece.  We 
do  not  find  there  that  close,  almost  necessary  relation  be- 
tween practice  and  environment  which  we  have  seen  in 
Tibet,  Egypt  and  Peru;  in  fact,  we  cannot  say  with  cer- 
tainty where  the  two  historic  Greek  forms  of  burial  orig- 
inated. Already  some  3000  years  before  the  Christian  era 
we  find  the  Minoan  civilization  in  the  ^Egean  world,  prac- 
ticing inhumation.  And  the  northern  Achaeans,  from 
whatever  source  they  came,  were  already  at  their  arrival 
in  Greece  practicing  cremation.  As  to  the  relation  between 
the  beliefs  and  practices  that  prevailed  on  Hellenic  soil,  we 
can  argue  only  by  analogy,  or  homology,  with  what  we 
have  seen  to  be  true  in  Egypt,  Peru,  Tibet  and  India ;  but 
it  is  far  more  reasonable  to  believe  that  the  same  relation 
holds  true  here  than  to  defend  the  other  horn  of  the  di- 
lemma. 

With  regard  to  the  Achaean  belief  in  a  heavenly  abode 
of  souls,  we  may  cut  the  matter  short  by  asserting  its 
rise  out  of  the  practice  of  cremation.  In  the  course  of  time, 
after  this  practice  had  become  the  rule  among  the  ancestors 
of  the  Homeric  Achaeans,  they  probably  came  to  feel  much 
the  same  regarding  it  as  did  the  Indian  of  California.  "It 
is  the  one  passion  of  his  superstition  to  think  of  the  soul  of 
his  departed  friend  as  set  free,  and  purified  by  the  flames ; 
not  bound  to  the  mouldering  body,  but  borne  up  on  the  soft 
clouds  of  smoke  toward  the  beautiful  sun."4  I  say  the 
Achaean  may  have  come  to  feel  in  this  way,  much  as  did 
the  Hindu;  but  this  was  not  the  original  motive  of  his 
practice.  His  thoughts  about  the  mouldering  body  of  his 
departed  friend  and  his  fancies  about  purification  were  not 
in  the  first  instance  inspired  by  a  desire  for  the  friend's 
welfare  after  death;  he  was  first  of  all  concerned  for  the 

*  Powers,  The  Indians  of  California,  pp.  181,  207. 


GREEK  IDEAS  OF  AN  AFTERWORLD.          75 

living,  especially  with  regard  to  the  sense  of  smell.  And 
however  transcendental  the  notion  of  purification  came  to 
be  by  reinterpretation  of  the  practice,  after  its  original 
motive  had  ceased  to  prevail — because  burial  came  to  be 
practiced  before  decomposition  had  set  in — the  very  associa- 
tion of  purity  with  cremation  betrays  the  original  motive 
of  the  practice,  just  as  did  the  use  of  spices  by  the  Egyp- 
tians. 

As  with  cremation  among  the  Achseans,  so  in  the  case 
of  inhumation  among  the  Minoans  and  Mycenaeans  we 
may  assert  that  the  practice  was  suggested,  and  passed 
through  its  primary  stage  of  development,  as  a  means  of 
escape  from  the  discomforting  odors  of  decomposition. 
And  as  the  belief  in  an  upper-world  abode  of  souls  devel- 
oped as  an  explanation  and  sanction  for  cremation,  so 
belief  in  an  underworld  developed  by  suggestion  from  the 
practice  of  inhumation.  To  make  good  the  claim  that  be- 
lief came  first  and  suggested  practice,  one  must  show  satis- 
factorily how  any  people  ever  could  have  associated  souls 
with  a  heavenly  or  with  an  underworld  abode  without  the 
practice  of  cremation  or  inhumation,  respectively,  or  at 
least  contact  with  some  people  who  did  practice  this  mode 
of  burial. 

The  belief  associated  with  cremation  never  became  so 
highly  elaborated  in  Greece  as  it  did  in  India,  and  for  very 
good  reasons.  For  in  the  first  place,  Greece  never  came  so 
completely  into  the  power  of  a  priestly  class  as  did  India; 
and  in  the  second  place,  the  practice  on  which  it  depended 
here  came  into  rivalry  with  the  already  established  prac- 
tice of  inhumation,  which  on  the  whole  was  cheaper.  To 
this  we  should  add  the  fact  that  the  social  institutions  of 
the  older  race  proved  to  be  the  more  persistent,  as  with 
the  Normans  and  Saxons  in  England,  whence  this  must 
have  been  especially  true  of  such  ideas  as  we  are  discussing. 
And  however  spectacular  and  interesting  the  act  of  crema- 


76  THE  MONIST. 

tion  became  among  the  Hellenes,  as  reflected  in  the  Ho- 
meric picture  of  the  funerals  of  Patroclus  and  Hector,  the 
accompanying  conception  of  the  soul  after  death  could  be 
but  very  vaguely  imaged,  as  in  the  case  of  India ;  while  the 
same  idea  accompanying  burial  in  the  ground,  in  cave- 
tombs,  cist-tombs,  and  rock-tombs,  as  the  so-called  "treas- 
ury of  Atreus"  was  capable  of  very  definite  imagery.  Thus, 
although  cremation  continued  to  be  practiced  side  by  side 
with  inhumation,  it  was  the  belief  associated  with  the  latter 
practice  that  possessed  the  more  definite  imaginative  ap- 
peal, and  that  finally  prevailed. 

Yet  the  upper-world  conception  of  the  soul  persisted 
and  influenced  the  belief  of  later  generations.  As  in  the 
first  instance  it  was  only  the  Achaean  masters  of  Hellas 
who  practiced  cremation,  while  the  subject  populace  in- 
humed its  dead;  and  since  in  the  classical  age  it  was  only 
the  wealthy  who  could  afford  cremation;  so  it  came  to  be 
believed  that  the  "good" — the  worthy  and  the  proud — at 
death  went  to  heaven  above,  while  the  poor  in  purse  and 
spirit  descended  into  hell.  Various  modifications  of  this 
composite  belief  have  grown  up  by  internal  suggestion  and 
by  accretions  from  foreign  practices  and  beliefs;  but  in 
the  last  analysis  each  belief  grew  out  of  a  practice,  and  the 
practice  originated  as  an  obvious  and  immediately  practical 
necessity. 

While  we  cannot  say  just  where  or  why  the  Minoans 
developed  inhumation  and  the  Achaeans  cremation,  or  why 
some  other  practice  did  not  arise  and  prevail  among  each 
people,  yet  it  is  perhaps  significant  that  cremation  was  the 
practice  of  the  northern  race,  like  the  aboriginal  Hindus, 
— a  people  who  had  more  need  of  fire  on  a  large  scale,  such 
as  would  be  necessary  for  the  cremation  of  human  bodies, 
a  people  with  whom  fire  was  necessarily  a  more  continuous 
object  of  experience  and  therefore  a  more  constant  agent 


GREEK  IDEAS  OF  AN  AFTERWORLD.  77 

of  purification  in  other  ways  also,  than  it  was  in  the  sunny 
southland  of  Crete  and  Hellas. 

Homer  was  the  poet  of  the  Achaean  overlords  of  Hellas. 
Yet  he  was  apparently  not  of  the  Achaean  race.  Although 
he  quite  consistently  presents  to  us  the  Achaean  mode  of 
burial,  his  idea  of  the  soul  and  its  abode  is  not  consistent 
with  the  practice  of  cremation.  He  thinks  of  the  cremated 
Heracles  as  having  a  corporate  existence  in  Olympus,  with 
lovely-ankled  Hebe  at  his  side;  yet  Heracles  must  also  be 
seen  of  Odysseus  in  the  house  of  Hades.  Homer  is  him- 
self aware  of  the  contradiction,  and  declares  it  to  be  but 
a  phantom  that  Odysseus  sees  there.  On  the  other  hand, 
Achaean  heroes — Patroclus  and  others  such  as  would  nat- 
urally have  been  cremated — he  unequivocally  represents  as 
being  in  the  populous  realm  of  Hades  in  the  distant  west. 
In  Homer's  references  to  the  realm  of  the  dead  we  discern 
the  unconscious  and  inextricable  mingling  of  at  least  three 
traditional  views  on  the  subject.  Nor  should  we  be  sur- 
prised at  this  when  we  note  that  the  entire  period  from  the 
Trojan  War  to  the  final  completion  of  the  Homeric  tales 
was  one  of  ethnic  amalgamation  between  at  least  the  two 
races  we  have  already  mentioned.  Our  view  of  this  process 
is  still  further  complicated,  and  yet  perhaps  much  illumi- 
nated, by  the  knowledge  of  a  continuous  intercourse  with 
the  west  coast  of  Asia  Minor  during  this  time,  such  that 
most  of  the  cities  that  laid  claim  to  Homer  were  of  this 
region. 

And  this  prompts  us  to  consider  how  the  notion  could 
have  arisen  that  the  dread  abode  of  souls  was  in  the  west. 
It  would  perhaps  be  interesting  to  point  to  the  west  as  the 
region  of  the  setting  sun,  to  associate  it  with  the  death  of 
the  day,  and  to  conjure  up  some  fancied  analogy  as  having 
been  indulged  in  by  the  aboriginal  authors  of  this  tradition. 
Yet  in  the  face  of  such  a  procedure  stands  the  fact  that  the 
west  has  always  been  the  land  of  allurement  and  promise 


78  THE  MONIST. 

to  which  Greek  no  less  than  Teuton  has  ever  turned  his 
eyes.  The  fact  is  that  if  the  association  of  the  west  is  an 
essential  element  of  the  belief,  as  it  appears  to  be,  then 
thoughts  of  the  west  were  inherently  involved  in  the  form 
of  burial  with  which  the  belief  was  genetically  associated. 

We  might  look  to  cremation  for  the  source  of  the  asso- 
ciation, if  anywhere  in  the  yEgean  world  the  prevailing 
winds  blew  to  the  westward,  thus  bearing  the  smoke  of  the 
funeral  pyre  in  that  direction.  But  such  is  not  the  case; 
and  besides,  neither  the  earthly  location  of  the  Odyssean 
afterworld  and  the  Islands  of  the  Blest,  nor  the  substantial, 
corporeal  nature  of  the  spirits  dwelling  there  would  permit 
of  this  conclusion. 

I  know  not  what  may  be  the  value  of  the  suggestion  I 
am  about  to  make  upon  this  subject ;  I  simply  present  it  as 
the  most  plausible  explanation  I  can  imagine  for  the  con- 
ception of  a  western  realm  of  the  dead.  I  have  by  no  means 
enumerated  all  the  methods  that  man  has  employed  for 
the  disposal  of  his  dead.  Fundamentally  there  is  but  one 
reason  for  disposing  of  the  dead  by  any  means,  and  that  is 
to  secure  a  separation  between  the  dead  and  the  living. 
Inhumation  and  cremation  are  merely  the  most  obvious 
and  most  universally  practicable  means  of  securing  this 
one  end. 

Now  one  of  the  simplest  modes  of  accomplishing  this 
object,  where  natural  facilities  permit,  is  what  is  called 
canoe-burial, — a  mode  in  which  the  body  of  the  dead  is 
placed  upon  a  log,  or  raft,  or  boat,  and  set  adrift  upon  the 
sea,  or  down  a  stream.  In  the  course  of  time  this  practice, 
just  as  any  other,  is  subject  to  elaboration  and  refinement, 
and  finally  to  mythical,  transcendental  interpretation.  I 
suggest  that  this  Hellenic  notion  of  a  western  realm  of  the 
dead  originated  on  the  western  coast  of  Asia  Minor.  Here 
all  rivers  flow  to  the  west;  out  to  the  westward  over  the 
sea  are  beautiful  islands  which  could  once  have  been  imag- 


GREEK  IDEAS  OF  AN  AFTERWORLD.  79 

ined  as  the  destination  of  bodies  set  adrift  on  the  rivers  of 
this  coast ;  and  finally,  when  these  islands  had  been  visited 
and  explored  and  the  fancy  exploded,  it  was  but  natural 
to  set  the  place  of  destination  of  the  dead  still  farther  to 
the  west  beyond  the  ^Egean  archipelago.  And  since  even 
by  Homer's  time  the  Hellenes  had  dim  fancies,  more  or 
less  substantiated,  of  extensive  coasts  in  the  distant  west, 
it  was  but  natural  that  the  earlier  notion  of  an  island  abode 
for  the  dead  had  to  give  way  to  fancies  of  a  more  con- 
tinental region.  But  as  the  primitive  occupants  of  this 
Asiatic  coast  had  grown  bolder  and  put  out  to  sea,  they  had 
perhaps  found  on  the  coasts  of  the  y£gean  islands  the  un- 
sightly wrecks  of  their  death-craft,  and  so  had  come  to  dis- 
continue the  practice.  It  is  not  necessary  to  suppose  that 
this  practice  was  current  in  the  time  of  Homer,  or  even 
of  the  Trojan  War;  mythical  fancies  may  survive  long 
after  the  conditions  that  fathered  them  have  ceased  to 
exist. 

Such  is  my  suggestion  for  explaining  the  notion  of  a 
western  abode  of  souls,  presented  on  the  assumption  that 
both  these  traditions  go  back  to  a  single  local  source.  Yet 
I  am  not  unmindful  that  the  coast  of  Epirus  and  Illyria 
furnish  the  natural  conditions  in  which  either  one  or  both 
may  have  arisen;  whence  we  should  have  to  suppose  that 
they  were  brought  into  Greece  by  the  Achaeans.  On  this 
assumption  we  should  have  to  suppose  further  that  these 
Achaean  adventurers,  after  leaving  their  native  abode  and 
the  conditions  supporting  their  native  mortuary  practice, 
took  to  cremation  as  a  new  means  of  disposing  of  their 
dead,  and  yet  retained  the  tradition  associated  with  the 
native  practice  of  canoe  burial.  This  would  help  to  account 
for  the  incongruities  in  the  Homeric  conception  of  the  con- 
dition of  souls  whose  bodies  had  been  burned;  it  would 
mean  that  they  had  not  yet  maintained  the  practice  long 
enough  to  have  invested  it  with  a  systematic  sanction  and 


8O  THE  MONIST. 

philosophy.  As  between  these  two  suggestions,  I  should 
probably  prefer  the  former.  As  yet  I  see  no  way  in  which 
archeology  may  help  us  here. 

In  any  case  the  tradition  of  a  western  abode  of  the  dead, 
which  had  already  been  started  and  which  had  by  this  time 
lost  all  direct  association  with  the  practice,  continued  and 
gathered  to  itself  the  Homeric,  and  Hesiodic,  and  Pindaric 
refinements  and  differentiae  which  we  have  already  noted. 
Such  is  the  regular  course  of  tradition.  It  is  undoubtedly 
in  this  way,  and  by  reference  to  the  same  kind  of  burial 
practice  in  Britain  that  the  traditional  pictufe  came  to  be 
built  up  of  the  black-hulled  ship  that  bore  "Elaine  the  fair, 
Elaine  the  beautiful"  down  the  Thames  to  Westminster; 
and  of  that  other  dusky  barge  that  bore  out  into  the  mystic 
lake  beyond  the  ken  of  mortal  man  all  that  was  mortal  of 
good  King  Arthur.  Such  a  social  background  is  probably 
necessary  for  the  historical  interpretation  of  the  death  voy- 
age of  Sinfiotli,  son  of  Sigmund,  away  "to  the  west" ;  and 
of  Balder  and  his  faithful  wife  Nanna,  laid  on  their  funeral 
pyre  on  the  deck  of  the  stately  ship  Ringhorn.  We  can 
understand  and  explain  how  a  traditional  practice  arises 
and  grows  by  social  concurrence,  and  how  a  belief  arises 
in  association  with  it,  all  conscious  association  with  the 
practice  being  gradually  lost.  But  to  explain  how  prac- 
tice should  arise  out  of  an  antecedent  belief,  and  how  that 
belief  should  first  have  arisen  as  a  purely  intellectual  con- 
ception without  sensuous  motivation — as  the  grin  without 
the  cat,  as  one  might  say — in  spite  of  some  three  thousand 
years  of  effort  upon  this  problem,  we  are  quite  as  far  from 
a  satisfactory  solution  as  ever. 

To  conclude,  then,  the  act  of  burial  by  early  peoples  is 
an  act  of  aversion  and  riddance,  even  as  the  traditional 
interpreters  of  the  act  have  claimed ;  but  the  primary  object 
of  the  riddance,  instead  of  being  a  metaphysical,  or  spirit- 
ual object,  is  a  real,  concrete,  sensuous  reality,  which  is 


GREEK  IDEAS  OF  AN  AFTER  WORLD.  8 1 

exactly  the  necessary  and  apposite  kind  of  motive  that  we 
should  expect.  If  only  Hobbes  had  hit  upon  this  formula! 
But  he  had  not  at  hand  the  rich  accumulation  of  anthro- 
pological data  that  we  now  possess.  And  even  Spencer 
and  Tylor,  with  all  the  data  at  their  command  and  with  all 
their  ability  to  analyze  and  organize  their  essential  ele- 
ments, made  the  same  mistake  as  Hobbes.  For  in  the  first 
place  they  made  belief  about  the  dead  a  result  of  secondary 
sensuous  experience,  instead  of  primary;  and  secondly, 
they  made  it  to  depend  upon  visual  instead  of  olfactory  ex- 
perience. The  sense  primarily  concerned  in  the  evolution 
of  religious  aversions  associated  with  ideas  of  the  dead  is 
undoubtedly  that  of  smell.  This  primary  aversion,  by  a 
traditional  transfiguration,  becomes  a  dread  or  fear  of  the 
dead  and  places  of  burial ;  and  only  when  man  requests  of 
his  most-used  sense  to  show  him  the  cause  of  the  aversion 
does  it  become  visualized.  And  then  only  is  it  that  dreams, 
visions,  apparitions,  reflections  and  other  illusory  visual 
phenomena  gain  a  superstitious  meaning. 

Thus  it  is  only  by  misinterpretation  of  the  act  of  avoid- 
ing or  allaying  the  noisome  odors  of  decomposition,  when 
the  real  motive  to  the  act  has  disappeared  from  view,  that 
a  people  can  ever  explain  its  burial  practice  as  a  spiritual 
"riddance"  or  "aversion,"  or  as  a  "laying  of  the  ghost." 
For  the  anthropologist  to  accept  this  secondary  aspect  of 
the  relation  between  belief  and  practice  as  being  primary, 
and  to  proceed  upon  this  assumption  to  the  explanation 
of  burial  practices  is  to  put  the  cart  before  the  horse.  Such 
reasoning  is  all  of  a  piece  with  myth;  it  is  reasoning  in  a 
circle,  and  will  never  get  us  anywhere  in  the  realm  of  scien- 
tific knowledge. 

For  such  reasons  as  I  have  given  above,  which  I  believe 
to  be  sound,  I  feel  reasonably  certain  that  my  primary 
assumption  of  an  obvious  and  constant  relation  between 
the  fact  of  death  and  beliefs  about  the  dead  is  justified; 


82  THE  MONIST. 

that  geographical  conditions  have  played  a  hitherto  un- 
recognized part  in  the  development  of  burial  practice  and 
belief  about  the  dead;  that  the  sense  of  smell  has  had  an 
unrecognized  share  in  the  development  of  religious  notions 
and  especially  religious  fears ;  that  the  Greek  notion  of  an 
underworld  abode  of  the  dead  grew  out  of  the  practice  of 
inhumation,  and  that  the  notion  of  a  heavenly  abode  of 
souls  in  like  manner  grew  out  of  the  practice  of  cremation. 
And  it  is  by  reason  of  the  satisfactory  corroboration  of 
my  reasoning  with  regard  to  inhumation  and  cremation 
that  I  suggest  a  primitive  practice  of  canoe-burial  on  the 
west  coast  of  Asia  Minor — or  possibly  the  Balkan  penin- 
sula— as  the  primary  motive  to  the  conception  of  a  western 
abode  of  souls,  whether  as  Islands  of  the  Blest  or  as  a  con- 
tinental realm  of  dark-browed  Hades. 

ORLAND  O.  NORRIS. 
YPSILANTI,  MICHIGAN. 


BERNARD  BOLZANO.* 

(1781-1848.) 

IN  BOLZANO  we  find  the  virtues  of  human  sympathy 
and  insight  coupled  with  the  austerer  virtues  of  the 
metaphysician  and  logician.  He  was  a  man  of  action  as 
well  as  a  man  of  ideas.  He  was  well  known  for  his  kindly 
disposition  and  his  broadmindedness.  He  possessed  not 
only  the  sympathy  with  the  poor  necessary  for  a  social 
reformer,  but  the  ability  to  develop  his  ideas  of  social  re- 
construction on  practical  lines.  Not  only  did  he  elaborate 
a  theory  of  an  ideal  state,  but  he  also  introduced  numerous 
reforms  in  the  actual  state  of  which  he  was  a  member.  He 
studied  theology  very  earnestly  as  a  young  man  and  later 
wrote  a  great  deal  on  the  subject.  Even  though  his  liberal 
views  brought  him  into  collision  with  those  on  whom  his 
livelihood  depended,  yet  he  courageously  continued  his 
teaching  and  writing,  always  making  it  his  aim  to  seek 
for  truth.  He  was  a  metaphysician  of  some  importance 
and  his  treatises  on  metaphysics  are  valuable,  not  only  for 
the  original  thought  which  they  contain,  but  also  for  his 
important  criticisms  of  Kant.  In  esthetics  his  work  is  by 
no  means  without  interest,  and  to  the  psychology  and 
ethics  of  his  day  he  made  very  valuable  contributions.  But 
preeminently  he  was  a  mathematician  and  logician.  In  his 

*  We  regret  that  owing  to  limited  time  and  the  uncertainties  of  trans- 
atlantic mail  service  The  Monist  is  compelled  to  go  to  press  without  receiving 
the  author's  imprimatur. 


84  THE  MONIST. 

work  on  mathematical  analysis  and  mathematical  logic, 
he  stood  out  from  all  the  other  thinkers  of  his  day.  He 
was  a  man  of  many  ideas  and  his  intellectual  equipment 
made  him  able  to  indicate  to  his  followers  the  most  fruitful 
lines  of  inquiry.  All  through  his  life  he  worked  for  the 
good  of  mankind,  helping  it  on  in  its  search  for  truth. 

Bernard  Bolzano  was  born  on  October  5,  1781,  at 
Prague.1  He  was  the  fourth  son  of  Bernard  Bolzano,  an 
upright  and  philanthropic  member  of  the  Italian  commun- 
ity at  Prague.  His  mother  was  a  very  pious  women.  He 
had  a  large  number  of  brothers  and  sisters,  the  majority 
of  whom  perished  in  childhood;  he  himself  was  a  sickly 
child.  In  his  early  youth  he  was  very  much  interested  in 
mathematics  and  philosophy.  His  education  was  of  the 
type  usual  at  the  end  of  the  eighteenth  century.  He  tells 
us  that  as  a  child  he  used  to  let  passion  completely  over- 
master him  because  he  believed  that  he  was  raging  not  at 
people  but  at  Evil  itself.  Bolzano  was  sent  to  one  of  the 
gymnasia  of  his  native  city,  where  he  did  not  distinguish 
himself  very  much,  and  later  proceeded  to  the  university 
there.  At  the  university  he  studied  philosophy  and  sub- 
sequently theology.  It  was  his  father's  wish  that  he  should 
be  a  business  man,  and  though  his  father  finally  gave  way 
he  showed  his  disapproval  of  his  son's  desire  to  continue 
his  studies  in  various  ways. 

Bolzano  had  been  brought  up  a  Roman  Catholic  and 
he  was  much  troubled  with  doubts  as  to  whether  he  should 
take  orders.  Finally,  however,  he  became  convinced  that 
difficult  problems,  such  as  the  authenticity  of  the  miracles, 
were  not  essential  parts  of  the  Catholic  faith,  and  as  in 
his  opinion  the  office  of  priest  offered  the  best  opportunity 
of  doing  good,  he  took  orders  in  1805.  At  the  same  time 
he  became  doctor  of  philosophy  at  Prague  University,  and 

1  Lebensbeschreibung  des  Dr.  B.  Bolzano  mit  einigen  seiner  ungedruckten 
Aufs'dtze  und  dem  Bildnisse  des  Verfassers;  eingeleitet  und  erldutert  von  dem 
Herausgeber  (J.  M.  Fesl),  Sulzbach,  1836. 


BERNARD  BOLZANO.  85 

was  appointed  professor  of  the  philosophic  theory  of  re- 
ligion. 

As  professor,  Bolzano  suffered  many  cramping  indig- 
nities which  surrounded  all  teachers  in  Roman  Catholic 
countries  at  that  time.  To  a  man  with  Bolzano's  sympathies, 
the  position  must  have  been  a  peculiarly  trying  one.  He 
had  a  great  love  for  young  people2  and  mixed  freely  with 
the  students.  He  was  particularly  sought  after  by  the 
students  because  of  his  liberal  views.  His  broad-minded 
interpretation  of  the  dogmas  of  the  Catholic  faith,  while 
provoking  the  distrust  of  the  authorities,  recommended  him 
to  the  younger  generation,  and  he  wielded  a  great  influence 
in  their  revolutionary  schemes  and  was  thought  by  many 
to  have  supported  them  with  an  enthusiasm  unbecoming 
in  a  professor.  At  any  rate,  relations  between  Bolzano 
and  the  authorities  grew  more  and  more  strained,  and 
finally,  as  he  would  not  recall  what  they  were  pleased  to 
call  his  "heresies,"  he  was  dismissed  on  the  grounds  that 
he  had  "failed  grievously  in  his  duties  as  priest,  as  precep- 
tor of  religion  and  of  youth,  and  as  a  good  citizen." 

After  his  dismissal  from  Prague,  two  t  ecclesiastical 
commissions  were  successively  appointed  by  the  Archbishop 
of  Prague  to  inquire  into  the  orthodoxy  of  his  teaching. 
In  the  first  commission,  the  majority  declared  that  Bol- 
zano's teaching  was  entirely  Catholic,  but  the  word  "en- 
tirely" was  deleted  at  the  wish  of  the  minority — which 
consisted  of  one  person.  This  decision  so  enraged  the  ob- 
scurantist party  that  a  large  amount  of  evidence  (not  a 
small  amount  of  which  was  "faked"  for  the  purpose)  was 
collected  and  put  before  the  second  commission.  In  1822 
Bolzano  made  two  declarations  in  writing  in  which  he 
stated  that  he  held  it  "dangerous,  even  with  the  best  in- 
tentions, for  a  man  to  seek  and  teach  new  points  of  view 

2  See  A.  Wishaupt,  Skizzcn  aus  dem  Leben  Bolsanos:  Beitrdge  zu  seiner 
Biographic  von  dessen  Arzte,  Leipsic,  1850,  pp.  19ff. 


86  THE  MONIST. 

as  proofs  of  the  truth  and  divine  nature  of  the  Christian 
Religion."3  The  commission  then  finally  collapsed.  Two 
years  later  Bolzano  was  pressed  for  a  public  recantation. 
The  Archbishop  of  Prague  brought  illicit  pressure  to  bear 
on  him  by  pleading  his  affection  for  him  and  by  declaring 
that  a  refusal  would  bring  him  to  the  grave.  Bolzano, 
however,  refused  to  recant  publicly,  but  solemnly  declared 
his  orthodoxy  in  writing. 

The  main  points  of  his  teaching  on  religion  are  set  out 
at  some  length  in  his  Lehrbuch  der  Religionswissenschaft.* 
He  defines  religion  as  the  aggregate  of  doctrines  which 
influence  man's  virtue  and  happiness.  He  then  proceeds 
to  discuss  what  seemed  to  him  the  most  perfect  religion, 
viz.,  the  Catholic  faith.  His  reason  for  so  regarding  the 
Catholic  faith  is  that  it  is,  in  his  opinion,  revealed  by  God. 
A  religion  is  divinely  revealed,  according  to  Bolzano,  if  it 
is  morally  beneficial  and  if  connected  with  it  there  are 
supernatural  events  which  have  no  other  use  than  that 
they  serve  to  demonstrate  this  religion.  In  the  first  chapter 
the  concepts  of  religion  in  general,  and  organized  religion 
in  particular,  are  discussed.  In  the  third  chapter  he  main- 
tains that  for  a  religion  to  be  true  it  must  be  revealed,  and 
then  he  proceeds  to  enunciate  the  characteristics  of  a  reve- 
lation. In  the  second  volume,  he  sets  out  to  prove  that  the 
Catholic  religion  possesses  the  highest  moral  usefulness 
and  that  its  origin  has  the  attestation  of  supernatural  oc- 
currences. He  discusses  the  evidence  for  Christ's  miracles 
and  the  genuineness  of  the  sources  and  points  out  the  pres- 
ence in  Christianity  of  the  external  characteristic  of  reve- 
lation. He  then  passes  on,  in  the  third  volume,  to  demon- 
strate in  some  detail  the  moral  usefulness  of  the  faith. 
After  a  discussion  of  the  Catholic  doctrine  of  the  sources 
of  knowledge  he  examines  the  various  doctrines  of  the 

3  Published  1836  (Sulzbach)  with  autobiography. 
*  Sulzbach,  1839  (4  volumes). 


BERNARD  BOLZANO.  87 

Catholic  church.  It  is  interesting  to  notice  that  he  regards 
the  doctrine  of  the  Trinity  as  entirely  reasonable,  and  com- 
pares the  Father  to  the  All,  the  Son  to  humanity,  and  the 
Holy  Ghost  to  the  individual  soul.  In  the  last  chapter  of 
this  volume  Bolzano  is  concerned  with  the  Catholic  system 
of  morals.  In  his  investigation  he  discusses  first  Catholic 
ethics  and  then  the  various  means  of  salvation  recom- 
mended by  the  church.  He  examines  each  of  the  sacra- 
ments in  turn.5 

After  his  dismissal  from  Prague,  Bolzano  wrote  a  very 
great  deal,  but  the  internal  censorship  prohibited  all  publi- 
cations in  his  name  and  even  in  some  cases  retained  the 
manuscript.  Bolzano  once  expressed  the  pious  hope  that 
some  day  he  might  be  allowed  to  publish  some  work  of  a 
purely  mathematical  nature !  After  he  left  Prague  he  lived 
chiefly  with  friends  at  Techobuz.  He  came  back,  finally, 
to  his  native  city  in  1841  and  continued  his  work  with 
vigor  until  his  death  in  1848. 

Though  it  was  in  mathematics  that  Bolzano  did  his 
most  important  work,  yet  in  other  subjects,  notably  in 
political  science,  his  work  is  of  considerable  value.  He  had 
very  great  sympathy  with  the  poor  and  was  anxious  to 
abolish  class  differences.  He  was  convinced  that  the  in- 
adequacy of  social  organizations  was  the  cause  of  poverty. 
He  never  wrote  very  much  on  the  matter,  but  made  it  the 
subject  of  many  of  his  professorial  addresses.  There  is, 
however,  one  short  manuscript8  in  which  he  sets  out  the 
main  points  of  his  political  theory.  Bolzano  himself  thought 
a  great  deal  of  this  manuscript  for  he  says  in  the  intro- 
duction: "And  small  as  is  the  number  of  these  pages,  yet 
the  author  thinks  he  may  be  allowed  to  attribute  some 
value  to  them.  Nay,  he  considers  that  this  little  book  is 

8  For  a  complete  list  of  his  theological  works  see  Bergmann,  Das  philo- 
sophische  Werk  Bernard  Bolzanos,  Halle,  1909,  p.  214. 

6  "Vom  besten  Staate,  MS.  in  the  Royal  Bohemian  Museum.  For  a  con- 
venient summary  of  the  MS.  see  Bergmann,  op.  cit.,  pp.  130ff. 


88  THE  MONIST. 

the  best  and  most  important  legacy  that  he  can  bequeath 
to  his  fellow  men  if  they  are  willing  to  accept  it." 

In  Bolzano's  ideal  state,  men  and  women  alike  are  to 
have  the  privilege  of  voting,  but  a  person  is  only  allowed 
to  vote  on  a  matter  of  which  he  has  some  knowledge  and 
in  which  he  has  some  interest.  Further,  the  right  of  voting 
is  liable  to  forfeiture  in  the  case  of  misconduct.  Any  citi- 
zen may  put  forward  a  suggestion.  The  suggestion  is 
examined  by  six  independent  citizens,  each  one  examining 
it  privately,  and  it  is  only  rejected  if  all  six  of  the  citizens 
reject  it — and  even  then  it  is  retained  by  the  state  for 
further  reference.  If  it  is  not  rejected,  a  general  vote  is 
taken,  and  if  there  is  a  majority  in  favor  of  it,  it  goes  to  a 
council7  which  is  composed  of  men  and  women  over  sixty 
years  of  age,  who  are  chosen  by  the  people  every  three 
years.  The  council  can  only  veto  the  decisions  of  the  people 
if  ninety  percent  of  the  council  are  against  it.  The  govern- 
ment is  the  administrative  body,  its  members  are  paid  and 
elected  by  the  people,  and  there  is  a  strict  limit  to  the 
length  of  time  that  they  may  remain  in  office.  The  govern- 
ment takes  special  care  to  prevent  private  individuals  com- 
bining in  their  own  interest.  Bolzano  looked  upon  war  as 
a  dreadful  misfortune  and  in  his  Utopia  war  is  only  to  be 
used  as  a  defensive  measure.  Bolzano  points  out  that 
internal  revolutions  are  unlikely,  for  they  arise  in  general 
from  one  of  two  causes — a  bad  constitution  or  poverty. 
Of  these,  poverty  is  to  be  non-existent  and  a  revolution 
due  to  the  first  cause  is  improbable  because  it  could  only 
be  brought  about  if  the  council  opposed  a  change  in  the 
constitution  which  the  people  considered  advisable.  But 
the  council  in  its  wisdom  would  not  taunt  the  people  but 
would  give  reasons  for  its  decision.  It  therefore  seems 
unlikely  that  the  people  would  rise  in  revolt,  all  the  more 
because  it  is  early  impressed  upon  the  young  that  a  good 

7  The  council  is  called  the  "Rat  der  Gepriiften." 


BERNARD  BOLZANO.  89 

citizen  does  not  work  against  the  government,  for  the 
government's  object  is  to  work  for  the  good  of  the  whole 
state. 

One  of  the  most  interesting  parts  of  the  manuscript 
deals  with  the  idea  of  property.  In  the  ideal  state  property 
is  only  desired  in  so  far  as  the  possession  of  it  contributes 
to  the  common  good.  The  only  valid  claim  of  a  man  to 
property  is,  therefore,  that  he  can  make  it  more  useful  to 
the  state  than  any  one  else  could.  The  fact  that  a  man 
may  possess  a  certain  thing  at  a  certain  time  is  not  a 
necessary  or  sufficient  reason  that  he  shall  possess  it  alto- 
gether. The  right  of  inheritance  is  not  recognized.  Things 
such  as  books,  paintings,  furniture  or  jewels,  are  given  to 
a  citizen  to  use  but  not  to  possess.  Further,  even  though  he 
may  have  established  his  claim  to  a  certain  object,  yet,  if 
at  any  subsequent  time  another  citizen  can  make  more  use 
of  it,  the  title  of  the  first  citizen  to  it  is  gone.  Moreover, 
the  state  does  not  offer  any  compensation  to  a  man  for 
depriving  him  of  anything.  Thus  a  man  whose  eyesight 
has  been  cured  has  his  glasses  taken  away  and  no  compen- 
sation is  made.  In  all  the  distribution  of  goods  the  govern- 
ment is  guided  entirely  by  the  principle  that  the  use  of  a 
certain  thing  should  be  granted  to  the  citizen  who  can 
render  it  most  useful  to  the  state  as  a  whole. 

The  ideals  of  the  state  are  freedom  and  equality.  There 
is  no  unequal  distribution  of  wealth.  However  there  is  not 
an  absolute  equality  of  owners,  for,  as  Bolzano  points  out, 
the  possibility  of  increasing  one's  property  is  a  powerful 
incitement  to  work.  But  there  are  limits  beyond  which 
a  man  cannot  increase  the  extent  of  his  property,  and  these 
limits  are  determined  by  the  consideration  of  the  good  of 
the  state  as  a  while.  There  are  "equal"  right  for  all  citi- 
zens, but  the  word  "equal"  is  not  to  be  interpreted  in  any 
narrow  sense.  Rather  there  is  an  adjustment  between  the 
rights  of  a  citizen  and  his  obligations,  between  his  strength 


9O  THE  MONIST. 

and  his  need.  The  government  aims  at  promoting  religious 
freedom.  No  religion  is  given  preferential  treatment  by 
the  state.  People  choose  their  own  ministers  of  religion  and 
support  them.  But  a  new  religion  may  not  be  preached 
without  permission,  for  some  might  not  be  able  to  grasp 
all  the  consequences  of  accepting  certain  doctrines  and  be- 
liefs. Further,  a  citizen  may  change  his  religion,  but  he 
must  first  bring  proof  that  he  has  studied  with  earnestness 
the  principles  of  the  religion  he  is  about  to  leave,  as  well 
as  of  the  one  which  he  desires  to  embrace. 

In  the  education  of  children  the  special  aim  is  the  de- 
velopment of  the  mind.  The  teachers  do  not  have  complete 
freedom  in  the  choice  of  what  the  children  are  taught. 
The  Council,  if  it  is  unanimous,  has  the  power  to  prevent 
the  teaching  of  any  particular  doctrine.  The  children's 
books  are  censored.  The  censor  is  responsible  directly  to 
the  government.  And  not  only  the  children's  books,  but 
all  the  books  in  the  state  are  censored  strictly. 

The  question  of  rewards  and  punishments  in  the  state 
is  treated  in  a  practical  way.  Rewards  are  to  consist  in 
public  recognition  of  merit,  and  punishments  are  not  ar- 
ranged on  a  definite  plan  but  are  modified  so  as  to  suit 
individual  cases.  There  is  however  a  special  proviso  that 
no  citizen  is  under  any  circumstances  to  be  imprisoned  for 
life. 

Bolzano  has  some  very  interesting  ideas  on  the  occu- 
pations of  the  people  in  his  Utopia.  To  begin  with,  the 
state  is  to  support  those  who  are  not  fit  to  work.  From 
those  who  are  fit,  the  state  demands  a  certain  fixed  amount 
of  work — the  fixed  amount,  of  course,  varying  from  one 
individual  to  another.  In  return  for  the  work  the  state 
distributes  goods.  Citizens  are  not  allowed  to  waste  their 
time  in  useless  or  pernicious  occupations — Bolzano  con- 
sidered newspapers  pernicious.  Neither  are  they  allowed 
to  do  things  in  any  but  the  quickest  and  most  satisfac- 


BERNARD  BOLZANO.  91 

tory  way.  Thus  they  are  not  allowed  to  thresh  with  a 
flail  when  a  threshing  machine  has  been  invented,  nor, 
presumably,  to  walk  when  there  is  a  tram.  One  interesting 
point  is  that  the  state  is  to  pay  compensation  for  damage 
done  by  nature.  Bad  weather  would  quickly  lose  its  terror 
for  farmers  in  Bolzano's  ideal  state.  Finally,  those  who 
wish  to  devote  their  lives  to  art  or  some  branch  of  learning 
are  supported  by  the  state  if  they  can  produce  evidence  to 
show  that  it  will  be  in  the  state's  interest  that  they  shall 
be  employed  in  this  way.  The  whole  theory  of  the  state 
is  peculiarly  fresh  and  in  many  respects  suggestive. 

But  Bolzano's  Utopia  is  only  a  practical  illustration  of 
his  general  ethical  principles.  The  guiding  principle  of 
his  inquiry  may  be  enunciated  as  follows:  Of  all  possible 
actions,  one  should  always  choose  that  one  which,  when 
all  consequences  have  been  considered,  produces  the  great- 
est amount  of  good  or  the  least  amount  of  evil,  for  the 
human  race  as  a  whole,  and  in  this  estimate  the  good  of 
individuals,  as  such,  is  to  be  left  out  of  consideration.  But 
Bolzano  points  out  that  if  this  principle  is  to  be  the  highest 
moral  law,  it  would  be  necessary  to  frame  a  definition  of 
good  and  bad  before  any  practical  applications  could  be 
made.  Further  since  he  holds  that  an  action  is  good  if  it 
is  an  action  which  we  ought  to  perform,  he  gets  back  im- 
mediately to  the  question:  What  ought  I  to  do?8 

There  then  remains  only  the  effects  of  action  on  the 
faculty  of  sensation.  Bolzano  argues  that,  since  one  can 
excite  only  either  pleasant  or  unpleasant  sensations  and 
since  no  one  would  hold  that  it  is  one's  duty  to  excite  un- 
pleasant sensations,  it  is  obviously  one's  duty  to  excite 
pleasant  sensations.  By  this  process  of  eliminating  every- 
thing except  the  faculty  of  sensation,  Bolzano  comes  to  the 
conclusion  that  the  highest  moral  duty  is  the  excitement 

8  For  an  interesting  and  valuable  criticism  of  Bolzano's  assertions  and 
deductions  mentioned  here,  see  Bergmann,  op.  cit.,  Part  V,  §  958. 


92  THE  MONIST. 

of  pleasant  sensations.  Not  the  least  interesting  part  of 
his  work  in  ethics  is  his  criticism  of  Kant's  categorical 
imperative.  He  urges  the  necessity  for  a  modification  in 
Kant's  principle  and  points  out  the  invalidity  of  Kant's 
theory  that  the  opposite  of  a  duty  involves  a  contradiction. 

Bolzano's  work  in  esthetics  is  not  without  interest.9 
His  theory  of  esthetics  is  the  result,  not  of  his  own  esthetic 
sensations,  but  of  a  painstaking  analysis  of  the  abstract 
idea.  His  definition  of  the  scope  of  the  subject  does  not 
make  it  coincide  with  the  theory  of  beauty  unless  we  include 
in  that  theory  not  only  the  sum  total  of  truths  directly  con- 
cerned with  beauty  but  also  all  those  which  stand  in  such 
a  relation  to  them  that  either  the  former  cannot  be  thor- 
oughly understood  without  the  latter  or  the  latter  without 
the  former.  To  get  at  his  concept  of  beauty,  he  eliminates 
goodness  and  attractiveness,  and  by  this  process  obtains 
a  first  criterion  of  beauty,  viz.,  all  beauty  is  pleasant,  i.  e., 
it  produces  pleasure  and  this  pleasure  arises  solely  from  the 
contemplation  of  the  object.  Further,  since  animals  are 
to  be  excluded  from  esthetic  enjoyment,  qualities  must  be 
introduced  which  they  do  not  possess,  e.  g.,  intelligence, 
judgment  and  reason.  Bolzano  then  comes  to  the  conclu- 
sion that  it  is  the  growth  of  these  qualities  in  us  that  is 
responsible  for  the  pleasure  we  find  in  beauty.  Together 
with  the  "Ueber  den  Begriff  des  Schonen"  in  the  Royal 
Bohemian  Museum,  there  is  another  short  treatise  of  Bol- 
zano's in  which  a  theory  of  laughter  is  elaborated.10  Bol- 
zano thought  that  laughter  was  caused  by  the  rapid  alter- 
nation of  pleasant  and  unpleasant  sensations  and  from  the 
fact  that  animals  and  infants  do  not  laugh  he  deduces  that 
laughter  is  not  entirely  physical.11 

In  his  metaphysics,  Bolzano  reveals  himself  as  "one  of 

9  See  Ueber  den  Begriff  des  Schonen,  Prague,  1843. 

10  Ueber  den  Begriff  des  L'dcherlichen,  1818. 

11  See  Bergmann,  op.  cit.,  Part  IV,  §  56. 


BERNARD  BOLZANO.  93 

the  acutest  critics  of  the  Kantian  philosophy  and  the  'ideal- 
ist' development  from  Fichte  to  Hegel."1  He  also  did 
some  important  original  work.  His  chief  book  on  the 
subject,13  entitled  Wissenschaftslehre:  Versuch  einer  aus- 
fiihrlichen  und  grosstenteils  neuen  Darstellung  der  Logik,1* 
is  divided  into  five  sections.  In  the  first  of  these  he  sets 
out  to  prove  that  objective  truth  exists  and  that  it  is  pos- 
sible for  us  to  have  knowledge  of  it ;  but  he  allows  that  in 
the  development  of  the  science  of  knowledge,  which  is  the 
most  fundament0!  of  the  sciences,  it  is  necessary  to  use 
some  psychological  methods  of  treatment.  In  the  second 
part,  the  "Theory  of  Elements,"  he  treats  successively 
ideas-in-themselves,  their  combination  into  propositions- 
in-themselves,  the  theory  of  true  propositions-in-themselves, 
and  finally  their  combination  into  syllogisms.  He  is  ex- 
tremely careful  to  distinguish  between  the  idea-in-itself 
and  the  conceived  idea.  The  concept  of  a  proposition-in- 
itself  is  produced  by  a  double  abstraction.  First  the  mean- 
ing of  the  proposition  and  the  words  conveying  the  mean- 
ing have  to  be  separated  from  each  other,  and  then  one  has 
to  forget  that  the  proposition  has  ever  been  in  anybody's 
mind.  By  this  means  we  get  to  the  concept  of  a  proposition- 
in-itself. 

In  the  distinction  that  he  draws  between  perception 
and  conception,  Bolzano  himself  says  that  he  owes  very 
much  to  Kant,  but  Bolzano  disagrees  with  him  in  the  use 
he  makes  of  this  distinction  in  his  theory  of  time  and 
space.  Bolzano  examines  in  some  detail  Kant's  theory  of 
time  and  space  and  his  theory  of  the  categories,  making 
some  very  acute  criticisms.  After  an  investigation  into  the 
theory  of  the  syllogism  and  a  discussion  of  the  function 


12  A.  E.  Taylor,  Mind,  October,  1915. 

For  a  criticism 
1905. 

Sulzbach,  1837. 


13  For  a  criticism  of  Bolzano's  theories  see  M.  Palagyi,  Kant  und  Bolzano. 
Halle,  1905. 


94  THE  MONIST. 

of  the  linguistic  expression  of  a  proposition,  the  "Theory 
of  Elements"  closes  with  a  criticism  of  previous  works  on 
the  subject.  Next  Bolzano  considers  the  appearance  in  the 
mind  of  propositions-in-themselves.  And  it  is  in  this  part 
of  his  work  in  particular  that  we  see  the  extent  and  depth 
of  his  learning.  He  treats  first  our  subjective  ideas,  then 
our  judgments,  then  the  relation  of  our  judgments  to 
truth,  and  finally  their  certainty  and  probability.  In  this 
investigation  Bolzano  uses  psychological  methods  to  some 
extent.  Then  after  the  fourth  part,  the  "Art  of  Inventing," 
he  comes  at  last  in  the  fifth  part  to  the  "Science  of  Knowl- 
edge Proper."  The  book  is  remarkable  as  much  for  its 
wealth  of  original  thought  and  the  clearness  of  expression 
as  for  the  important  criticisms  of  earlier  works  on  the 
subject. 

But  important  as  is  Bolzano's  work  in  metaphysics, 
ethics,  esthetics,  and  theology,  it  is  preeminently  as  a  math- 
ematician that  he  should  be  remembered.  Now  there  are 
two  ways  of  looking  at  mathematics.  One  can  look  upon 
it  as  Huxley  did:  "Mathematics  may  be  compared  to  a 
mill  of  exquisite  workmanship,  which  grinds  you  stuff  to 
any  degree  of  fineness."  On  the  other  hand,  one  can  look 
upon  mathematics  as  a  real  and  genuine  science  and  then 
the  applications  are  only  interesting  in  so  far  as  they  con- 
tain and  suggest  problems  in  pure  mathematics.  From  the 
second  point  of  view  the  most  important  business  of  the 
mathematician  is  to  examine  and  strengthen  the  founda- 
tions of  mathematics  and  to  purify  its  methods.  In  addi- 
tion to  these  points  of  view  which  may  be  called  the  prac- 
tical and  the  philosophical,  a  third  point  of  view  has  sprung 
up  in  the  last  century  which  may  be  called  the  purely  logical 
point  of  view.  Whitehead  describes  this  new  point  of  view 
in  the  words,  "Mathematics  in  its  widest  significance  is  the 
development  of  all  types  of  formal,  necessary,  deductive, 


BERNARD  BOLZANO.  95 

reasoning/'1  In  this  purely  logical  system,  it  is  proposed 
to  treat  any  special  development  of  mathematics  with  the 
help  of  a  definite,  logically  connected  complex  of  ideas, 
and  the  mathematician  is  not  to  be  satisfied  to  solve  par- 
ticular problems  with  the  help  of  any  methods  which  may 
casually  present  themselves,  however  ingenious  these  meth- 
ods may  be.  Clear  definitions  and  unambiguous  axioms 
must  be  framed  and  then  by  rigorous  reasoning  the  the- 
orems of  the  subject  are  to  be  deduced. 

We  find  examples  of  the  first  and  second  points  of  view 
among  the  Greeks.  It  is  said  of  Pythagoras  that  "he 
changed  the  occupation  with  this  branch  of  knowledge  into 
a  real  science,  inasmuch  as  he  contemplated  its  foundation 
from  a  higher  point  of  view  and  investigated  the  theorems 
less  materially  and  more  intellectually,"15  and  of  Plato 
that  "he  filled  his  writings  with  mathematical  discussions, 
showing  everywhere  how  much  geometry  there  is  in  phi- 
losophy." Just  as  mathematics  among  the  Greeks  had  its 
origin  in  the  geometry  invented  by  the  Egyptians  for 
practical  surveying  purposes,  so  the  mathematics  of  the 
seventeenth  and  eighteenth  century  received  its  stimulus 
from  the  practical  researches  of  Kepler,  Newton  and  La- 
place. But  in  this  same  fragment  of  Eudemus  we  find 
it  recorded  that  Euclid  tried  to  revise  the  methods  used 
and  "put  together  the  elements,  arranging  much  for  Eude- 
mus, finishing  much  for  Thaetetus ;  he  moreover  subjected 
to  rigorous  proofs  what  had  been  negligently  demonstrated 
by  his  predecessors." 

This  same  work  that  Euclid  did  for  Greek  mathematics 
three  hundred  years  B.  C,  the  new  school  of  nineteenth 
century  mathematicians  performed  for  European  mathe- 

10  A.  N.  Whitehead,  A  Treatise  on  Universal  Algebra,  Cambridge,  1898, 
preface,  p.  vi. 

16  Extract  from  a  fragment  preserved  by  Proclus ;  generally  attributed 
to  Eudemus  of  Rhodes  who  belongs  to  the  peripatetic  school  and  wrote  treat- 
ises on  geometry  and  astronomy.  See  extracts  in  J.  T.  Merz,  History  of 
European  Thought  in  the  Nineteenth  Century,  Vol.  II,  p.  634. 


g  THE  MONIST. 

matics.  The  researches  of  Newton  had  suggested  a  wealth 
of  material  for  mathematical  treatment.  Newton  a'nd 
Leibniz  had  stumbled  across  the  powerful  methods  of  the 
calculus,  which  were  of  tremendous  practical  importance; 
but  as  Klein  says,  "the  naive  intuition  was  especially  active 
during  the  period  of  the  genesis  of  the  calculus,"17  and  in 
the  great  call  for  powerful  methods  the  theoretical  side  was 
almost  entirely  overshadowed.  For  example  Newton  as- 
sumed the  existence  of  the  velocity  of  a  moving  point  at 
every  point  of  its  path,  not  troubling  whether,  as  subse- 
quent investigation  has  shown  to  be  the  case,  there  might 
not  be  continuous  functions  having  no  derivative.  The 
great  work  then  of  this  new  school  was  to  investigate  the 
validity  of  the  methods  used  in  the  two  previous  centuries. 
This  was  no  easy  task,  and  it  is  only  now  after  one  hundred 
years  that  the  theory  of  the  subject  is  being  put  on  a  logic- 
ally satisfactory  basis.  The  most  important  ideas  round 
which  the  greater  part  of  the  work  in  mathematics  cen- 
tered, are  those  of  continuity  and  infinity.  The  importance 
of  these  concepts  became  apparent  from  the  work  done  on 
infinite  series.  A  particularly  simple  example  of  series, 
viz.,  decimal  fractions,  was  in  use  as  early  as  the  sixteenth 
century,  but  Leibniz  was  the  first  mathematician  to  have 
any  idea  of  the  importance  of  series  in  mathematics.  Be- 
fore his  time  it  had  not  been  realized  that  an  infinite  series 
can  only  have  a  meaning  under  certain  circumstances.  Un- 
fortunately Leibniz  came  to  the  conclusion  that  the  sum 
of  the  series 

i  —  i  -|-  i  —  i. . . .ad  inf. 

is  5^2  >18  and  so  exercised  a  somewhat  baneful  influence  on 

17  Evanston  Colloquium;  Lectures  on  Mathematics  delivered  September, 
1893,  Lecture  VI. 

18  Euler  in  1755  (Instit.  Calc.  Diff.)  defined  the  sum  of  this  series  to  be 
J4-    In  the  recent  theory  of  divergent  series  (due  in  great  measure  to  E.  Borel 
see  his  Legons  sur  les  series  divergentes,  Paris,  1901)  one  way  of  denning  the 
formal  sum  of  a  divergent  series  So/i  is  as  the  limit,  when  it  exists,  of  *2>anXn 
as  x  tends  to  unity  through  values  less  than  unity.     This  definition  has  the 


BERNARD  BOLZANO.  97 

subsequent  mathematical  developments  of  the  theory  of 
infinite  series.  However  it  was  left  to  the  genius  of  Bol- 
zano19 to  enunciate  for  the  first  time  the  necessary  and 
sufficient  conditions  for  the  convergence  of  an  infinite  se- 
ries. In  1804  Bolzano  published  his  Betrachtungen  iiber 
einige  Gegenstdnde der Element ar geometric  (Prague),  and 
in  1810  his  Beytrdge  zu  einer  begrundeteren  Darstellung 
derMathematik  (Prague).  In  1816  he  published  an  impor- 
tant tract  on  the  binomial  theorem.  In  this  tract  his  work 
on  convergency  is  of  great  value  and  his  investigation  for 
a  real  argument  (which  he  everywhere  presupposes)  is  very 
satisfactory.  Bolzano  comments  on  the  unrestricted  use 
of  infinite  series  which  was  common  at  the  time.  In  1812 
Gauss  had  published  an  investigation  into  the  circum- 
stances under  which  the  hypergeometric  series  converges, 
and  in  1820  Cauchy  delivered  some  extremely  important 
lectures  on  analysis  at  the  College  de  France,  where  he 
was  the  leader  of  a  group  of  young  mathematicians.  Thus 
Bolzano,  Gauss  and  Cauchy  were  the  pioneers.  In  his 
book,  Der  binomische  Lehrsatz  und  als  Folgerung  aus 
ihm  der  polynomische  und  die  Reihen,  die  zur  Berechnung 
der  Logarithmen  und  Exponential  gross  en  dienen,  genauer 
als  bisher  erweisen  (Prague),  Bolzano  has  made  a  valu- 
able criticism  of  earlier  investigations.  It  is  remarkable 
that  his  writings,  though  of  great  importance,  received 
comparatively  little  attention  at  the  time.  According  to 
Merz,  he  had  not,  like  Cauchy,  "the  art  peculiar  to  the 
French  of  refining  their  ideas  and  communicating  them  in 

merit  of  simplicity  and  also  of  "consistency,"  i.  e.,  When  the  series  So»  con- 
verges, its  sum  is  still  the  limit  as  x  tends  to  unity  through  smaller  values,  of 
^anxn  if  this  limit  exists. 

Defining  the  formal  sum  in  this  way  the  sum  of  the  series  1  — 1  +  1... 
ad  inf.  is  Vz. 

19  Accounts  of  Bolzano's  mathematical  work  were  given  by  Otto  Stolz 
(Math.  Ann.,  Vol.  XVIII,  1881,  pp.  255-279;  Vol.  XXII,  1883,  pp.  518-519) 
and  on  pp.  37-39  of  the  notes  at  the  end  of  the  reprint  of  Bolzano's  "Rein 
analytischer  Beweis"  of  1817  in  No.  153  of  Ostivald's  Klassiker. 


98  THE  MONIST. 

the  most  appropriate  and  taking  manner."2  In  his  Rein 
analytischer  Beweis  (1817)  Bolzano  tells  us  that  it  is  very 
much  better  to  publish  one's  mathematical  work  in  separate 
treatises ;  in  this  way  there  is  more  chance  of  getting  acute 
criticism.  Consequently  we  find  his  mathematical  work 
scattered  about  in  various  small  treatises.21  Also  he  tells 
us  that  one  of  his  treatises  had  the  misfortune  not  to  be 
noticed  by  some  of  the  learned  periodicals  and  in  others  to 
be  criticized  only  superficially. 

In  1842,  in  the  course  of  some  work  on  the  undulatory 
theory  of  light,  he  made  a  prophecy  which  is  extremely 
interesting  in  the  light  of  the  invention  of  spectrum  anal- 
ysis and  the  researches  of  Sir  W.  Huggins,  Kirchhoff,  and 
others.  He  said:  "I  foresee  with  confidence  that  use  will 
hereafter  be  made  of  it  in  order  to  solve,  by  observing 
the  changes  which  the  color  of  stars  undergoes  in  time, 
the  questions  as  to  whether  they  move,  with  what  velocity 
they  move,  how  distant  they  are  from  us  and  much  else 
besides."  But  let  us  return  to  the  most  important  part  of 
Bolzano's  mathematical  investigations. 

In  1817  Bolzano  published  a  paper  we  have  already 
mentioned  entitled  "Rein  analytischer  Beweis  des  Lehr- 
satzes:  dass  zwischen  je  zwei  Werthen,  die  ein  entgegen- 
gesetztes  Resultat  gewahren,  wenigstens  eine  reele  Wurzel 
der  Gleichung  liege."  This  paper  is,  in  a  way,  his  most 
important  work  and  is  a  triumph  of  careful  and  subtle 
mathematical  analysis.  His  central  theorem,  as  the  title 
indicates,  is  as  follows :  If  in  an  equation  f(x)  —  o,  x  =  a 
makes  f(x)  positive  and  x  =  p  makes  f(x)  negative,  then 
there  is  at  least  one  real  root  of  the  equation  f(x)  =  o 
between  a  and  p.  Before  he  begins  his  constructive  work 
he  criticizes  very  acutely  the  previous  attempts  of  La- 
grange  and  others.  He  points  out  the  errors  that  had 

»o  Op.  cit.,  Vol.  II,  p.  709. 

81  For  complete  list  see  Bergmann,  op.  cit.,  pp.  213-214. 


BERNARD  BOLZANO.  99 

been  made  by  previous  investigators  and  he  emphasizes 
once  more  the  great  importance  of  freeing  mathematical 
analysis  from  the  intuitional  treatment  to  which  it  had 
formerly  been  subjected.  In  order  to  prove  his  main  theo- 
rem, Bolzano  found  it  necessary  to  introduce  the  concept 
of  the  continuity  of  a  function,  the  notion  of  the  upper 
limit  of  a  variate  and  some  important  work  on  infinite 
series.  His  method  is  briefly  as  follows: 

1.  He  introduces  the  concept  of  "continuity."    A  func- 
tion is  said  to  be  "continuous"  for  the  value  x  if  the  differ- 
ence between  /(jr-f-co)  and  f(x)  can  be  made  less  than 
any  assigned  number,  however  small,  if  only  CD  is  taken 
sufficiently  small. 

2.  He  discusses  the  convergence  of  infinite  series  and 
makes  the  following  important  statement.     "If  the  differ- 
ence between  the  value  of  the  sum  of  the  first  n  terms  and 
the  first  n-\-r  terms  of  a  series  can  be  made  as  small  as  we 
please,  for  all  values  of  r,  if  only  we  take  n  large  enough, 
then  there  is  one  number  X  and  only  one  such  that  the 
sum  of  the  first  p  terms  approaches  ever  more  and  more 
nearly  to  X  as  p  increases."    Unfortunately  his  proof  of 
this  theorem  is  not  rigorous  and  his  discussion  only  renders 
the  existence  of  X  probable. 

3.  From  his  work  on  infinite  series  Bolzano  passes  on 
to  an  extremely  important  theorem  in  which  he  introduces 
the  new  idea  of  an  upper  limit.     And  the  theorem,  as  it 
occurs  in  this  paper,  gains  in  importance  from  the  fact  that 
the  method  used  is  one  of  fundamental  importance  in  anal- 
ysis.   The  theorem  runs  as  follows :  "If  un  be  such  a  num- 
ber that  the  property  M  holds  for  all  values  of  x  which 
are  less  than  un,  and  if  the  property  does  not  hold  for 
all  values  of  x  without  exception,  then  of  all  the  num- 
bers un  satisfying  this  condition  there  is  one  (say  U)  which 
is  greater  than  all  the  others."    This  theorem,  which  might 
appear  obvious  to  those  who  allow  their  geometrical  in- 


IOO  THE  MONIST. 

tuitions  to  cloud  their  mathematical  ideas,  is  proved  by 
Bolzano  with  great  care  and  completeness.  The  method 
used  in  the  proof  was  used  a  great  deal  by  Weierstrass 
and  is  now  known  as  the  "Bolzano-Weierstrass"  process. 
As  the  method  is  of  such  great  importance,  we  will  indicate 
roughly  the  way  it  is  used  in  the  proof  of  this  theorem. 
It  will  be  convenient  to  call  ;tr's  which  have  the  property  M 
"suitable"  x's  and  ^r's  which  do  not  have  the  property  M 
"unsuitable"  .ar's;  and  further  to  call  a  number  N  a  "suit- 
able" number  if  all  x's  which  are  less  than  N  have  the 
property  M,  and  to  call  a  number  N  an  "unsuitable"  num- 
ber if  there  are  some  values  of  x,  less  than  N,  which  do 
not  have  the  property  M.  Now  it  is  obvious  that  there  is 
a  positive  number  D,  such  that  un  -)-D  is  an  unsuitable 
number.  Then,  bisecting  the  interval  between  un  and 
un  -f-  D,  we  get  the  number  un  -\-  D/2 ;  bisecting  the  inter- 
val between  un  and  un  -f-  D/2  the  number  un  -\-  D/22 ;  and 
so  on.  When  either  all  the  numbers  un  -\-  D/2r  for  r  = 
i,  2,  3.  ..  are  unsuitable  or  there  is  a  number  R  such 
that  un  -f  D/2R  is  an  unsuitable  and  un  -f  D/2R— l  a  suit- 
able number.  In  the  first  case  the  existence  of  U  is  estab- 
lished, U  being  equal  to  un.  In  the  second  case  we  repeat 
the  process,  dividing  the  interval  between  un  -\-  D/2R— l 
and  un  -f  D/2R.  Again,  either  all  the  numbers  un  +  D/2R 
+  D/2R+J,  s  —  i,  2.  . .  .  are  unsuitable  or  there  is  a  num- 
ber S  such  that  un  -f-  D/2R  -f-  D/2R— s  is  an  unsuitable 
and  un  +  D/2R  -f  D/2R+S  —  i  a  suitable  number.  We 
continue  the  same  process :  if  it  does  not  terminate  we  get 
finally  to  an  infinite  series 

UH  +  D/2R  +  D/2S  -f  D/2?  -f . . . 

and  since  R,  S,  T.  .  .  are  positive  integers  the  series  ob- 
viously satisfies  the  conditions  of  the  theorem  in  paragraph 
(2)  above,  and  so  there  is  a  definite  limit  to  which  it  tends. 


BERNARD  BOLZANO.  IOI 

this  limit  being  the  "upper  limit"  U  in  question.  The 
existence-theorem  for  an  upper  limit  is  thus  established. 

4.  Bolzano  next  attacks  the  following  theorem:  "f(x) 
and  <p(.tr)  are  continuous  functions  of  x  and  for  x  =  a, 
f(x)  <cp(.r)  and  for  x  —  |3,  f(x)  >q>(^r)  :  then  there  is 
a  value  of  x  between  a  and  (3  for  which  f(x)  =  <$(x)" 
We  will  indicate  the  method  Bolzano  uses  to  prove  it  and 
we  shall  see  exactly  why  he  found  it  necessary  to  establish 
the  existence  of  an  "upper  limit."     Bolzano  shows  that, 
since  f(x)  and  q>(x)  are  continuous,  there  is  a  number  co 
such  that  all  numbers  less  than  it  satisfy  the  relation 
cp(<x  +  co)  >  /(a  +  co).     Such  a  number  we  may  call  as 
in  paragraph  (3)  a  "suitable"  number.    Then  from  a  direct 
application  of  the  theorem  about  an  upper  limit  he  estab- 
lishes the  existence  of  an  upper  limit,  say  U,  for  all  suit- 
able numbers.     It  is  then  easy  to  show  that  /(a  -}-  U) 
cannot  be  less  than  cp(a  -f-  U)  and  cannot  be  greater  than 
qp(a  +  U)  and  is  therefore  equal  to  (p(a  -f-  U).     In  this 
kind  of  way  Bolzano  proves  the  existence  of  the  value  of 
x  between  a  and  P  giving  f(x)  =  (p(^). 

5.  Finally  Bolzano  proves  that  an  expression  of  the 
form 

a  +  bxm  +  cxn+ .  . .  +  pxr, 

in  which  m,  n,.  .  .r  are  positive  integers,  is  continuous. 
Then  by  means  of  an  easy  application  of  a  slightly  modi- 
fied form  of  the  theorem  in  (4)  he  proves  that  there  is  at 
least  one  real  root  between  a  and  [3.  The  whole  paper  is 
extremely  valuable  and  it  is  interesting  to  see  how  Bolzano 
was  led  from  his  central  theorem  to  the  theorem  in  (4),  to 
the  concept  of  "continuity"  and  the  idea  of  an  "upper 
limit,"  and  in  the  existence-theorem  for  the  upper  limit  to 
the  question  of  the  convergence  of  series. 

In  mathematical  logic  and  in  the  theory  of  infinite  num- 
bers, Bolzano's  work  was  also  of  great  importance.    Bol- 


IO2  THE  MONIST. 

zano's  definition  of  the  continuum  is  of  some  interest  in 
itself.  He  defines  a  continuum  as  a  set  of  points  such  that 
every  point  has  another  point  also  belonging  to  the  set  as 
near  to  it  as  we  please."  This  is  expressed  in  modern 
phraseology  by  saying  that  the  continuum  is  a  set  of  points 
which  is  "everywhere  dense."  The  name  continuum  is 
now  used  (after  Cantor)  only  for  a  set  of  points  which  is 
not  only  "everywhere  dense"  but  also  "perfect."  A  set  of 
points  is  "perfect"  when  every  convergent  sequence  has  a 
limit  which  is  itself  a  number  belonging  to  the  set,  and 
conversely  when  every  number  is  the  limit  of  properly 
chosen  convergent  sequences  of  numbers  themselves  be- 
longing to  the  set.23  Thus  Bolzano  would  call  the  set  of 
rational  numbers  a  "continuum,"  but  this  set  is  not  perfect 
and  is  therefore  not  a  "continuum"  in  the  modern  sense 
of  the  word.  In  his  work  on  infinite  numbers  Bolzano 
anticipated  to  some  extent  the  work  of  Georg  Cantor.  An 
"infinite"  collection  is  defined  to  be  a  collection  which  has 
no  last  term.24  He  proves  that  the  number  of  natural 
numbers  and  the  number  of  real  numbers  is  infinite,  and 
he  sees  (§49)  that  the  number  of  these  two  collections  is 
different.  Bolzano  also  recognizes  the  fact  that  it  is  pos- 
sible to  arrange  the  points  in  two  lines  of  different  lengths 
so  that  each  point  of  one  collection  corresponds  to  one 
single  point  of  the  other  collection  and  vice  versa,  no  point 
being  left  without  a  corresponding  point.  This  brilliant 
idea  of  a  one-one  correspondence  went  a  long  way  toward 
dispersing  the  cloud  of  mystery  which  hung  over  the  con- 
temporary infinite  number.  Leibniz  had  stated  the  diffi- 
culty quite  plainly.  Every  number  can  be  doubled,  he  said, 
therefore  the  number  of  natural  numbers  and  the  number 
of  even  natural  numbers  is  the  same.  Therefore  the  whole 

22  Paradoxien  des  Unendlichen,  Leipsic,  1851,  2d  ed.,  Berlin,  1889,  §  38. 

23  See  E.  W.  Hpbson,  The  Theory  of  Functions  of  a  Real  Variable  and 
the  Theory  of  Fourier's  Series,  Cambridge,  1907,  p.  49. 

24  Paradoxien  des  Unendlichen,  §  9. 


BERNARD  BOLZANO.  1 03 

is  equal  to  the  part — which  is  absurd.  Bolzano  realized 
that  there  is  no  real  contradiction  in  this.  This  same  idea 
of  the  one-one  correspondence  between  points  belonging 
to  certain  sets  of  points  has  led  to  the  modern  idea  of  "re- 
flexiveness"  of  infinite  numbers.  The  property  of  "re- 
flexiveness"20  together  with  that  of  "non-inductiveness,"a 
which  disposes  of  all  attempts  to  count  up  infinite  collec- 
tions or  identify  the  number  of  terms  in  an  infinite  collec- 
tion with  the  ordinal  number  of  the  last,  has  removed  all 
serious  difficulties  and  has  helped  to  make  it  possible  to 
put  the  concept  of  an  infinite  number  on  a  logical  founda- 
tion.27 Defining  "similar"  classes  as  classes  whose  terms 
have  a  one-one  relation  to  each  other  and  the  "cardinal 
number"  or  "power"  of  a  class  as  the  class  of  all  similar 
classes,  we  see  immediately  that  the  class  of  natural  num- 
bers and  the  class  of  even  natural  numbers  have  the  same 
cardinal  numbers.  Thus  Bolzano  was  quite  right  in  seeing 
no  contradiction  in  Leibniz's  statements. 

From  these  few  references  to  isolated  theorems  and 
statements  in  Bolzano's  work,  it  is  seen  that  he  had  most 
of  the  ideas  essential  in  the  modern  view  of  mathematics, 
and  that  in  mathematics  at  least  Bolzano's  work  has  been 
a  source  of  inspiration  to  those  who  came  after  him. 
Whether  in  his  theology,  his  ethics,  his  political  science, 
his  metaphysics,  or  his  mathematics,  the  desire  for  clear- 
ness of  concepts  was  always  his  aim.  Even  the  parts  of 
his  work  which  are  no  longer  of  intrinsic  interest,  e.  g., 
his  esthetics  or  his  theory  of  laughter,  have  an  interest 
for  us  in  that  they  show  us  the  methods  he  used  in  seeking 

25  A  number  is  said  to  be  "reflexive"  if  it  is  not  increased  by  adding  one 
to  it.  See  B.  Russell,  Our  Knowledge  of  the  External  World  as  a  Field  for 
Scientific  Method  in  Philosophy,  Chicago  and  London,  1914,  p.  190. 

28  A  number  is  said  to  be  "non-inductive"  if  it  does  not  possess  deductive 
properties.  See  B.  Russell,  op.  cit.,  p.  195. 

27  Cf.  the  definitions  "that  which  cannot  be  reached  by  mathematical  in- 
duction starting  from  1"  and  "that  which  has  parts  which  have  the  same  num- 
ber of  terms  as  itself,"  B.  Russell,  The  Principles  of  Mathematics.  Cambridge. 
1903,  Vol.  I,  p.  368. 


IO4  THE  MONIST. 

for  truth.  That  there  is  objective  truth  and  that  we  can 
have  knowledge  of  it — this  was  the  thesis  which  he  set 
before  him  in  his  work.  In  mathematics  especially  his 
work  was  needed,  for  whereas  idealists  maintained  that 
mathematics  deals  only  with  appearances,  empiricists  in- 
sisted that  mathematics  could  only  approximate  to  the 
truth.  Bolzano's  life  work  was  to  start  mathematicians 
on  the  right  way  to  refute  both  the  idealists  and  the  em- 
piricists. His  method  of  strictly  logical  analysis  of  the 
ideas  of  continuity  and  the  infinite  was  the  clue  which  was 
followed  up  by  all  the  great  mathematical  logicians  and 
mathematical  analysts  of  the  nineteenth  century,  until 
finally  the  fundamental  thesis  has  been  proved  that  all 
concepts  of  pure  mathematics  are  wholly  logical.  Thus 
Bolzano  was  one  of  the  first  to  suspect  and  in  this  he  was 
a  worthy  successor  of  the  great  Leibniz.  Unlike  most 
mathematicians  of  his  day,  Bolzano  did  not  in  his  thirst 
for  results  succumb  to  D'Alembert's  maxim,  Allez  en 
avant,  la  foi  vous  viendra. 

We  live  in  days  when  some  of  the  contradictions  and 
paradoxes  which  have  perplexed  the  human  race  since  the 
days  of  Zeno  are  being  finally  cleared  up.  Do  not  let  us 
forget  the  work  of  Bolzano  who  with  painstaking  endeavor 
sowed  the  seeds  of  this  great  revolution  in  mathematical 
ideas. 

DOROTHY  MAUD  WRINCH. 

CAMBRIDGE,  ENGLAND. 


A  MEDIEVAL  INTERNATIONALIST. 

A  RBITRATION,  a  league  of  peace  and  a  council  of 
1~Y  conciliation  seem  to  be  very  modern  suggestions  as 
methods  of  avoiding  war  between  civilized  nations.  Some 
hints  of  these,  however,  can  be  found  in  Kant's  Perpetual 
Peace  and  in  the  grand  dessein  as  expounded  by  the  Abbe 
de  S.  Pierre.  These  schemes  belong  to  the  Revolutionary 
and  Renaissance  periods.  But  even  before,  in  the  Middle 
Ages,  similar  schemes  are  to  be  found  in  the  work  of 
Petrus  de  Bosco  (Pierre  Dubois). 

The  political  acuteness  of  this  brilliant  thinker  can  only 
be  understood  by  allowing  for  the  fact  that  he  had  listened 
at  Paris  to  "that  most  prudent  friar  Thomas  Aquinas"1 
and  by  remembering  that  he  wrote  while  the  official  poli- 
ticians were  engineering  war  after  war  for  no  purpose. 
His  work  on  international  politics  is  contained  in  the  un- 

printed  Summaria  brevis abbreviations  guerrarum 

and  in  the  "De  recuperatione  Terre  Sancte,"  published 
(1891)  in  the  Collection  des  Textes.  I  propose  to  sum- 
marize and  comment  upon  the  latter,  not  as  of  merely 
archeological  interest,  but  as  an  early  attempt  to  grapple 
with  the  same  political  problem  which  we  now  face. 

The  treatise  is  supposed  to  deal  with  a  plan  for  recov- 
ering the  Holy  Land  and  is  addressed  in  1306  to  Edward  I, 
"King  of  England  and  Scotland,  Lord  of  Ireland  and  Duke 
of  Aquitaine,"  as  a  great  legislator  and  one  who  was 

1  Par.  63,  De  recup.  Terre  Sancte.    (In  medieval  Latin  final  ae  becomes  <?.) 


IO6  THE  MONIST. 

specially  interested  in  a  new  crusade.  But  this  initial  pur- 
pose of  the  treatise,  even  if  it  was  intended  by  the  author 
as  more  than  a  mere  captatio  benevolentiae ,  is  certainly 
subordinated  to  the  general  problem  of  international  policy 
among  the  European  states.2  The  order  of  the  argument 
is  confused,  the  author  continually  going  back  to  a  subject 
after  he  has  left  it  for  some  other.  He  writes  well,  but  too 
eagerly  to  be  as  exact  as  the  philosophers  of  his  day.  He 
is  genuinely  excited  by  the  pressing  importance  of  estab- 
lishing peace.  I  shall,  therefore,  not  follow  the  order  of 
the  treatise,  but  state  first  the  nature  of  the  problem  as  it 
appears  to  Dubois  and  then  his  suggestions  for  solution. 

War  between  European  countries  and  kings,  says  Du- 
bois, is  the  chief  hindrance  to  "having  time  for  progress 
in  morality  and  knowledge."  War  breeds  war  until  war 
becomes  a  habit.8  The  deaths  of  one  war  cause  speedy 
preparations  for  revenge.4  "We  should  seek  a  general  peace 
and  pray  God  for  it,  that  by  peace  and  in  time  of  peace  we 
may  progress  in  morality  and  the  sciences,  since  we  cannot 
otherwise ;  as  the  Apostle  feels  when  he  says :  'The  peace  of 
God  which  passeth  all  understanding  keep  your  hearts  and 
your  minds:'  your  minds,  which  are  reasonable  souls,  are 
not  kept  but  are  often  destroyed  by  wars,  discords  and  civil 
brawls  which  are  like  wars,  and  by  the  continuance  of  all 
such.  Therefore,  as  far  as  he  can,  every  good  man  should 
avoid  and  flee  them ;  and  when  he  takes  to  war,  being  un- 
able otherwise  to  obtain  his  rights,  he  ought  as  much  as 
possible  to  shorten  it ....  Thus  universal  peace  is  the  end 
we  seek."8 

2  Guillaume  de  Nogaret  uses  the  same  pious  cover  for  his  scheme  of 
social  reform.  One  had  to  bow,  so  to  speak,  to  the  crusading  ideal  and  then 
one  was  free  to  suggest  anything ! 

8  Quanto  frequentius  bella  committunt,  tanto  magis  appetunt  committere, 
hoc  consuetudine  magis  quam  emendatione  deputantes."  Par.  2. 

4  "Ad  bellum  et  vindictam  voluntariam  se  preparant." 
6  Par.  27,  in  line. 


A  MEDIEVAL  INTERNATIONALIST.  IO7 

It  is  agreed  that  peace  is  desirable;  but,  says  Dubois, 
"since  it  is  proved  that  neither  the  Scriptures,  nor  sermons 
drawn  from  the  Scriptures,  nor  the  elegant  lamentations 
and  exhortations  of  preachers  have  been  sufficient  to  stop 
frequent  wars  and  the  temporal  and  eternal  death  of  so 
many  human  beings  which  have  resulted,  why  should  there 
not  be  found  at  last  a  new  remedy  for  militarism  (reme- 
dium  manus  militaris),  as  for  example  a  judiciary  backed 
by  force  (justicia  necessario  compulsiva)  ?"  (par.  109). 
"This  is  an  argument,"  he  declares,  "to  which  a  reply  is 
impossible  morally  and  politically  speaking."  Peace  has 
come  within  states  by  vis  coactiva:  so  also  it  will  come 
between  states.  One  could  not  have  a  clearer  statement 
of  political  judgment  upon  the  evidence.  The  author  him- 
self says  that  he  depends  upon  experience  for  his  opinions : 
and  he  declares  that  exhortations  to  peace  and  praise  of  its 
excellencies  and  even  rhetorical  attacks  on  war  are  polit- 
ically valueless.  They  have  been  tried  and  they  have  failed. 

Before  speaking,  however,  of  the  means  by  which  peace 
is  to  be  established  between  states,  we  must  notice  the 
plan  which  is  not  suggested  by  Pierre  Dubois.  The  gov- 
erning ideal  of  medieval  politics,  unity,  led  many  to  look 
for  peace  through  subordination  to  one  overlord.  "Now 
there  is  no  sane  man,  I  think,"  Dubois  writes,  (par.  63), 
"who  could  think  it  likely  that  in  this  latest  age  (in  hoc 
•fine  saeculorum)  there  could  be  one  monarch  of  the  whole 
world  in  temporal  affairs  who  would  rule  all  and  whom 
as  superior  all  would  obey.  For,  if  there  were  any  attempt 
at  this  there  would  be  wars,  seditions  and  discords  without 
end;  nor  would  there  be  any  one  who  could  allay  them  by 
reason  of  the  number  of  different  nations,  the  distance  and 
distinction  between  countries  and  the  natural  inclination 
of  men  to  diverge.  Although  some  have  been  popularly 
called  "lords  of  the  world"  nevertheless  I  think  that  since 
the  countries  were  settled  there  never  has  been  any  one 


IO8  THE  MONIST. 

whom  all  obeyed."  That  passage,  if  it  seems  to  condemn 
Dante  as  a  homo  non  sane  mentis,  certainly  shows  an  his- 
torical acumen  and  a  political  judgment  far  superior  to  the 
opinions  of  the  De  Monarchia.  Dubois  recognizes  the  im- 
possibility of  arriving  at  peace  by  means  of  the  conquest  by 
one  state  of  all  other  states.  He  sees  that  world-power  is 
nonsense. 

It  must  be  admitted,  however,  that  from  the  passages 
of  the  Summaria  brevis  which  have  been  commented  upon 
by  M.  de  Wailly  and  Ernest  Renan,  one  might  judge  that 
Dubois  hoped  for  a  domination  in  Europe  of  the  French 
king.  He  held,  indeed,  that  it  should  be  arrived  at  by  dip- 
lomacy and  not  by  war,  but  in  the  above  passage  of  the 
De  recuperatione  he  seems  to  condemn  not  merely  any 
special  means,  but  dreams  of  domination  by  a  single  lord. 

Inconsistency  may  be  urged  against  him,  and  yet  it  must 
be  remembered  that  here  he  is  writing  to  the  English  king 
and  also  that  he  may  very  well  have  felt  uncertain  as  to 
how  the  vis  coactiva  above  the  warring  states  might  be 
established,  even  if  he  held  quite  clearly  to  the  notion  that 
the  ultimate  supremacy  of  one  monarch  was  impossible. 
But  let  us  turn  to  the  definite  political  means  he  suggests 
for  establishing  peace  between  European  states. 

The  means  by  which  such  peace  is  to  be  arrived  at  are : 
First:  International  arbitration  and  the  establishment  of 
an  international  judiciary.  This  is  to  begin  by  a  general 
council  (par.  3),  a  preliminary  to  all  medieval  and  early 
Renaissance  plans  for  reform.  But  what  is  unusual  in 
Pierre  Dubois  is  the  statement  that  the  difficulty  of  arran- 
ging matters  is  due  to  the  fact  that  the  cities  of  Italy,  for 
example,  and  the  various  princes  acknowledge  no  superior. 
"Before  whom  then,"  he  asks,  "can  they  bring  their  dis- 
putes? It  can  be  answered  that  the  council  should  estab- 
lish elected  arbiters  (arbitros)  religious  or  others,  prudent, 
experienced  and  trustworthy  men."  These  are  to  select 


A  MEDIEVAL  INTERNATIONALIST. 

three  prelates  and  three  others  for  either  party  to  the  dis- 
pute. They  are  to  be  well  paid  and  such  as  are  not  likely 
to  be  corrupted  by  affection,  hate,  fear,  greed  or  otherwise. 
They  are  to  meet  at  a  suitable  place,  to  have  presented  to 
them  in  a  summary  and  clear  form,  without  minor  and 
unimportant  details,  the  pleas  of  either  side.  They  are 
to  take  evidence  from  witnesses  and  documents,  each  wit- 
ness being  examined  by  at  least  two  trustworthy  and  care- 
ful members  of  the  "jury."  The  depositions  are  to  be  writ- 
ten and  preserved.  "For  the  decision,  if  it  is  expedient, 
they  are  to  have  assessors  (assessores)  who  are  thought 
by  them  most  trustworthy  and  best  trained  in  the  divine, 
the  canon  and  the  civil  law."6 

Secondly,  these  decisions  must  be  made  effective.  The 
Holy  See  is  recognized  as  an  influnce,  but  excommunica- 
tion had  better  not  be  used.  "Temporal  punishment,  al- 
though incomparably  less  than  eternal,  will  be  more 
feared."7  The  suggestions  in  detail  of  Pierre  Dubois  are 
perhaps  a  little  comic,  but  we  must  allow  for  the  situation. 
In  the  first  place  any  group  making  war  shall,  after  the  war 
is  over,  be  removed  bodily  and  sent  to  colonize  the  Holy 
Land !  If  they  do  not  oppose  the  movement,  they  may  take 
some  of  their  property  with  them.  The  author  feels  that 
it  may  be  difficult.  He  then  goes  on  as  to  other  measures. 
Suppose,  he  says,  that  the  Duke  of  Burgundy  declares  war 
against  the  King  of  France, — the  king  should  then  institute 
an  economic  boycott8  and  by  a  general  council  the  same 
boycott  should  be  declared  by  all  Europe.  Active  military 
measures  should  be  taken  to  devastate  the  country  so  that 
the  whole  people  should  feel  it:  Dubois,  it  seems,  would 
adopt  extreme  measures  to  prevent  war  spreading,  his  main 

8  Par.  12,  De  recup.  Terre  Sancte. 

7  Excommunication  is  to  be  used  (§  101)  but  not  depended  upon  by  itself. 
Any  one  refusing  to  enter  the  league  of  peace  (pacts  universalis  federa)  is  to 
be  immediately  attacked. 

8  Prohibebit  quod  nullus  ad  terras  eorum  deferat  victualia,  arma,  merces 
et  alia  quaecumque  bona,  etiam  quacumque  causa  sibi  debita,"  (par.  5). 


I IO  THE  MONIST. 

point  being  that  in  whatever  corner  it  broke  out  the  whole 
of  Europe  should  act  together  and  at  once  to  stop  it. 

The  reader  may  feel  that  this  is  hopelessly  unpractical, 
since  we  could  not  act  thus  against  any  great  country  or 
against  any  combination  of  countries.  But  we  must  re- 
member ( i )  that  Dubois  supposes  Europe  to  be  one  polit- 
ical system  (respublica  Christicolarum)  able  to  act  in  con- 
cert at  least  in  some  issues,  and  (2)  that  every  war  begins, 
according  to  him,  in  some  comparatively  small  group.  Thus 
practically,  if  Europe  had  adopted  strong  economic,  even 
without  military,  action  during  the  Balkan  wars  of  1912 
and  1913,  the  war  of  1914  might  never  have  occurred. 
And  surely  it  is  not  unpractical  to  suggest  that  all  civilized 
countries  should  act  together  in  the  case  of  any  conflict 
breaking  out  such  as  that  of  1912.  Deal  effectively  with 
the  small  conflicts  and  the  first  difficulty  is  met  with  regard 
to  the  larger.  But  one  can  imagine  the  horror  of  medieval 
diplomatists  if  all  the  states  were  asked  to  prevent  any  small 
wars  by  direct  intervention  of  enforcing  arbitration.  Even 
to-day  all  the  schemes  for  rearranging  international  politics 
start  from  the  present  almost  universal  war.  I  cannot  help 
feeling,  however,  that  Dubois  was  right.  Our  schemes  for 
doing  without  war  must  inculcate  combined  action  in  small 
wars.  Deal  effectively  with  them  and  we  may  never  have 
to  deal  at  all  with  war  between  great  states.  It  is  the 
spark,  not  the  conflagration,  that  we  must  consider  first: 
and  perhaps  European  diplomacy  was  more  futile  in  1912 
than  in  July  1914,  although  the  results  of  inaction  did  not 
show  themselves  till  August,  1914.  But  let  us  return  to 
the  general  thesis  and  omit  further  applications  of  it. 

After  details  as  to  raising  funds  for  a  common  force 
and  plans  for  a  common  advance  on  the  Holy  Land,  Dubois 
recalls  himself  to  his  main  interest,  "a  general  peace."  In 
the  third  place  therefore,  he  says  that  no  external  measures 
will  be  effective  until  the  religious  attitude  is  changed. 


A  MEDIEVAL  INTERNATIONALIST.  Ill 

This  opens  an  elaborate  project  for  the  reform  of  the 
Roman  church.  Dubois  says  (par.  29)  if  the  pope  really 
wants  to  stop  war  "he  must  begin  with  his  brothers  the 
cardinals  and  his  fellow  bishops."  They  are  always  going 
to  war  (ipsi  guerras  movent).  Their  attitude  is  quarrel- 
some even  in  England  and  France  where  they  do  not  ac- 
tually fight.  The  monks  are  as  bad.  But  the  whole  attack 
is  common  to  many  writers  of  the  date  of  Pierre  Dubois. 
His  remedies  are  extreme.  First  he  suggests  that  if  the 
pope  had  no  "temporal  power/'  no  one  need  to  go  to  war 
for  him  and  that  would  be  a  beginning;  and  next,  he 
actually  proposes  the  confiscation  of  ecclesiastical  property 
by  states  and  the  use  of  the  wealth  for  common  European 
civilization  !8  But  how  ? 

The  fourth  suggestion  of  Pierre  Dubois  is  that  the 
money  should  be  spent  in  education.10  The  purpose  of  the 
education,  according  to  the  general  thesis  as  to  the  taking 
of  the  Holy  Land,  is  directed  by  the  general  need  of  non- 
military  contact  with  the  East.  It  is  urged  that  you  can  only 
hold  the  East  effectively  by  intellectual  superiority  to  it. 

Then  begins  a  long  and  elaborate  scheme  of  education, 
primary  and  secondary.  University  education  is  implied 
but  not  dealt  with  in  detail.  All  this  is  to  occur  in  the  Holy 
Land.  It  is  a  well-known  medieval  trick  for  writing  a 
Utopia.  In  1223  'The  Complaint  of  Jerusalem"  gave  a 
plan  for  reconstructing  European  society  under  the  guise 
of  a  scheme  for  an  Eastern  kingdom.  So  here  Dubois, 
appearing  to  speak  of  what  ought  to  be  done  when  the  Holy 
Land  is  established  as  a  state,  is  really  speaking  of  the 
remedies  which  ought  to  be  applied  in  Europe.  In  the 
matter  of  education  he  is  as  original  as  in  politics,  but  what 
is  most  interesting  to  us  now  are  the  hints  for  bringing 

•  Par.  57.  "Que  tendit  ad  reformationem  et  unitatem  veram  totius  rei- 
publice  catholicorum." 

10  Par.  60,  "Studentcs  et  corum  doctorcs  vivent  de  bonis  dictorum  priora- 
tum.  etc," 


112  THE  MONIST. 

the  European  nations  together.  Colleges  for  boys  and  for 
girls  are  to  be  established  where  "modern  languages"  are 
to  be  taught — "the  literary  idioms,  especially  of  Europe, 
that  by  these  scholars  trained  to  speak  and  write  the  lan- 
guages of  all,  the  Roman  church  and  the  princes  of  Europe 
should  be  able  to  communicate  with  all  men."  Some  are 
also  to  be  taught  medicine,  some  surgery — the  girls  also 
(par.  61)  ;  and  these  girls,  in  the  medieval  fashion  perhaps, 
are  to  be  married  to  foreigners,  even  Orientals  (ditioribus 
Orientalibus  in  uxores  dari).  I  need  not  detail  the  plans 
for  intermarriage  and  colonization,  among  which  is  in- 
serted a  suggestion  for  a  married  clergy  (par.  102).  A 
long  section  follows  upon  the  utility  of  scientific  knowledge 
"according  to  brother  Roger  Bacon"  (par.  79)  and  upon 
the  variety  of  human  knowledge  in  general.  There  are 
interesting  hints  as  to  the  transformation  of  convents  into 
girls'  schools,  and  as  to  military  reform,  for  example  the 
institution  of  definite  uniforms  (par.  16).  But  all  these 
do  not  bear  directly  upon  his  plans  for  peace  and  we  may 
therefore  omit  them  here.  His  boldness  of  conception  is  clear. 

The  other  element  in  his  Utopia,  which  is  to  establish 
peace,  is  a  modification  of  the  processes  of  law  (par.  90 f.). 
The  processes  must  be  shortened  according  to  a  definite 
plan;  but  the  detail  need  not  concern  us  here.  The  fact 
remains  that  he  saw  that  social,  educational  and  religious 
reform  within  the  state  are  all  means  for  the  attainment 
of  international  peace. 

The  closing  section  of  the  work  (110-142)  are  ad- 
dressed to  Philip,  king  of  France,  who  is  asked  to  send  the 
preceding  to  Edward  I.  Dubois  urges  the  economic  gain 
from  the  abolition  of  wars,  and  in  the  meantime  the  insti- 
tution of  various  military  reforms — as  for  example  the 
regular  payment  of  troops.  It  is  amusing  to  note  that  the 
author  feels  the  danger  to  himself  from  the  powers  that 
be,  if  his  projects  are  made  too  public.  He  therefore  asks 


A  MEDIEVAL  INTERNATIONALIST.  113 

both  Edward  I  and  Philip  to  consider  his  ideas  more  or  less 
privately;  and  he  hints  that  one  who  does  not  happen  to 
hold  popular  opinions  may  suffer  even  physical  assault. 

So  far  as  we  know  nothing  evil  happened  to  Pierre 
Dubois.  He  was  a  lawyer  who  worked  first  for  the  king  of 
France  and  afterward,  when  he  wrote  the  De  recupera- 
tione,  in  the  service  of  Edward  I  in  Guyenne.  He  seems 
to  have  represented  the  central  government  in  either  case, 
and  to  have  found  his  chief  opponents  among  the  church- 
men. He  is  known  as  the  author  of  a  popular  pamphlet 
in  French  against  papal  claims,  as  the  writer  of  a  few 
short  Latin  treatises,  and  as  the  elected  representative  of 
Coutances  at  the  Etats  Generaux  which  met  in  Tours  in 
1308.  After  that  nothing  is  known  of  him. 

More  than  six  hundred  years  have  gone  since  the  trea- 
tise of  Pierre  Dubois  was  forgotten :  and  one  may  well  rub 
one's  eyes  in  wonder  at  what  is  now  occurring  in  Europe. 
Perhaps  we  are  dreaming.  The  practical  man  will  say 
that  the  old  plans  for  political  reform  are  by  current  events 
proved  to  be  valueless ;  that  the  internationalists  are  shown 
by  the  facts  to  be  unable  to  understand  real  politics.  And 
yet  one  would  have  thought  that  any  plan  might  have  been 
better  worth  trying  than  one  which  has  brought  us  to  our 
present  pass.  However  that  may  be  we  should  not  despair 
too  soon.  Ecclesiastical  reformation  was  suggested  for 
hundreds  of  years  before  Europe  arrived  at  the  compara- 
tively tolerant  situation  in  religion  now  established.  Polit- 
ical reformation  may  be  more  difficult,  but  the  work  of  its 
forerunners  is  important.  Si  Lyra  non  lyrasset,  Lutherus 
non  saltasset:  so  also  in  politics,  the  effective  reformer  is 
taught  by  his  predecessors  who  found  the  circumstances  of 
their  time  too  strong  for  them. 

C.  DELISLE  BURNS, 

LONDON,  ENGLAND. 


CLASS,  FUNCTION,  CONCEPT,  RELATION.1 

IN  my  Grundlagen  der  Arithmetik  of  1884  I  have  tried 
to  make  it  seem  probable  that  arithmetic  is  a  branch 
of  logic  and  need  not  borrow  any  ground  of  proof  what- 
ever from  experience  or  intuition.  The  actual  demonstra- 
tion of  my  thesis  is  carried  out  in  my  Grundge seize  of  1893 
and  1903  by  the  deduction  of  the  simplest  laws  of  numbers 
by  logical  means  alone.  But  to  make  this  proof  convincing, 
considerably  higher  claims  must  be  made  for  deduction 
than  is  habitually  done  in  arithmetic.2  A  set  of  a  few 
methods  of  deduction  has  to  be  fixed  beforehand,  and  no 
step  may  be  taken  which  is  not  in  accordance  with  them. 
Consequently,  when  passing  over  to  a  new  judgment  we 
must  not  be  satisfied,  as  mathematicians  seem  nearly  al- 
ways to  have  been  hitherto,  with  saying  that  the  new  judg- 
ment is  evidently  correct,  but  we  must  analyze  each  step  of 
ours  into  the  simple  logical  steps  of  which  it  is  composed, 
— and  often  there  are  not  a  few  of  these  new  steps.  No 
hypothesis  can  thus  remain  unnoticed.  Every  axiom  which 
is  needed  must  be  discovered,  and  it  is  just  the  hypotheses 
which  are  made  tacitly  and  without  clear  consciousness 
that  hinder  our  insight  into  the  epistemological  nature  of 
a  law. 

In  order  that  such  an  undertaking  be  crowned  with 
success,  the  concepts  which  we  need  must  naturally  be  con- 

1  [Translated  from  the  Grundgesetze  der  Arithmetik  by  Johann  Stachel- 
roth  and  Philip  E.  B.  Jourdain.] 

2  Grundlagen,  pp.  102-104. 


CLASS,  FUNCTION,  CONCEPT,  RELATION.  1 15 

ceived  distinctly.  This  is  true  especially  in  what  concerns 
the  thing  that  mathematicians  denote  by  the  word  "ag- 
gregate" (Menge).  It  seems  that  Dedekind,  in  his  book 
Was  sind  und  was  sollen  die  Zahlenf3  of  1888,  uses  the 
word  "system"  to  denote  the  same  thing.  But  in  spite  of 
the  exposition  which  appeared  four  years  earlier  in  my 
Grundlagen,  a  clear  insight  into  the  essence  of  the  matter 
is  not  to  be  found  in  Dedekind's  work,  though  he  often  gets 
somewhat  near  it.  This  is  the  case  in  the  sentence:4  "Such 
a  system  S  is  completely  determined  if  of  everything  it  is 
determined  whether  it  is  an  element  of  S  or  not.  Hence 
the  system  S  is  the  same  as  the  system  T  (in  symbols 
S  =  T)  if  every  element  of  S  is  also  element  of  T  and  every 
element  of  T  is  also  element  of  S."  In  other  passages,  on 
the  other  hand,  Dedekind  strays  from  the  point.  For  in- 
stance:5 "It  very  frequently  happens  that  for  some  reason 
different  things  a,  b,  c, . .  .  can  be  considered  from  a  com- 
mon point  of  view,  can  be  put  together  in  the  mind,  and  we 
then  say  that  they  form  a  system  S."  Here  a  presentiment 
of  the  correct  idea  is  contained  in  the  words  "common 
point  of  view";  but  the  "putting  together  in  the  mind"  is 
not  an  objective  characteristic.  In  whose  mind,  may  I 
ask?  If  they  are  put  together  in  one  mind  and  not  in 
another  do  they  then  form  a  system?  What  is  to  be  put 
together  in  my  mind  must  doubtless  be  in  my  mind.  Then 
do  not  things  outside  myself  form  systems?  Is  a  system 
a  subjective  formation  in  each  single  soul?  Is  then  the 
constellation  Orion  not  a  system?  And  what  are  its  ele- 
ments? The  stars,  the  molecules,  or  the  atoms?  The  fol- 
lowing sentence8  is  remarkable:  "For  uniformity  of  ex- 
pression it  is  advantageous  to  admit  the  special  case  that  a 
system  S  is  composed  of  a  single  (one  and  one  only)  ele- 
ment a:  the  thing  a  is  an  element  of  S,  but  every  thing 

8  [English  translation  under  the  title  Essays  on  the  Theory  of  Numbers, 
Chicago  and  London,  1901.     See  especially  p.  45.] 

*  [Ibid.,  p.  45.]  6  [Ibid.]  e 


Il6  THE  MONIST. 

different  from  a  is  not  an  element  of  S."  This  is  after- 
ward7 understood  in  such  a  way  that  every  element  ^  of  a 
system  S  can  be  itself  regarded  as  a  system.  Since  in  this 
case  element  and  system  coincide,  it  is  here  quite  clear  that, 
according  to  Dedekind,  the  elements  are  the  proper  con- 
stituents of  a  system.  Ernst  Schroder  in  his  lectures  on 
the  algebra  of  logic8  goes  a  step  in  advance  of  Dedekind  in 
drawing  attention  to  the  connection  of  his  systems  with 
concepts,  which  Dedekind  seems  to  have  overlooked.  In- 
deed, what  Dedekind  really  means  when  he  calls  a  system 
a  "part"  of  a  system9  is  that  a  concept  is  subordinated  to  a 
concept  or  an  object  falls  under  a  concept.  Neither  Dede- 
kind nor  Schroder  distinguish  between  these  cases  because 
of  a  mistake  in  point  of  view  which  is  common  to  them 
both.  In  fact,  Schroder  also,  at  bottom,  considers  the  ele- 
ments to  be  what  really  make  up  his  class.  An  empty  class 
should  not  occur  with  Schroder  any  more  than  an  empty 
system  with  Dedekind.  But  the  need  arising  from  the 
nature  of  the  matter  makes  itself  felt  in  a  different  way 
with  each  writer.  Dedekind  says:10  "On  the  other  hand, 
we  intend  here  for  certain  reasons  wholly  to  exclude  the 
empty  system,  which  contains  no  element  at  all,  although 
for  other  investigations  it  may  be  convenient  to  invent 
(erdichten)  such  a  system."  Thus  such  an  invention  is 
permitted;  it  is  only  desisted  from  for  certain  reasons. 
Schroder  dares  to  invent  an  empty  class.  Apparently  then 
both  agree  with  many  mathematicians  in  holding  that  we 
may  invent  anything  we  please  that  does  not  exist, — even 
what  is  unthinkable;  for  if  the  elements  form  a  system, 
then  the  system  is  annulled  at  the  same  time  as  the  ele- 

7  [Ibid.,  p.  46.] 

8  Vorlesungen  uber  die  Algebra  der  Logik  (exakte  Logik),  Vol.  I,  Leipsic, 
1890,  p.  253.     [This  reference  of  Frege  seems  wrong  and  it  should  perhaps 
rather  be  to  such  a  page  as  p.  100.  Cf.  also  Frege's  later  critical  study :  "Kriti- 
sche  Beleuchttmg  einiger  Punkte  in  E.  Schroders  Vorlesungen  uber  die  Al- 
gebra der  Logik,"  Archiv  fur  systematische  Philosophic,  Vol.  I,  1895,  pp.  433- 
456.] 

9  [Op.  cit.,  p.  46:]  10  [Ibid.,  pp.  45-46.] 


CLASS,  FUNCTION,  CONCEPT,  RELATION. 

ments.  As  to  where  the  limits  of  this  license  lie  and 
whether  indeed  there  are  any  such  limits,  without  any 
doubt  we  will  not  find  much  clearness  and  agreement; — 
and  yet  the  correctness  of  a  proof  may  depend  on  such 
questions.  I  believe  I  have  settled  them  in  a  way  that  is 
final  for  all  intelligent  persons,  in  my  Grundlagen11  and  in 
my  lecture  "Ueber  formale  Theorien  der  Arithmetik."1 
Schroder  invents  his  zero-class  and  thus  gets  into  diffi- 
culties.13 We  do  not  find,  then,  a  clear  insight  into  the 
matter  with  either  Schroder  or  Dedekind ;  but  still  the  true 
position  of  affairs  is  seen  whenever  a  system  is  to  be  de- 
termined. Dedekind  then  brings  forward  properties  which 
a  thing  must  have  in  order  to  belong  to  a  system,  i.  e.,  he 
defines  a  concept  by  its  characteristics.14  If  now  a  concept 
is  made  up  of  characteristics  and.  not  of  the  objects  falling 
under  the  concept,  there  are  no  difficulties  to  be  urged 
against  an  empty  concept.  Of  course  in  this  case  an  ob- 
ject (Gegenstand)  can  never  also  be  a  concept,  and  a  con- 
cept under  which  only  one  object  falls  must  not  be  confused 
with  this  object.  Thus  we  are  finally  left  with  the  result 
that  the  number  datum  contains  an  assertion  about  a  con- 
cept.10 I  have  traced  back  number  to  the  relation  of  simi- 
larity18 (Gleichzahligkeit}  and  similarity  to  univocal  cor- 
respondence (eindeutige  Zuordnung).  Of  "correspond- 
ence" much  the  same  holds  as  of  "aggregate"  (Menge). 
Nowadays  both  words  are  often  used  in  mathematics,  and 

"  Pp.  104-108. 

l2Sit2ungsberichte  der  Jenaischen  Gesellschaft  fur  Medicin  und  Natur- 
wissenschaft,  July  17,  1885. 

13Cf.  E.  G.  Husserl,  Gottinger  gelehrte  Anzeigen,  1891,  No.  7,  p.  272,— 
where,  however,  the  difficulties  are  not  solved. 

14  On  concept,  object,  property,  and  characteristics,  cf.  my  Grundlagen, 
pp.  48-50,  60-61,  64-65,  and  my  essay  "Ueber  Begriff  und  Gegenstand,"  Viertel- 
jahrsschrift  fur  wissenschaftliche  Philosophic,  Vol.  XVI,  1892,  pp.  192-205. 

15  See  Grundlagen,  pp.  59-60. 

16  [The  same  idea  and  word  were  used  by  Dedekind  (op.  cit.,  p.  53)  ;  and 
the  same  idea  but  with  the  name  "equivalence"  was  used  by  Georg  Cantor  (cf. 
Contributions  to  the  Founding  of  the  Theory  of  Transfinite  Numbers,  Chicago 
and  London,  1915,  pp.  40,  86).] 


Il8  THE  MONIST. 

very  oftep  there  is  lacking  an  insight  into  what  is  intended 
to  be  denoted  by  them.  If  my  opinion  is  correct  that  arith- 
metic is  a  branch  of  pure  logic,  then  a  purely  logical  ex- 
pression has  to  be  chosen  for  "correspondence."  I  choose 
the  word  "relation."  Concept  and  relation  are  the  founda- 
tion stones  upon  which  I  erect  my  structure. 

But  even  when  concepts  have  been  grasped  quite  pre- 
cisely, it  would  be  difficult — nearly  impossible  in  fact — to 
satisfy  the  demands  we  have  had  to  make  of  a  process  of 
proof  without  some  special  means  of  help.  Now  such  a 
means  is  my  ideography  (Begriffsschrift),  the  explanation 
of  which  will  be  my  first  problem.  The  following  remarks 
may  be  noticed  before  we  proceed  farther.  It  is  not  pos- 
sible to  define  everything,  hence  it  must  be  our  endeavor 
to  go  back  to  the  logically  simple  which  as  such  cannot 
properly  be  defined.  I  must  then  be  satisfied  with  referring 
by  hints  to  what  I  mean.  Before  all  I  have  to  strive  to  be 
understood,  and  therefore  I  will  try  to  develop  the  subject 
gradually  and  will  not  attempt  at  first  a  full  generality 
and  a  final  expression.  The  frequent  use  made  of  quota- 
tion marks  may  cause  surprise.  I  use  them  to  distinguish 
the  cases  where  I  speak  about  the  sign  itself  from  those 
where  I  speak  about  its  denotation.  Pedantic  as  this  may 
appear,  I  think  it  necessary.  It  is  remarkable  how  an 
inexact  mode  of  speaking  or  writing  which  perhaps  was 
originally  employed  only  for  greater  convenience  or  brev- 
ity and  with  full  consciousness  of  its  inaccuracy,  may, 
when  that  consciousness  has  disappeared,  end  by  confusing 
thought.  Has  it  not  happened  that  number  signs  have  been 
mistaken  for  numbers,  names  for  the  things  named,  the 
mere  auxiliary  means  for  the  real  end  of  arithmetic  ?  Such 
experiences  teach  us  how  necessary  it  is  to  make  the  high- 
est demands  of  exactitude  in  manner  of  speech  and  writing. 
And  I  have  taken  pains  at  least  to  do  justice  to  such  de- 
mands wherever  it  seemed  to  be  of  importance. 


CLASS,  FUNCTION,  CONCEPT,  RELATION. 

If17  we  are  asked  to  give  the  original  meaning  of  the 
word  "function"  as  used  in  mathematics,  we  easily  fall 
into  saying  that  a  function  of  x  is  an  expression  formed  by 
means  of  the  notations  for  sum,  product,  power,  difference, 
and  so  on,  of  "x"  and  definite  numbers.  This  attempt  at  a 
definition  is  not  successful  because  a  function  is  here  said 
to  be  an  expression,  a  combination  of  signs,  and  not  what 
the  combination  stands  for.  Then  probably  another  at- 
tempt would  be  made  with  "denotation  (Bedeutung)  of  an 
expression"  instead  of  "expression."  But  there  appears 
the  letter  "x"  which  indicates  a  number,  not  as  the  sign 
"2"  does,  but  indefinitely.  For  different  number-signs 
which  we  put  in  the  place  of  "#•",  we  get,  in  general,  differ- 
ent denotations.  Suppose  for  example,  that  in  the  ex- 
pression "(2  +  3.^)^",  instead  of  'V  we  put  the  num- 
ber-signs "o",  "i",  "2",  "3",  one  after  the  other;  we  then 
get  as  corresponding  denotations  the  numbers  o,  5,  28,  87. 
Not  one  of  these  denotations  can  claim  to  be  our  function. 
The  essence  of  the  function  is  in  the  correspondence  that 
it  establishes  between  the  numbers  whose  signs  we  put  for 
"#•"  and  the  numbers  which  then  appear  as  denotations 
of  our  expression, — a  correspondence  which  is  represented 
to  intuition  by  the  course  of  the  curve  whose  equation  is, 
in  rectangular  coordinates,  "y=  (2  -f-  3--*^)^'".  In  gen- 
eral, then,  the  essence  of  the  function  lies  in  the  part  of  the 
expression  which  is  outside  the  "x".  The  expression  of 
a  function  needs  completion  (ist  ergdnzungsbedurftig)  and 
is  not  satisfied  (ungesdttigt) .  The  letter  "x"  only  serves 
to  keep  places  open  for  a  numerical  sign  which  is  to  com- 
plete the  expression,  and  thus  makes  known  the  special 
kind  of  need  for  completion  that  constitutes  the  peculiar 
nature  of  the  function  indicated  above.  In  what  follows, 

17  Cf.  my  lecture  Funktion  und  Begriff,  Jena,  1891,  and  my  essay  "Ueber 
Begriff  und  Gegenstand"  cited  above.  My  Begriff 'sschrift  of  1879  now  no 
longer  represents  my  standpoint,  and  thus  should  only  be  used  with  caution 
to  illustrate  what  I  said  here. 


I2O  THE  MONIST. 

the  Greek  letter  "£"  will  be  used18  instead  of  the  letter 
'V.  This  keeping  open  is  to  be  understood  in  this  way: 
All  places  in  which  "!•"  stand  must  always  be  filled  by  the 
same  sign  and  never  by  different  ones.  I  call  these  places 
argument-places  and  that  whose  sign  or  name  takes  these 
places  in  a  given  case  I  call  argument  of  the  function  for 
this  case.  The  function  is  completed  by  the  argument; 
I  call  what  it  becomes  on  completion  the  value  of  the  func- 
tion for  the  argument.  We  thus  get  a  name  of  the  value 
of  a  function  for  an  argument  when  we  fill  the  argument- 
places  in  the  name  of  the  function  with  the  name  of  the 
argument.  Thus,  for  example,  "(2H~3-12)1"  is  a  name 
of  the  number  5,  composed  of  the  function-name  "(2  + 
3-?2)l"  and  "i".  The  argument  is  not  to  be  reckoned  in 
with  the  function,  but  serves  to  complete  the  function  which 
is  unsatisfied  by  itself.  If  in  the  following  an  expression 
like  "the  $(?)"  is  used,  it  is  always  to  be  observed  that 
the  only  service  rendered  by  "\"  in  the  notation  of  the 
function  is  that  it  makes  the  argument-places  recognizable ; 
it  does  not  imply  that  the  essence  of  the  function  becomes 
changed  when  any  other  sign  is  substituted  for  "§". 

To  the  fundamental  operations  of  calculation  mathe- 
maticians added,  as  function-forming,  the  process  of  pro- 
ceeding to  the  limit  as  exemplified  by  infinite  series,  differen- 
tial quotients  and  integrals ;  and  finally  the  word  "function" 
was  understood  in  such  a  general  way  that  the  connection 
between  value  of  function  and  argument  was  in  certain 
circumstances  no  longer  expressed  by  signs  of  mathemat- 
ical analysis,  but  could  only  be  denoted  by  words.  Another 
extension  consisted  in  admitting  complex  numbers  as  argu- 
ments and  consequently  also  as  function-values.  In  both 
directions  I  have  gone  still  farther.  While,  indeed,  the 

18  Nothing,  however,  is  fixed  by  this  for  our  ideography.  The  "£"  never 
appears  in  the  developments  of  the  ideography  itself,  and  I  only  use  it  in  my 
exposition  of  it  and  in  illustrations. 


CLASS,  FUNCTION,   CONCEPT,  RELATION.  121 

signs  of  analysis  were  hitherto  on  the  one  hand  not  always 
sufficient,  they  were  on  the  other  hand  not  all  employed 
in  the  formation  of  function-names.  For  instance,  "^  =  4" 
and  "|  >  2"  were  not  allowed  to  count  as  names  of  func- 
tions; but  I  do  so  allow  them.  But  that  indicates  at  the 
same  time  that  the  domain  of  function-values  cannot  re- 
main limited  to  numbers ;  for  if  I  take  as  arguments  of  the 
function  |2  — 4  the  numbers  o,  I,  2,  3,  in  succession,  I  do 
not  get  numbers.  I  get:  "o2  =  4",  "i2  =  4",  "22  =  4", 
"3*  =  4",  which  are  expressions  of  one  true  and  some  false 
thoughts.  I  express  this  by  saying  that  the  value  of  the 
function  |2  — 4  is  either  the  "truth-value  (Wahrheits- 
werth)  of  the  true  or  of  the  false."1  From  this  it  can  be 
seen  that  I  do  not  intend  to  assert  anything  by  merely 
writing  down  an  equation,  but  that  I  only  designate  (be- 
zeichne)  a  truth-value,  just  as  I  do  not  intend  to  assert 
anything  by  simply  writing  down  "22"  but  only  designate 
a  number.  I  say:  "The  names  "2*  =  4"  and  "3  >  2"  denote 
the  same  truth-value"  which  I  call  for  short  the  true.  In 
the  same  manner  "3*  =  4"  and  "i>2"  denote  the  same 
truth-value,  which  I  call  for  short  the  false  just  as  the 
name  "22"  denotes  the  number  4.  Accordingly  I  say  that 
the  number  4  is  the  "denotation"  of  "4"  and  of  "22",  and 
that  the  true  is  the  "denotation"  of  "3  >  2".  But  I  dis- 
tinguish the  "meaning"  (Sinn)  of  a  name  from  its  "de- 
notation" (Bedeutung).  The  names  "22"  and  "2  +  2" 
have  not  the  same  meaning,  nor  have  "22  =  4"  and  "2  -f-  2 
=  4".  The  meaning  of  the  name  of  a  truth-value  I  call 
a  "thought"  (Gedanken).  I  say  further  that  a  name  "ex- 
presses" (ausdriickt)  its  meaning  and  "denotes"  its  de- 
notation. I  "designate"  (beseichne)  by  a  name  what  it 
means. 

The  function  ?2  =  4  can  thus  have  only  two  values,  the 

19  I  have  shown  this  more  exhaustively  in  my  essay  "Ueber  Sinn  und  Be- 
deutung" in  the  Zeitschrift  fur  Phihs.und  phil.  Kritik,  Vol.  C,  1892,  pp.  25-50). 


122  THE  MONIST. 

true  for  the  arguments  -f-  2  and  —  2  and  the  false  for  all 
other  arguments. 

Also  the  domain  of  what  is  admitted  as  argument  must 
be  extended, — indeed,  to  objects  quite  generally.  Objects 
(Gegenstdnde)  stand  opposed  to  functions.  I  therefore 
count  as  an  object  everything  that  is  not  a  function ;  thus, 
examples  of  objects  are  numbers,  truth-values,  and  the 
ranges  (Werthverldufe)  to  be  introduced  further  on.  The 
names  of  objects — or  proper  names — are  not  therefore 
accompanied  by  argument-places,  but  are  satisfied  like  the 
objects  themselves. 

I  use  the  words,  "the  function  $(|)  has  the  same  range 
as  the  function  ^P(^)",  as  denoting  the  same  as  the  words, 
"the  functions  $(|)  and  ^(\)  have  the  same  value  for  the 
same  argument."  This  is  the  case  with  the  functions 
|2  =  4  and  ^.^2=i2,  at  least  if  numbers  are  taken  as 
arguments.  But  we  can  also  imagine  the  signs  of  evolution 
and  multiplication  denned  in  such  a  manner  that  the  func- 
tion (|2  =4)  =  (3.1=  12)  has  the  value  of  the  true  for 
any  argument  whatever.  Here  an  expression  of  logic  may 
be  used :  "The  concept  square-root  of  4  has  the  same  ex- 
tension as  the  concept  something  of  which  three  times  its 
square  is  12."  With  those  functions  whose  value  is  always 
a  truth-value  we  can  therefore  say  "extension  of  the  con- 
cept" instead  of  "range  of  the  function,"  and  it  seems  suit- 
able to  say  that  a  concept  (Be griff)  is  a  function  of  which 
the  value  is  always  a  truth-value. 

Hitherto  I  have  only  dealt  with  functions  of  a  single 
argument,  but  we  can  easily  pass  over  to  functions  with 
two  arguments.  Such  functions  are  doubly  in  need  of 
completion.  A  function  with  one  argument  is  obtained 
when  a  completion  by  means  of  one  argument  has  been 
effected.  Only  by  means  of  a  repeated  completion  do  we 
arrive  at  an  object,  and  this  object  is  then  called  the  "value" 
of  the  function  for  the  pair  of  arguments.  Just  as  the 


CLASS,  FUNCTION,  CONCEPT,  RELATION.  123 

letter  "|"  served  with  functions  of  one  argument,  I  use 
here  the  letters  "|"  and  "£"  in  order  to  indicate  the  two- 
fold non-satisfaction  of  a  function  of  two  arguments,  as, 
for  example,  in  "(|  +  £)*  +  £"•  By  replacing  "£"  by  "i", 
for  example,  we  satisfy  the  function  in  such  a  way  that  we 
have  in  (|-|-i)2-fi  a  function  with  only  one  argument. 
This  manner  in  which  we  use  the  letters  "§"  and  "£"  must 
always  be  kept  in  mind  when  an  expression  like  "the  func- 
tion ¥(1,  £)"  occurs.20  I  call  the  places  in  which  "!=" 
stands  "^-argument-places",  and  those  in  which  "£"  stands 
"^-argument-places".  I  say  that  the  ^-argument-places 
are  "related"  (verwandt)  to  one  another,  and  also  the 
^-argument-places  to  one  another,  and  I  say  that  a  ^- 
ment-place  is  not  related  to  a  ^-argument-place. 

The  functions  with  two  arguments  £  =  £  and 
have  as  value  always  a  truth-value — at  least  if  the  signs 
"="  and  ">"  are  defined  in  a  suitable  manner.  I  shall 
call  such  functions  "relations".  In  the  first  relation,  for 
example,  I  stands  to  I,  and  in  general  every  object  to  itself; 
in  the  second,  for  example,  2  stands  to  I.  I  say  that  the 
object  F  "stands  in  the  relation  *P(|,  £)  to"  the  object  A, 
if  $(T,  A)  is  the  true.  I  say  that  the  object  A  "falls 
under"  the  concept  $(?)»  if  3>(A)  is  the  true.  It  is  pre- 
sumed, of  course,  that  both  the  functions  $(£)  and 
have  always  truth-values  as  values.21 


I  have  already  said  above  that  no  assertion  is  to  lie  as 

2<>  Cf .  note  18. 

21  Here  there  is  a  difficulty  which  may  easily  obscure  the  true  position  of 
things  and  thus  rouse  distrust  of  the  correctness  of  my  view.  If  we  compare 
the  expression  "the  truth-value  of  the  circumstance  that  A  falls  under  the  con- 
cept *(£)"  with  "4>(A)",  we  see  that  to  the  "*(A)"  properly  corresponds  "the 
truth-value  of  the  circumstance  that  (A)  falls  under  the  concept  $( I)"  and  not 
"the  concept  *(£)"•  The  last  words  do  not  therefore  really  designate  a  concept 
(in  my  sense  of  the  word),  though  they  have  the  appearance  of  doing  so  in 
our  linguistic  form.  With  regard  to  the  constrained  position  in  which  language 
here  finds  itself,  cf.  my  essay  "Ueber  Begriff  und  Gegenstand"  mentioned  in 
note  14. 


124  THE  MONIST. 

yet  in  a  mere  equation;  by  "2  -{-3  =  5"  only  a  truth-value 
is  designated  and  it  is  not  stated  which  of  the  two  it  is. 
Again,  if  I  write  "(2-f3  =  5)  =  (2  =  2)"  and  presup- 
pose that  we  know  that  2  =  2  is  the  true,  yet  I  would  not 
have  asserted  by  that  the  sum  of  2  and  3  is  5,  but  I  would 
only  have  designated  the  truth-value  of  the  circumstance 
that  "2  +  3  =  5"  denotes  the  same  as  "2  =  2".  Thus  we 
need  a  special  sign  to  assert  that  something  or  other  is 
true.  For  this  purpose  I  write  what  I  call  a  "sign  of 
assertion"  just  before  the  name  of  the  truth-value,  so  that 
if  this  sign  is  written  just  before  "2*  =  4","  it  is  asserted 
that  the  square  of  2  is  4.  I  make  a  distinction  between 
"judgment"  (Urtheil)  and  ''thought"  (Gedanken),  and 
understand  by  "judgment"  the  recognition  of  the  truth 
of  a  "thought."  I  shall  call  the  ideographic  representation 
of  a  judgment  by  means  of  the  sign  of  assertion  an  "ideo- 
graphic theorem"  or  more  shortly  a  "theorem."  I  regard 
this  sign  of  assertion  as  composed  of  a  vertical  line,  which 
I  call  "line  of  judgment"  (Urtheilsstrich) ,  and  a  short 
straight  horizontal  line  proceeding  from  the  middle  of 
the  vertical  line  and  going  toward  the  right,  which  I  will 
simply  call  the  "horizontal  line"  (Wagerechte).  In  my 
Begriffsschrift  I  called  this  last  line  the  "line  of  content" 
(Inhaltsstrich)  and  at  that  time  I  expressed  by  the  words 
"judicable  content"  (beurtheilbarer  Inhalt)  what  I  have 
now  arrived  at  distinguishing  into  truth-value  and 
thought.23  The  horizontal  line  most  often  occurs  in  com- 
bination with  other  signs,  as  it  does  here  with  the  line  of 
judgment,  and  is  thus  guarded  against  confusion  with  the 
minus  sign.  Wherever  it  occurs  by  itself  it  must  be  made 
somewhat  longer  than  the  minus  sign  for  purposes  of  dis- 

22 1  often  use  here  the  notations  of  sum,  product,  and  power  in  order 
conveniently  to  form  examples  and  to  facilitate  understanding  by  means  of 
hints,  although  these  signs  are  not  yet  defined  in  this  place.  But  we  must  keep 
in  view  the  fact  that  nothing  is  founded  on  the  denotations  of  these  signs. 

23  Cf.  my  essay  "Ueber  Sinn  und  Bedeutung"  cited  above. 


CLASS,  FUNCTION,  CONCEPT,  RELATION. 

tinction.  I  regard  it  as  a  name  of  a  function  in  the  way 
that  "A"  preceded  by  this  sign  denotes  the  true  if  A  is  the 
true,  and  the  false  if  A  is  not  the  true.  Of  course  the  sign 
"A"  must  denote  an  object]  names  without  denotation 
may  not  occur  in  our  ideography.  The  above  arrangement 
is  made  so  that  "A"  preceded  by  a  horizontal  line  denotes 
something  under  all  circumstances  if  only  "A"  denotes 
something.  If  not,  "\"  preceded  by  a  horizontal  line  would 
not  denote  a  concept  with  sharp  boundaries, — and  thus 
would  not  denote  a  concept  in  my  sense.  I  here  use  capital 
Greek  letters  as  names  denoting  something  without  my 
saying  what  their  denotations  are.  In  the  actual  develop- 
ments of  my  ideography  they  will  not  occur  any  more  than 
"|"  and  "f '.  The  above  "\"  preceded  by  a  horizontal 
line  denotes  a  function  whose  value  is  always  a  truth- 
value  or,  by  what  I  have  said,  a  concept.  Under  this  con- 
cept falls  the  true  and  this  only.  Thus  "2*  =  4"  preceded 
by  a  horizontal  line  denotes  the  same  thing  as  "2*  =  4", 
namely  the  true.  In  order  to  do  away  with  brackets, 
I  lay  down  that  all  which  stands  to  the  right  of  the 
horizontal  line  is  to  be  regarded  as  a  whole  which  stands 
at  the  argument-place  of  the  function  denoted  by  "5"  Pre~ 
ceded  by  a  horizontal  line,  unless  brackets  forbid  this. 
The  sign  "2*  — 5"  preceded  by  a  horizontal  line  denotes 
the  false  and  thus  the  same  as  "22  =  5",  whereas  "2"  pre- 
ceded by  a  horizontal  line  denotes  the  false,  and  thus  some- 
thing different  from  the  number  2.  If  "A"  is  a  truth- 
value,  A  preceded  by  a  horizontal  line  is  the  same  truth- 
value,  and  thus  the  equation  of  "A"  to  "A"  preceded  by  a 
horizontal  line  denotes  the  true.  But  this  equation  denotes 
the  false  is  A  is  not  a  truth-value ;  so  that  we  can  say  that 
it  denotes  the  truth-value  of  the  circumstances  that  A  is  a 
truth-value. 

Thus  the  function  "$(!)"  preceded  by  a  horizontal 
line,  denotes  a  concept  and  the  function  "*?(!,£)"  pre- 


126  THE  MONIST. 

ceded  by  a  horizontal  line,  denotes  a  relation,  whether  or 
not  $(!)  is  a  concept  and  \P(|,£)  is  a  relation. 

Of  the  two  signs  out  of  which  the  sign  of  assertion  is 
composed  the  line  of  judgment  alone  contains  the  assertion. 

We  need  no  sign  to  declare  that  a  truth-value  is  the 
false,  if  only  we  have  a  sign  by  which  either  truth-value 
is  changed  into  the  other.  This  sign  is  also  indispensable 
on  other  grounds.  I  now  lay  down  that  the  value  of  the 
function  denoted  by  "|"  preceded  by  a  horizontal  line  from 
the  middle  of  which  hangs  a  small  vertical  line  directed 
downward  and  called  the  "line  of  denial"  (V erneinungs- 
strich),  so  that  the  whole  is  like  a  sign  of  assertion  turned 
round  on  its  face,  is  to  denote  the  false  for  every  argu- 
ment for  which  the  value  of  the  function  denoted  by  "|" 
preceded  by  a  horizontal  line  is  the  true.  For  all  other 
arguments  the  function  under  definition  is  to  be  the  true. 
The  function  thus  defined  may  be  called  "the  negation  of 
§",  and  thus  its  value  is  always  a  truth-value;  it  is  a  con- 
cept under  which  all  objects  fall  with  the  single  exception 
of  the  true.  From  this  it  follows  that  horizontal  lines, 
whether  or  not  they  form  part  of  a  sign  of  negation,  can 
be  combined  with  immediately  preceding  or  following 
simple  horizontal  lines  in  such  a  way  that  the  latter,  so  to 
speak,  lose  their  separate  existence  and  melt  into  the  former 
(Verschmelzung  der  Wagerechten). 

Thus  "the  negation  of  22  =  $"  denotes  the  true;  and 
thus  we  may  put  the  sign  of  assertion  so  as  to  join  on  to  the 
left  of  the  sign  of  negation.  We  may  assert,  too,  the 
negation  of  2. 

I  have  already  used  the  sign  of  equality  to  form  ex- 
amples, but  it  is  necessary  to  lay  down  something  more 
accurate  about  it.  The  sign  "F  =  A"  is  to  denote  the 
true  if  F  is  the  same  as  A,  and  the  false  in  all  other  cases. 

In  order  to  dispense  with  brackets  as  far  as  possible, 
I  lay  down  that  all  which  stands  on  the  left  of  the  sign 


CLASS,  FUNCTION,  CONCEPT,  RELATION.  127 

of  equality  as  far  as  the  nearest  horizontal  line  is  to  denote 
the  ^-argument  of  the  function  §  =  £,  in  so  far  as  brackets 
do  not  forbid  this;  and  that  all  which  stands  on  the  right 
of  the  sign  of  equality  as  far  as  the  next  sign  of  equality  is 
to  denote  the  ^-argument  of  that  function  in  so  far  as 
brackets  do  not  forbid  this. 

GOTTLOB  FREGE. 
JENA,  GERMANY. 


A  CHINESE  POET'S  CONTEMPLATION  OF  LIFE. 

INTRODUCTION. 

MY  attention  has  repeatedly  been  called  to  the  poetry  of  Su 
Tung  P'o  (also  briefly  named  "Su  Hsi"),  especially  to  his 
thoughtful  meditation  on  an  excursion  by  boat  to  the  Scarlet  Cliff. 
In  this  poem  he  comments  on  the  transiency  of  life,  and  referring  to 
the  law  of  change  as  represented  by  the  phases  of  the  moon  he  finds 
the  underlying  permanence  symbolized  by  the  river  which  remains 
the  same  although  its  waters  pass  on  without  a  halt. 

The  original  was  kindly  furnished  me  by  Mr.  Sawland  J.  Shu, 
president  of  the  Technological  College  at  Nanking,  while  a  literal 
translation  was  procured  through  Prof.  Frederick  G.  Henke  from 
Mr.  W.  T.  Tao  and  another  one  from  Prof.  King  Shu  Liu,  of  the 
University  of  Nanking.  Professor  Henke  further  informed  me 
on  the  authority  of  Prof.  William  F.  Hummel  that  a  prose  trans- 
lation by  Prof.  Herbert  A.  Giles  was  published  in  the  University 
of  Nanking  Magazine  and  republished  together  with  other  Chinese 
poems  collected  in  the  volume  entitled  Gems  of  Chinese  Literature. 

Professor  Giles  says  that  Su  Tung  P'o  was  "even  a  greater 
favorite  with  the  Chinese  literary  public"  than  the  famous  Ou-Yang 
Hsiu.1  So  we  may  regard  Su  Tung  P'o  as  easily  a  genius  of  first 
rank.  Professor  Giles  says  of  him: 

"Under  his  hands,  the  language  of  which  China  is  so  proud 
may  be  said  to  have  reached  perfection  of  finish,  of  art  concealed. 
In  subtlety  of  reasonings,  in  the  lucid  expression  of  abstractions, 
such  as  in  English  too  often  elude  the  faculty  of  the  tongue,  Su 
Tung  P'o  is  an  unrivalled  master." 

Even  a  rough  translation  of  his  poems  will  impress  the  reader 

1  Ou-Yang  Hsiu  lived  1017-1072  A.  D.  Professor  Giles  says  of  him:  "A 
leading  statesman,  historian,  poet,  and  essayist  of  the  Sung  dynasty.  His 
tablet  is  to  be  found  in  the  Confucian  temple,  an  honor  reserved  for  those 
alone  who  have  contributed  to  the  elucidation  or  dissemination  of  Confucian 
truth." 


A  CHINESE  POET'S  CONTEMPLATION  OF  LIFE.         I2Q 

with  the  versatility  as  well  as  the  profundity  of  his  poetic  flights, 
and  here  I  venture  to  present  his  famous  poem  on  "The  Scarlet 
Cliff"  in  English  blank  verse  which  seems  to  be  the  appropriate 
form  for  this  kind  of  thought.  I  hope  that  it  will  be  a  fair  example 
of  Chinese  literature  in  its  noblest  accomplishment. 

There  are  some  people  who  have  little  appreciation  of  the 
beauties  of  Chinese  literature  and  have  nothing  but  ridicule  or  even 
contempt  for  it.  With  reference  to  one  of  these  haughty  scoffers 
Professor  Giles  adds  with  grim  humor: 

"On  behalf  of  his  (Su  Tung  P'o's)  honored  manes  I  desire  to 
note  my  protest  against  the  words  of  Mr.  Baber,  recently  spoken 
at  a  meeting  of  the  Royal  Geographical  Society,  and  stating  that 
'the  Chinese  language  is  incompetent  to  express  the  subtleties  of 
theological  reasoning,  just  as  it  is  inadequate  to  represent  the 
nomenclature  of  European  science.'  I  am  not  aware  that  the  nomen- 
clature of  European  science  can  be  adequately  represented  even  in 
the  English  language ;  at  any  rate,  there  can  be  no  comparison 
between  the  expression  of  terms  and  of  ideas,  and  I  take  it  the 
doctrine  of  the  Trinity  itself  is  not  more  difficult  of  comprehension 
than  the  theory  of  'self -abstraction  beyond  the  limits  of  an  external 
world,'  so  closely  reasoned  out  by  Chuang  Szu.  If  Mr.  Baber  merely 
means  that  the  gentlemen  entrusted  with  the  task  have  proved 
themselves  so  far  quite  incompetent  to  express  in  Chinese  the  subtle- 
ties of  theological  reasoning,  then  I  am  with  him  to  the  death." 

Mr.  K.  S.  Liu  sends  with  his  translation  these  further  remarks 
concerning  Su  Hsi,  the  classical  philosopher  of  Chinese  belles  lettres : 

"This  poem  was  composed  by  Su  Hsi,  a  famous  Chinese  poet 
who  flourished  1036-1101.  Owing  to  the  intrigues  of  his  political 
enemies  he  was  exiled  to  Hwang-Cheo,  a  place  in  the  province  of 
Hu  Peh.  While  there  he  made  a  visit  to  a  place  called  Chi  Pi 
(literally  Red  Wall),  made  famous  by  the  battle  which  took  place 
there  between  Tsao-tsao  and  Cheo-yu  (two  historical  characters  in 
the  period  of  the  Three  Kingdoms).  The  poem  is  an  account  of 
this  visit  and  a  description  of  the  feelings  it  aroused  in  him.  Like 
many  other  poets  who  consider  poetry  an  embodiment  in  symbols 
of  one's  inner  spiritual  experiences,  he  shows  in  the  poem,  first, 
the  ephemeral  nature  of  human  existence  with  all  its  paraphernalia, 
and  then  how  in  the  contemplation  of  nature  one  can  transcend 
the  mutations  of  time  and  be  one  with  the  eternal  order.  In  this 
state  one  can  rise  above  the  vicissitudes  of  life." 


I3O  THE  MONIST. 

The  poem  begins  by  giving  the  date  of  Su  Hsi's  excursion  to 
the  Scarlet  Cliff.  The  year  reads  in  Chinese  characters  fan  siih, 
and  we  here  encounter  the  difficulty  of  reproducing  the  Chinese 


3 


V  ^t  a  *  ** 


w 

C/5 

w 


£  <•$  TJQ  *g  *$£ 

o  'A 


a  \ 


method  of  determining  chronology.  For  this  they  make  use  of  the 
sexagenary  cycle  by  repeating  five  times  the  twelve  branches  and 
six  times  the  ten  stems  (see  the  author's  Chinese  Thought,  p.  4). 


A  CHINESE  POET'S  CONTEMPLATION  OF  LIFE.         13! 


The  meaning  of  jan  (pronounced  zhan)  is  the  "germ  in  the 
womb,"  and  it  "denotes  the  ninth  of  the  ten  stems  ;  it  is  connected 
with  the  north  and  running  water."  It  means  "great,  full"  and  also 


•=: 


"to  flatter  and  adulate."  As  the  ninth  of  the  ten  stems  it  denotes 
swollen  water,  hence  we  translate  it  "billow."  The  other  character 
siih  which  is  the  eleventh  of  the  twelve  branches  denotes  in  its 


132  THE  MONIST. 

horary  significance  the  hour  7-9  P.  M.,  called  the  "dog  hour."  We 
here  translate  it  by  "hound."  To  Chinamen  this  denotation  of  the 
year  is  very  familiar,  but  it  is  difficult  to  reproduce  its  exact  sig- 
nificance in  a  poetic  translation  in  English.  The  "billow  hound" 
year  corresponds  in  our  chronology  to  1082  A.  D.,  which  is  the 
fifty-eighth  year  in  the  sexagenary  cycle  under  the  Sung  dynasty. 
The  latter  being  a  matter  of  course  in  the  poet's  day  is  not  men- 
tioned in  the  Chinese  text. 

39       38       37       36       35        34       33 

*• 


»   t 


> 

*'         J   * 


'  i 


CHINESE  TEXT. 

The  songs  "To  the  Bright  Moon"  and  "To  the  Modest  Maid" 
mentioned  in  the  poem  are  probably  the  odes  known  as  I,  XII,  8 
and  I,  III,  17  of  the  Shih  King,  the  canonical  collection  of  ancient 
Chinese  songs.  In  the  translation  of  William  Jennings  (The  Shi 
King,  pages  151  and  69)  they  read  as  follows: 

To  the  Bright  Moon. 
O  Moon  that  climb'st  effulgent! 
O  ladylove  most  sweet! 


A  CHINESE  POET'S  CONTEMPLATION  OF  LIFE.         133 

Would  that  my  ardor  found  thee  more  indulgent! 
Poor  heart,  how  dost  thou  vainly  beat! 

O  Moon  that  climb'st  in  splendor! 

O  ladylove  most  fair! 
Couldst  thou  relief  to  my  fond  yearning  render! 

Poor  heart,  what  charing  must  thou  bear! 

0  Moon  that  climb'st  serenely! 
O  ladylove  most  bright! 

Couldst  thou  relax  the  chain  I  feel  so  keenly! 
Poor  heart,  how  sorry  is  thy  plight! 

To  the  Modest  Maid. 

A  modest  maiden,  passing  fair  to  see, 
Waits  at  the  corner  of  the  wall  for  me. 

1  love  her,  yet  I  have  no  interview: — 

I  scratch  my  head — I  know  not  what  to  do. 

The  modest  maid — how  winsome  was  she  then, 
The  day  she  gave  me  her  vermilion  pen ! 
Vermilion  pen  was  never  yet  so  bright — 
The  maid's  own  loveliness  is  my  delight. 

Now  from  the  pasture  lands  she  sends  a  shoot 
Of  couchgrass  fair;  and  rare  it  is,  to  boot. 
Yet  thou,  my  plant  (when  beauties  I  compare), 
Art  but  the  fair  one's  gift,  and  not  the  Fair! 

There  is  some  doubt,  according  to  Professor  Giles,  whether 
the  Scarlet  Cliff  visited  by  Su  Hsi  was  really  the  place  of  battle  as 
the  latter  assumes,  but  the  poem  remains  of  the  same  significance 
even  if  Su  Hsi  was  mistaken,  and  we  need  feel  no  concern  about  it. 

p.  c. 

THE  SCARLET  CLIFF. 

It  was  the  Billow-Hound  year  of  House  Sung: 
The  seventh  moon  was  on  the  wane,  when  I 
Was  down  stream  drifting  in  a  boat  with  friends 
On  an  excursion  to  the  Scarlet  Cliff. 


134  THE  MONIST. 

The  evening  breeze  so  gently  blew  that  scarce 

The  water  rippled  on  its  smooth  expanse. 

I  rilled  the  cups  and  bade  my  friends  to  sing 

The  ode  'To  the  Bright  Moon,"  and  then  they  chanted 

The  lay  melodious  "To  the  Modest  Maid." 

Slowly  the  moon  rose  o'er  the  eastern  hills, 
And  passed  between  the  Wain  and  Capricorn, 
Shedding  her  silver  beams  upon  the  water, 
To  link  our  world  below  with  heaven  above. 

In  such  surroundings,  infinite  in  charm, 

Our  skiff  was  freely  gliding, — traveling 

Unchecked  through  space,  unmindful  whither  bound; 

Like  gods  we  moved  in  a  transcendent  realm: 

I  poured  out  a  libation  for  our  joy, 

And  beating  time  on  our  boat's  wooden  rim, 

I  sang  these  verses  in  sad  exaltation : 

"Our  olive  boat  with  orchid  oars  propelled, 
Breaks  splashing  through  the  moonlit  glittering 

wave; 

In  lovelorn  loneliness  I  here  am  held, 
From  friends  who  now  lie  buried  in  the  grave." 

One  of  my  guests  accompanied  the  song 

Upon  his  flageolet,  with  proper  notes 

To  suit  the  music  to  the  sentiment 

Of  plaintive  moods,  in  sounds  that  wove  unbroken 

Their  silken  threads  around  our  company. 

The  music  stirred  the  dragon  in  the  deep 

And  moved  the  the  boatswain's  widow  unto  tears. 

"And  why  is  that?"  I  asked  in  pensive  query 

My  cherished  guest.     "Why  does  thy  magic  art 

So  powerfully  affect  us  all?"     Said  he: 


A  CHINESE  POET'S  CONTEMPLATION  OP  LIFE.         135 

"Few  stars  are  seen  and  yet  the  moon  shines  bright, 
To  southern  lands  the  raven  wings  his  flight. 

"Was  this  not  uttered  here  by  Tsao  Meng  Te, 

Here,  eastward  of  Hsia-K'ou,  west  of  Wu-Chang, 

Where  hill  and  stream  in  wild  luxuriance  blend? 

'T  is  here  Meng  Te  was  routed  by  Chou  Yii. 

Before  him  lay  Ching-chou.    Kiang-ling  he  conquered, 

And  eastward  did  he  push  upon  the  river; 

His  warships,  prow  to  stern,  stretched  thousand  miles, 

The  banners  of  his  troops  darkened  the  sun. 

Then  a  libation  he  poured  out,  and  nearing 

The  Scarlet  Cliff,  the  hero  of  his  age, 

On  horseback,  clad  in  armor,  spake  those  words! 

Yet  where  is  he  to-day?    And  what  are  we? 

To-day  we  fish  and  gather  fuel  here 

On  river  isles  where  shrimps  are  our  companions 

And  deer  our.  friends.    We  paddle  here  about 

In  frail  canoe  and  drink  companionship 

From  flasks  of  gourd.    How  transient  is  the  life 

Of  creatures  as  ephemeral  as  we. 

Tossed  o'er  the  ocean  like  a  husk  of  straw, 

We  are  mere  twinklings  on  the  river  Time; 

Oh,  could  I  be  the  stream  itself  which  rolls 

Incessantly  and  without  end!    Alas! 

Could  I  but  clasp  the  bright  and  beauteous  moon 

Close  to  my  heart  and  dwell  with  her  in  heaven ! 

Yet  unfulfilled  remain  my  deep-felt  yearnings 

Which  find  expression  in  melodious  strain." 

"But  you  my  friend,"  replied  I  questioning, 
"Do  you  well  comprehend  the  mystery 
Of  this  great  river  and  the  changing  moon? 
Past  flows  the  water  but  'tis  never  gone; 
The  moon  is  waning,  but  again  'twill  wax. 


136  THE  MONIST. 

So  I  with  this  great  world,  all  in  a  change — 

E'en  Heaven  and  Earth  are  transient  constantly — 

Myself,  and  also  thou,  in  this  same  sense 

Viewed  as  a  whole,  live  on  eternally. 

Why  then  lament?    Thou  long'st  for  what  thou  hast!" 

And  further  musing  on  life's  complex  problems 

Continued  I:  "Whate'er  our  senses  hold 

Is  owned  by  him  who  feels  it,  who  enjoys  it. 

For  nothing  can  I  take  unless  I  own  it, 

The  bracing  breeze,  the  landscape  of  the  river, 

The  moon  above  the  valleys,  gorgeous  sights 

Enrapturing  the  eye,  and  all  the  sounds 

Which  greet  the  ear,  all  are  enjoyed  by  me. 

All  these  are  mine,  and  without  let  or  hindrance 

Are  they  the  gifts  of  God,  unstintedly 

Given  to  man — indeed  to  all  mankind. 

And  we  enjoy  them  now." 

He  smiled  approval — 

My  friend;  he  threw  away  the  dregs  of  wine 
And  had  his  cup  refilled  up  to  the  brim. 

Thus  finishing  our  feast  we  laid  us  down 
To  rest  among  the  scattered  cups  and  plates, 
While  in  the  distant  east  dim  streaks  of  light 
Appeared  as  heralds  of  another  day. 


CRITICISMS  AND  DISCUSSIONS. 
LEIBNIZ  AND  LOCKE. 

John  Locke,  the  founder  of  the  sensationalist  school,  who  form- 
ulated the  principle  of  his  philosophy  in  the  statement  Nihil  est  in 
intellectu  quod  non  antea  fuerit  in  sensu,  and  who  therefore  on  the 
one  hand  denied  innate  ideas  and  on  the  other  claimed  that  all 
knowledge  rises  from  experience,  devotes  to  an  investigation  of 
truth  Chapter  V,  and  also  part  of  Chapter  VI  of  his  famous  work 
On  the  Human  Understanding  from  which  we  make  the  following 
extracts : 

"  'What  is  truth?'  was  an  inquiry  many  ages  since;  and  it  being 
that  which  all  mankind  either  do  or  pretend  to  search  after,  it  cannot 
but  be  worth  our  while  carefully  to  examine  wherein  it  consists ;  and 
so  acquaint  ourselves  with  the  nature  of  it,  as  to  observe  how  the 
mind  distinguishes  it  from  falsehood. 

"Truth  then  seems  to  me,  in  the  proper  import  of  the  word,  to 
signify  nothing  but  the  joining  or  separating  of  signs,  as  the  things 
signified  by  them  do  agree  or  disagree  one  with  another.  The  join- 
ing or  separating  of  signs  here  meant,  is  what  by  another  name  we 
call  'proposition.'  So  that  truth  properly  belongs  only  to  propo- 
sitions :  whereof  there  are  two  sorts,  viz.,  mental  and  verbal ;  as  there 
are  two  sorts  of  signs  commonly  made  use  of,  viz.,  ideas  and  words... 

"We  must,  I  say,  observe  two  sorts  of  propositions  that  we  are 
capable  of  making: 

"First,  Mental,  wherein  the  ideas  in  our  understandings  are, 
without  the  use  of  words,  put  together  or  separated  by  the  mind 
perceiving  or  judging  of  their  agreement  or  disagreement. 

"Secondly,  Verbal  propositions,  which  are  words,  the  signs  of 
our  ideas,  put  together  or  separated  in  affirmative  or  negative  sen- 
tences. By  which  way  of  affirming  or  denying,  these  signs,  made 
by  sounds,  are,  as  it  were,  put  together  or  separated  one  from  an- 


138  THE  MONIST. 

other.  So  that  proposition  consists  in  joining  or  separating  these 
signs,  according  as  the  things  which  they  stand  for  agree  or  dis- 
agree .... 

"When  ideas  are  so  put  together  or  separated  in  the  mind,  as 
they  or  the  things  they  stand  for  do  agree  or  not,  that  is,  as  I 
may  call  it  'mental  truth.'  But  truth  of  words  is  something  more, 
and  that  is  the  affirming  or  denying  of  words  of  another,  as  the 
ideas  they  stand  for  agree  or  disagree:  and  this  again  is  twofold; 
either  purely  verbal  and  trifling  or  real  and  instructive,  which  is 
the  object  of  real  knowledge. . . . 

"Though  our  words  signify  nothing  but  our  ideas,  yet  being 
designed  by  them  to  signify  things,  the  truth  they  contain,  when 
put  into  propositions,  will  be  only  verbal  when  they  stand  for  ideas 
in  the  mind  have  not  an  agreement  with  the  reality  of  things. 
And  therefore  truth,  as  well  as  knowledge,  may  well  come  under 
the  distinction  'verbal'  and  'real';  that  being  only  verbal  truth 
wherein  terms  are  joined  according  to  the  agreement  or  disagree- 
ment of  the  ideas  they  stand  for,  without  regarding  whether  our 
ideas  are  such  as  really  have  or  are  capable  of  having  an  existence 
in  nature.  But  then  it  is  they  contain  real  truth  when  these  signs 
are  joined  as  our  ideas  agree;  and  when  our  ideas  are  such  as  we 
know  are  capable  of  having  an  existence  in  nature:  which  in  sub- 
stances we  cannot  know  but  by  knowing  that  such  have  existed. 

"Truth  is  the  marking  down  in  words  the  agreement  or  dis- 
agreement of  ideas  as  it  is.  Falsehood  is  the  marking  down  in  words 
the  agreement  or  disagreement  of  ideas  otherwise  than  it  is.  And 
so  far  as  these  ideas  thus  marked  by  sounds  agree  to  their  arche- 
types, so  far  only  is  the  truth  real.  The  knowledge  of  this  truth 
consists  in  knowing  what  ideas  the  words  stand  for,  and  the  per- 
ception of  the  agreement  or  disagreement  of  those  ideas,  according 
as  it  is  marked  by  those  words .... 

"Certainty  is  twofold;  certainty  of  truth,  and  certainty  of 
knowledge.  Certainty  of  truth  is,  when  words  are  so  put  to- 
gether in  propositions  as  exactly  to  express  the  agreement  or  dis- 
agreement of  the  ideas  they  stand  for,  as  really  it  is.  Certainty  of 
knowledge  is,  to  perceive  the  agreement  or  disagreement  of  ideas, 
as  expressed  in  any  proposition.  This  we  usually  call  'knowing,'  or 
'being  certain  of  the  truth  of  any  proposition.'  " 

His  great  critic  Leibniz  wrote  a  voluminous  book1  to  refute 

1New  Essays  Concerning  Human  Understanding.     Translated  by  A.  G. 
Langley.    2d  ed.,  Chicago  and  London,  1916. 


CRITICISMS   AND   DISCUSSIONS.  139 

Locke's  sensationalism,  pointing  out  that  what  Locke  called  re- 
flection was  not  a  product  of  sensation.  He  amended  Locke's  prin- 
ciple to  read :  Nihil  est  in  intellectu  quod  non  antea  fuerit  in  sensu, 
nisi  intellectus  ipse,  and  this  amendment  upset  Locke's  very  lucid 
but  superficial  arguments.  According  to  Leibniz  the  senses  furnish 
us  the  material  for  positive  knowledge  but  they  offer  nothing  but 
particular  instances,  not  methods,  nor  principles,  nor  general  truths. 
Brutes  have  the  same  sensations  as  man,  but  brutes  can  never  attain 
to  necessary  propositions.  These  conceptions  of  necessary  propo- 
sitions are  innate  in  the  human  mind.  The  human  mind  is  not  a 
tabula  rasa,  but  contains  certain  principles  which,  in  the  measure 
that  experience  furnishes  the  occasion,  develop  into  ideas  of  eternal 
and  necessary  verities. 

From  this  standpoint  Leibniz  distinguishes  two  kinds  of  truths, 
necessary  truths  and  contingent  truths ;  the  former  are  the  eternal 
verities  as  instanced  by  mathematics,  the  latter  the  knowledge  of  par- 
ticular facts  furnished  by  experience.  God  is  the  ultimate  source 
of  both  kinds  of  truth;  the  eternal  verities  correspond  to  his  in- 
tellect, the  contingent  truths  to  his  will.  The  former  are  such  and 
can  not  be  different  because  God  is  such ;  the  latter  could  be  different 
but  are  not  because  God  willed  them  to  be  as  they  are  and  not  other- 
wise. Necessary  truths  reveal  to  us  what  is  possible  and  what  impos- 
sible. Thus, e.g., a  regular  decahedron  (i.e.,  a  figure  bounded  by  ten 
equal  plane  surfaces)  is  impossible,  and  "all  intelligible  ideas  have 
their  archetype  in  the  eternal  possibilities  of  things." 
In  reply  to  Locke's  view  of  certainty,  Leibniz  says: 
"Our  certitude  would  be  small,  or  rather  nothing,  if  it  had  no 
other  basis  of  simple  ideas  than  that  which  comes  from  the  senses. 
Have  you  forgotten,  sir,  how  I  have  shown  that  ideas  are  originally 
in  our  mind,  and  that  indeed  our  thoughts  come  to  us  from  the 
depths  of  our  own  nature,  other  creatures  being  unable  to  have  an 
immediate  influence  upon  the  soul?  Besides,  the  ground  of  our 
certitude  in  regard  to  universal  and  eternal  truths  is  in  the  ideas 
themselves,  independently  of  the  senses,  just  as  ideas  pure  and  in- 
telligible do  not  depend  on  the  senses,  for  example,  those  of  being, 
unity,  identity,  etc.  But  the  ideas  of  sensible  qualities,  as  color, 
savor,  etc.,  (which  in  reality  are  only  phantasms)  come  to  us  from 
the  senses,  i.  e.,  from  our  confused  perceptions.  And  the  basis  of 
the  truth  of  contingent  and  particular  things  is  in  the  succession 


I4O  THE  MONIST. 

which  causes  these  phenomena  of  the  senses  to  be  rightly  united 
as  the  intelligible  truths  demand." 

It  is  not  our  intention  to  criticize  any  one  of  the  philosophers 
but  we  wish  to  point  out  how  far  and  in  what  respect  we  agree  with 
Leibniz's  views  as  here  outlined.  We  select  Leibniz  because  his 
philosophy  is  less  onesided  than  any  other,  and  has  incorporated  all 
considerations,  religious,  scientific,  mathematical  and  historical. 
What  he  calls  innate  ideas  reflecting  the  eternal  and  necessary  truths 
whose  source  lies  in  God,  we  denote  as  the  purely  formal  and  we 
have  shown  that  purely  formal  conceptions  have  been  gained  by 
abstraction.  Man  alone  has  the  faculty  of  abstraction  and  so  he 
alone  is  capable  of  producing  and  operating  with  purely  formal  con- 
ceptions such  as  numbers,  geometrical  figures,  the  notion  of  mathe- 
matical or  pure  space,  logical  syllogisms,  the  formulas  of  causation 
and  of  the  conservation  of  substance  and  energy.  The  principle 
pervading  the  function  of  these  concepts  is  called  reason,  and  reason 
truly  reflects  the  cosmic  order,  which  is  due  to  the  efficiency  of 
purely  formal  interrelations — the  so-called  purely  formal  laws.  Our 
senses  furnish  us  particulars  only,  and  these  particulars,  which  are 
innumerable  isolated  sense-impressions,  would  remain  a  chaos  of 
disconnected  items  if  they  were  not  classified  and  systematized  ac- 
cording to  purely  formal  laws.  The  point  overlooked  by  Leibniz 
and  also  later  on  by  Kant  is  the  question  as  to  the  origin  of  mind. 
The  framework  of  reason,  man's  logical  faculty,  his  notion  of 
numbers  and  of  space  relations  have  indeed  originated  through  ex- 
perience as  Locke  claimed,  but  it  was  experience  in  a  wider  sense 
than  either  Locke  or  Leibniz  conceived  it  to  be.  Experience  in  those 
days  meant  sense-experience,  or  the  purely  sensory  element  of  sen- 
tient creatures.  In  this  sense  Leibniz  is  right  that  no  amount  of 
sense-impressions  can  bring  forth  an  eternal  or  universal  or  neces- 
sary idea.  Locke  on  the  other  hand,  conscious  of  the  fact  that  man 
was  in  possession  of  universal  and  necessary  concepts  and  admitting 
no  other  source  of  knowledge  than  experience,  insisted  on  the  prop- 
osition that  all  ideas,  even  the  most  complicated  ones,  were  derived 
from  sensations,  as  which  he  understands  experience  to  be. 

Now  it  is  obvious  that  there  is  nothing  purely  sensory,  Sen- 
sations are  possessed  of  forms  and  the  formal  impresses  itself  to- 
gether with  sense  impressions  upon  sentient  creatures.  We  have  on 


CRITICISMS  AND  DISCUSSIONS.  14! 

other  occasions  set  forth  how  sensory  impressions  are  by  a  mechan- 
ical necessity  so  grouped  that  they  are  registered  together,  the  par- 
ticular ones  being  subsumed  under  the  more  general  so  that  all  of 
them  build  up  a  well-arranged  system  constituting  a  logical  frame- 
work of  types.  This  framework  is  the  mind  which  is  built  up  not 
of  mere  sensations,  but  of  the  interrelations  of  sense-impressions 
according  to  their  various  forms.  Experience  in  the  current  sense 
includes  the  form  of  the  sensory,  and  in  this  sense  the  faculty  of 
conceiving  purely  formal  relations  has  indeed  arisen  from  expe- 
rience. 

The  sensationalist  school  identifies  the  sense  element  of  our 
knowledge  with  the  formal  and  overlooks  their  radical  difference. 
We  must  insist  against  the  sensationalist  school  that  everything 
formal  is  radically  different  from  the  sensory.  The  sensory  is  al- 
ways particular  while  the  formal  can  be  generalized.  By  leaving 
out  of  sight  everything  particular  our  thought  can  operate  in  a  field 
of  pure  relations,  and  we  can  exhaust  all  their  possibilities.  We  can 
say  what  is  possible  as  well  as  what  is  impossible  and  (all  inter- 
ference of  unexpected  particulars  being  excluded)  we  can  also  say 
what  result  will  always  be  obtained  under  definite  given  conditions. 
We  can  exhaust  all  possibilities  of  the  purely  formal  and  can  sys- 
tematize the  whole  field.  What  will  always  be,  is  called  "necessary," 
and  so  these  propositions  which  are  inevitable  are  called  by  Leibniz 
"eternal  truths." 

We  agree  with  Leibniz  that  the  source  of  these  eternal  truths 
is  God;  nay  we  go  one  step  further  in  definiteness  and  claim  that 
the  eternal  verities,  of  which  our  human  notions  of  eternal  truths 
are  mental  reflections,  are  God  himself.  All  depends  on  our  defi- 
nition of  God.  Together  with  the  whole  cosmic  order  the  necessary 
truths  constitute  an  eternal  omnipresence,  an  efficient  system  of 
norms  which  mould  the  world  and  determine  all  things.  They  form 
a  kind  of  spiritual,  or  purely  formal  organism,  a  superpersonal 
presence  which  is  the  ultimate  raison  d'etre  and  determinant  of  all 
things,  the  cosmos  in  its  entirety  as  well  as  all  particular  events  that 
happen  in  the  course  of  its  being. 

Any  one  who  has  once  grasped  the  deep  significance  of  the 
purely  formal  will  have  liberated  his  mind  forever  of  the  super- 
stitious, mystical  or  allegorical  conceptions  of  the  deity,  but  he  will 
at  the  same  time  understand  the  truth  that  underlies  the  God-idea 
and  thus  he  will  know  the  real  nature  of  the  true  God,  whose  exist- 


142  THE  MONIST. 

ence  is  not  a  matter  of  belief,  but  a  scientific  certainty.  All  former 
proofs  of  the  existence  of  God  were  necessarily  failures,  because  in 
all  cases  the  attempt  was  made  to  prove  the  existence  of  an  anthro- 
pomorphic God  with  arguments  that  prove  the  true  God,  the  eternal 
norm  of  being,  and  here  the  argument  breaks  down,  because  it  no 
longer  applies  to  the  idea  of  an  anthropomorphic  God. 

Leibniz  has  not  overcome  the  mystical  conception  both  of  God 
and  truth.  He  has  unfortunately  adopted  the  very  primitive  con- 
ception of  an  atomic  nature  of  reality  which  is  described  in  his 
monadology.  It  is  strange  that  a  man  of  his  caliber  did  not  see 
how  contradictory  is  the  idea  of  God  as  the  central  monad.  On  the 
other  hand  his  theory  is  vindicated  if  we  interpret  his  God  to  be  the 
universal  and  omnipresent  norm  that  regulates  every  event  and 
constitutes  the  cosmic  order  of  the  world. 

Insisting  on  the  unity  of  the  soul,  Leibniz  conceived  all  unities 
as  local  units,  and  these  innumerable  local  units,  the  monads,  were 
conceived  as  centers  of  force  endowed  with  feeling  and  an  entelechy, 
which  means  that  they  were  capable  of  pursuing  purposes.  At  the 
same  time  Leibniz  held  them  to  be  separate  entities,  so  as  to  render 
their  cohesion  and  interaction  a  profound  problem  which  could  be 
solved  only  by  the  bold  hypothesis  of  the  preestablished  harmony. 

The  problem  of  unity  together  with  all  problems  of  combina- 
tion and  configuration  belongs  in  the  domain  of  pure  form.  Com- 
bination of  several  parts  working  in  cooperation  constitute  a  unity 
and  introduce  something  new.  It  did  not  exist  before  and  will  break 
to  pieces  again,  but  the  law  of  its  combination  remains  forever  and 
constitutes  the  eternal  background  of  its  existence.  The  sensation- 
alist school  misses  the  main  point  of  all  philosophical  considerations 
and  thus  loses  the  essence  of  the  significance  of  religion ;  but  Leib- 
niz who  discovers  the  weak  spot  in  their  arguments  has  not  suc- 
ceeded in  persenting  a  satisfactory  solution  of  the  problem  but  ends 
in  proclaiming  a  mystical  God-conception  and  a  dogmatic  proclama- 
tion of  a  preestablished  harmony.  p.  c. 

EXISTENTS  AND  ENTITIES.1 

That  we  must  distinguish  between  what  we  may  call  "having 
existence"  and  "having  entity  or  being"  becomes  evident  when  we 
look  somewhat  closely  at  ordinary  mathematical  propositions.  A 
class  (or  system,  or  aggregate)  M  is  said  to  "exist"  when  it  has 

'Cf.  Monist,  Jan.  1910,  Vol.  XX,  p.  114,  note  85. 


CRITICISMS  AND  DISCUSSIONS.  143 

at  least  one  member;2  whereas,  when  mathematicians  speak  of, 
for  example,  "the  existence  of  roots  of  an  equation"  or  "the  exist- 
ence of  the  definite  integral  of  a  continuous  function,"  they  use 
the  word  "existence"  in  another  sense :  the  roots  or  the  integral  are 
not  classes,  but  individuals  constructed  out  of  mathematical  con- 
cepts to  supply  an  answer  to  certain  questions.  We  can,  of  course, 
consider  such  an  individual  as  the  member  of  the  class  (N)^ 
whose  sole  member  is  this  individual,  and  can  then  consider  the 
second  kind  of  mathematicians'  existence-proofs  as  proofs  of  the 
existence  of  the  class  N ;  but  we  should,  for  the  sake  of  clearness, 
avoid  speaking  of  the  "existence"  of  the  member3  of  N,  and  use 
some  such  word  as  "entity"  or  "being"  instead. 

Mr.  B.  Russell4  has  thus  distinguished  being  and  existence  in 
1901 :  "Being  is  that  which  belongs  to  every  conceivable  term,  to 
every  possible  object  of  thought — in  short  to  everything  that  can 
possibly  occur  in  any  proposition,  true  or  false,  and  to  all  such 
propositions  themselves.  Being  belongs  to  whatever  can  be  counted. 
If  A  be  any  term  that  can  be  counted  as  one,  it  is  plain  that  A  is 
something,  and  therefore  that  A  is.  'A  is  not'  must  always  be 
either  false  or  meaningless.  For  if  A  were  nothing,  it  could  not 
be  said  not  to  be;  'A  is  not'  implies  that  there  is  a  term  A  whose 
being  is  denied,  and  hence  that  A  is.  Thus  unless  'A  is  not'  be  an 
empty  sound,  it  must  be  false — whatever  A  may  be,  it  certainly  is. 
Numbers,  the  Homeric  gods,  relations,  chimeras,  and  four-dimen- 
sional spaces  all  have  being,  for  if  they  were  not  entities  of  a  kind, 
we  could  make  no  propositions  about  them.  Thus  being  is  a  gen- 
eral attribute  of  everything,  and  to  mention  anything  is  to  show 
that  it  is. 

"Existence,  on  the  contrary,  is  the  prerogative  of  some  only 
amongst  beings.  To  exist  is  to  have  a  specific  relation  to  existence 
— a  relation,  by  the  way,  which  existence  itself  does  not  have.  This 
shows,  incidentally,  the  weakness  of  the  existential  theory  of  judg- 
ment— the  theory,  that  is,  that  every  proposition  is  concerned  with 
something  that  exists.  For  if  this  theory  were  true,  it  would  still 
be  true  that  existence  itself  is  an  entity,  and  it  must  be  admitted  that 
existence  does  not  exist.  Thus  the  consideration  of  existence  itself 

fCf.,  e.g.,  Dedekind,  Was  sind  und  was  sollcn  die  Zahlen?  2d  ed.,  Braun- 
schweig, 1893,  pp.  5,  12;  or  Essays  on  the  Theory  of  Numbers,  Chicago,  1901, 
pp.  49,  58;  Russell,  The  Principles  of  Mathematics,  Cambridge,  1903,  pp.  21,32. 

*  Of  course,  the  member  of  N  may  be  itself  a  class  and  may  thus  "exist," 
but  we  obviously  need  not  consider  this  further. 

'Mind.  N.  S.,  Vol.  X,  No.  39,  1901,  pp.  310-311. 


144  THE  MONIST. 

leads  to  non-existential  propositions,  and  so  contradicts  the  the- 
ory...." 

This  doctrine  was  repeated  in  Mr.  Russell's  Principles  of 
Mathematics;5  the  existence-theorems  of  mathematics  were  said8 
to  be  "proofs  that  the  various  classes  defined  are  not  null,"  and  the 
earlier  statement7  that  these  theorems  are  proofs  "that  there  are 
entities  of  the  kind  in  question"  must  not  be  taken  to  mean  what  it 
apparently  expresses. 

While  Mr.  Russell  emphasized  the  distinction  between  entity 
and  existence,  it  does  not  seem  that  at  that  time  he  quite  realized 
the  full  bearings  of  the  question,  at  least  in  mathematics.  He  at- 
tributed a  denotation  to  every  term  that  can  possibly  occur  in  a 
proposition.  Thus  "the  round  square"  had  a  denotation,  and  the 
only  further  existence-question  in  logic  and  mathematics  was 
whether  the  numbers — at  least  such  as  were  defined  as  classes — , 
classes  of  spaces,  and  so  on,  could  be  proved  to  "exist," — whether 
members  of  the  classes  in  question  could  be  constructed  by  logical 
methods  provided  that  the  initial  postulates  are  granted. 


Before  going  on  to  discuss  the  clear  separation  of  the  impor- 
tant question  of  entity  from  the  less  important  question  of  existence, 
which  came  in  Mr.  Russell's  later  works,  we  will  refer  to  the  very 
strong  tendency,  even  among  logicians  and  mathematicians,  to  at- 
tribute a  denotation  to  every  denoting  phrase. 

Thus,  H.  MacColl8  remarked  that  a  symbol  which  corresponds 
to  nothing  in  our  universe  of  admitted  realities,  has,  nevertheless, 
"like  everything  else  named,"  a  symbolical  entity.  In  his  sixth 
paper  on  "Symbolic  Reasoning,"9  MacColl  attempted  to  give  a 
simple  theory  of  the  existential  import  of  propositions. 

By  elt  e2,  e3,. . . .,  he  denoted  "our  universe  of  real  existences," 
and  by  oi}  oz,  o3,.  . . .,  "our  universe  of  non-existences,  that  is  to 
say,  of  unrealities,  such  as  centaurs,  nectar,  ambrosia,  fairies,  with 
self-contradictions,  such  as  round  squares,  square  circles,  fiat  spheres, 

"Pp.  449-450;  cf.  pp.  43,  71. 

'Ibid.,  p.  497. 

''Ibid.,  p.  vii. 

'Symbolic  Logic  and  its  Applications,  London,  1906,  p.  42;  MacColl  here 
and  elsewhere  used  the  word  "existence"  where  we  use  "entity."  Cf.  Mind, 
N.  S.,  Vol.  XI,  1902,  pp.  356-357. 

•Mind,  N.  S.,  Vol.  XIV,  1905,  pp.  74-81 ;  cf.  Symbolic  Logic  and  its  Appli- 
cations, pp.  5,  76-78. 


CRITICISMS  AND  DISCUSSIONS.  145 

etc." ;  the  "symbolic  universe,  or  universe  of  discourse,"  S,  may 
consist  either  wholly  of  realities,  wholly  of  unrealities,  or  partly  of 
realities  and  partly  of  unrealities. ...  If  A  denotes  an  individual  or 
a  class,  any  intelligible  statement  0(A)  containing  the  symbol  A, 
implies  that  the  individual  or  class  represented  by  A  has  a  symbolic 
existence;  but  whether  the  statement  <£(A)  implies  that  that  which 
A  denotes  has  a  real  or  unreal  or  (if  a  class)  partly  real  and  partly 
unreal  existence,  depends  upon  the  context." 

We  will  pass  over  the  discussion  between  Messrs.  MacColl 
and  A.  T.  Shearman10  on  the  interpretation  of  the  Boolian  equation 
"O  =  OA,"  and  come  to  Mr.  Russell's  articles  of  1905,11  in  which 
the  theory  of  non-entity  was,  it  seems,  for  the  first  time  treated 
satisfactorily. 

The  sense  in  which  the  word  "existence"  is  used  in  symbolic 
logic  is  a  definable  and  purely  technical  sense.  To  say  that  A 
exists  means  that  A  is  a  class  which  has  at  least  one  member.  Thus 
whatever  is  not  a  class  does  not  exist  in  this  sense;  and  among 
classes  there  is  just  one  that  does  not  exist,  namely,  the  null-class. 
MacColl's  two  universes  of  existences  and  non-existences  are  not 
to  be  distinguished  in  symbolic  logic,  and  each  of  them  is  identical 
with  the  null-class.  There  are  no  centaurs ;  "x  is  a  centaur"  is 
false  whatever  value  we  give  to  x,  even  when  we  include  values 
which  do  not  "exist"  in  the  meaning  which  occurs  in  philosophy 
and  daily  life,  such  as  numbers  or  propositions. 

"The  case  of  nectar  and  ambrosia  is  more  difficult,  since  these 
seem  to  be  individuals,  not  classes.  But  here  we  must  presuppose 
definitions  of  nectar  and  ambrosia :  they  are  substances  having  such 
and  such  properties,  which,  as  a  matter  of  fact,  no  substances  do 
have.  We  have  thus  merely  a  defining  concept  for  each,  without 
any  entity  to  which  the  concept  applies.  In  this  case,  the  concept 
is  an  entity,  but  it  does  not  denote  anything.  . .  .These  words  [such 
as  nectar  and  ambrosia]  have  a  meaning,  which  can  be  found  by 
looking  them  up  in  a  classical  dictionary,  but  they  have  not  a  deno- 
tation: there  is  no  entity,  real  or  imaginary,  which  they  point  out." 

"Mind,  N.  S.,  Vol.  XIV,  1905,  pp.  78-79,  295-296,  440,  578-580;  Vol.  XV, 
1906,  pp.  143-144;  and  Shearman's  book  The  Development  of  Symbolic  Logic; 
a  Critical-Historical  Study  of  the  Logical  Calculus,  London,  1906,  pp.  161-171. 

u  "The  Existential  Import  of  Propositions,"  Mind,  N.  S.,  Vol.  XIV,  1905, 
pp.  398-401 ;  "On  Denoting,"  ibid.,  pp.  479-493. 


146  THE  MONIST. 

The  last  sentence  refers  to  Frege's12  distinction  of  Sinn  (meaning) 
and  Bedeutung  (denotation). 

A  point  of  passing  interest  in  connection  with  an  attempt  at  the 
solution  of  a  mathematical  paradox,  referred  to  later,  is  this  sen- 
tence in  MacColl's  reply:13  "I  may  mention,  as  a  fact  not  wholly 
irrelevant,  that  it  was  in  the  actual  application  of  my  symbolic  sys- 
tem to  concrete  problems  that  I  found  it  absolutely  necessary  to 
label  realities  and  unrealities  by  special  symbols  e  and  o,  and  to 
break  up  the  latter  class  into  separate  individuals,  o1}  oz,  o3,  etc., 
just  as  I  break  up  the  former  into  separate  individuals  el}  e2,  e3,  etc." 

When  a  phrase  which  in  form  is  denoting,  and  yet  does  not 
denote  anything, — e.  g.,  "the  present  king  of  France," — occurs  in  the 
statement  of  a  proposition,  the  question  as  to  the  interpretation 
of  propositions  in  whose  verbal  expression  this  phrase  occurs  arises, 
and  Mr.  Russell,  in  the  article  "On  Denoting"  referred  to,  suc- 
ceeded in  assigning  a  meaning  to  every  proposition  in  whose  verbal 
expression  any  denoting  phrases — whether  they  appear  to  denote 
something  or  nothing  at  all,  e.  g.,  everything,  nothing,  something, 
a  man,  every  man,  no  man,  the  father  of  Charles  II,  the  present 
king  of  France — occur.  It  is  not  necessary  to  assume  that  denoting 
phrases  ever  have  any  meaning  in  themselves. 

The  theory  of  MacColl  and  the  allied  theory  of  Meinong  were 
rejected  by  Mr.  Russell1*  because  they  conflict  with  the  law  of  con- 
tradiction. If  any  grammatically  correct  denoting  phrase  stands  for 
an  object  although  such  objects  may  not  subsist,  such  objects  are 
apt  to  infringe  the  law  of  contradiction.  Thus  it  is  contended  that 
the  round  square  is  round,  and  also  not  round. 

To  solve  the  paradoxes  that  appear  in  the  mathematical  theory 
of  aggregates,  Mr.  Russell  treated  classes  and  relations  in  the  same 
way  as  he  treated  denoting  phrases.15 

Poincare,  among  others,  recognized  that  all  the  paradoxes  of 
the  modern  theory  of  aggregates,  such  as  those  of  Burali-Forti, 
Russell  and  Richard,  arise  from  a  kind  of  vicious  circle  which  may 
be  expressed,  in  the  language  of  Peano,  thus:  Everything  which 

u  "Ueber  Sinn  und  Bedeutung,"  Zeitschr.  fur  Phil  und  phil.  Kritik,  Vol. 
C,  1892,  pp.  25-50. 

"Mind,  N.  S,  Vol.  XIV,  1905,  p.  401. 

14  Ibid.,  pp.  491,  482-483. 

u  "On  Some  Difficulties  in  the  Theory  of  Transfinite  Numbers  and  Order 
Types,"  Proc.  Land.  Math.  Soc.  (2),  Vol.  IV,  1906,  pp.  29-53  (cf.  especially 
the  part  on  the  "No-Classes  Theory")  ;  "Les  Paradoxes  de  la  Logique,"  Rev. 
de  Metaphys.  et  de  Morale,  Vol.  XIV,  1906,  pp.  627-650. 


CRITICISMS  AND  DISCUSSIONS.  147 

contains  an  apparent  variable  must  not  be  one  of  the  possible 
values  of  this  variable.16  But  Poincare  did  not»perceive  that  if  we 
wish  to  avoid  such  vicious  circles  we  must  have  recourse  to  a 
fundamental  re-moulding  of  logical  principles,  more  or  less  anal- 
ogous to  the  "no  classes"  theory.  To  have  shown  this  seems  to 
be  one  of  Mr.  Russell's  greatest  merits ;  simply  because  practically 
all  the  other  mathematicians  who  have  interested  themselves  in  the 
paradoxes  did  not  realize  this  important  fact.  Thus,  said  Mr. 
Russell,17  the  method  by  which  Poincare  tried  to  avoid  the  vicious 
circle  consists  in  saying  that  when  we  assert  that  "all  propositions 
are  true  or  false,"  which  is  the  law  of  the  excluded  middle,  we 
exclude  tacitly  the  law  of  the  excluded  middle  itself.  The  difficulty 
is  to  make  this  tacit  exclusion  legitimate  without  falling  into  the 
vicious  circle.  If  we  say,  "All  propositions  are  true  or  false,  ex- 
cepting the  proposition  that  every  proposition  is  true  or  false,"  we 
do  not  avoid  the  vicious  circle.  For  this  is  a  judgment  bearing  on 
all  propositions,  viz.:  "All  propositions  are  either  true  or  false,  or 
identical  with  the  proposition  that  all  propositions  are  true  or  false." 
And  that  supposes  that  we  know  the  meaning  of  "all  propositions 
are  true  or  false,"  where  all  has  no  exception.  That  comes  to  de- 
fining the  law  of  the  excluded  middle  by :  "All  propositions  with  the 
exception  of  the  law  of  the  excluded  middle  are  true  or  false," 
where  the  vicious  circle  is  flagrant.  We  must,  then,  find  a  means 
to  formulate  the  law  of  the  excluded  middle  in  such  a  way  that  it 
does  not  apply  to  itself. 

On  the  details  of  the  new  construction  of  logic  in  such  a  way 
that  the  paradoxes  are  avoided  while  nearly  all  of  the  work  of 
Cantor  on  the  transfinite  is  preserved,  we  must  refer  to  Mr.  Russell's 
works  of  1908  and  1910.18  Mr.  Russell's  method  of  avoiding  the 
paradoxes  in  question  is  by  what  he  called  the  "theory  of  types," 
and  the  object  of  this  theory  was  shortly  described  by  Dr.  White- 
head  and.  Mr.  Russell19  as  follows :  "The  vicious  circles  in  question 
arise  from  supposing  that  a  collection  of  objects  may  contain  mem- 
bers which  can  only  be  defined  by  means  of  the  collection  as  a 

MWe  may  also  express  this  principle  as  follows:  A  collection  of  objects 
may  not  contain  members  which  can  only  be  defined  by  means  of  the  collection 
as  a  whole. 

v  Rev.  de  Metaphys.  et  de  Morale,  Vol.  XIV,  pp.  644-645. 

""Mathematical  Logic  as  Based  on  the  Theory  of  Types,"  Amer.  Journ. 
of  Math.,  Vol.  XXX,  1908,  pp.  222-262;  A.  N.  Whitehead  and  B.  Russell, 
Principia  Mathematica,  Vol.  I,  Cambridge,  1910,  pp.  39-88. 

"  Op.  cit.,  p.  39. 


148  THE  MONIST. 

whole.  Thus,  for  example,  the  collection  of  propositions  will  be 
supposed  to  contain  a  proposition  stating  that  'all  propositions  are 
either  true  or  false.'  It  would  seem,  however,  that  such  a  statement 
could  not  be  legitimate  unless  'all  propositions'  referred  to  some 
already  definite  collection,  which  it  cannot  do  if  new  propositions 
are  created  by  statements  about  'all  propositions.'  We  shall,  there- 
fore, have  to  say  that  statements  about  'all  propositions'  are  mean- 
ingless. More  generally,  given  any  set  of  objects  such  that,  if  we 
suppose  the  set  to  have  a  total,  then  such  a  set  cannot  have  a  total. 
By  saying  that  a  set  has  'no  total/  we  mean,  primarily,  that  no 
significant  statement  can  be  made  about  'all  its  members.'  Propo- 
sitions, as  the  above  illustration  shows,  must  be  a  set  having  no 
total.  The  same  is  true,  as  we  shall  shortly  see,  of  propositional 
functions,  even  when  these  are  restricted  to  such  as  can  significantly 
have  as  argument  a  given  object  a.  In  such  cases,  it  is  necessary 
to  break  up  our  set  into  smaller  sets,  each  of  which  is  capable  of  a 
total.  This  is  what  the  theory  of  types  aims  at  effecting."20 

*       *       * 

In  the  next  place,  we  shall  go  back  four  or  five  years  in  time, 
and  see  how  the  distinction  between  entity  and  existence  became 
necessary  in  a  mathematical  investigation  which  is  somewhat  famil- 
iar to  me.  If  I  consider,  at  rather  greater  length  than  it  deserves, 
my  own  work  of  1903  and  190421  on  the  contradiction  of  Burali- 
Forti  and  its  bearings  on  the  theory  of  well-ordered  aggregates, 
it  is  merely  because  familiarity  with  this  investigation  enables  me 
to  point  out  a  small,  unobserved  merit  which  it  has,  in  distinguishing 
entity  from  existence,  and  also  to  give  yet  another  illustration  of 
the  tendency — which  seems  particularly  common  with  mathemati- 
cians— of  holding  to  the  belief  in  the  being  or  existence  or  sub- 
sistence in  some  sense,  of  a  non-entity. 

Burali-Forti  had  found,  in  1897,  the  now  well-known  contra- 
diction arising  from  the  fact  that  'the  ordinal  type  of  the  whole 
series  of  (finite  and  transfinite)  ordinal  numbers'  appears  both  to 
be  and  not  to  be  the  greatest  ordinal  number.  From  this  I  con- 
cluded, in  1903,  that  there  are  no  such  things  as  "the  type"  and 

"The  theory  of  logical  types  was  described,  in  ordinary  language,  in  op, 
cit.,  pp.  39-68;  and  the  theory  of  denoting  was  explained  in  the  chapter  on 
"Incomplete  Symbols"  (ibid.,  pp  69-88). 

"A  general  account  of  these  investigations  is  contained  in  my  paper, 
written  in  Peano's  international  (uninflected)  Latin:  "De  Infinite  in  Mathe- 
matica,"  in  Revista  de  Mathematica,  VoL  VIII. 


CRITICISMS  AND  DISCUSSIONS.  149 

"the  cardinal  number"  of  the  series  just  referred  to.  Hence,  by  a 
tacit  use  of  an  axiom  afterwards  stated  explicitly  by  Zermelo,  I  con- 
cluded that  every  aggregate  which  has  a  cardinal  number  and  every 
series  which  has  a  type  can  be  well-ordered.  The  use  of  Zermelo's 
axiom  was,  with  me  as  with  most  mathematicians,  unrecognized; 
it  occurred  in  some  work  of  Mr.  G.  H.  Hardy's  on  which  I  based 
my  argument;  and  I  was  really  concerned,  not  so  much  with  the 
proof  that  every  aggregate  can  be  well-ordered,  as  with  the  proof 
that  the  series  (W)  of  ordinal  numbers  has  no  type. 

The  matter  becomes  simpler  to  express  when  we  consider 
classes  instead  of  series.  My  contention,  then,  was  that  there  is 
no  such  thing  as  "the  cardinal  number  of  the  class  of  ordinal  num- 
bers" seems  to  represent.  But  if  we  adopt,  as  I  adopted,  the  Frege- 
Russell  definition  of  the  cardinal  number  of  a  class  u  as  the  class 
of  those  classes  which  are  similar  to  (can  be  put  in  a  one-one  cor- 
respondence with)  u,  there  arises  a  difficulty.  The  cardinal  number 
of  the  class  w  of  ordinal  numbers  is  the  class  of  those  classes  which 
are  similar  to  w,  and  this  class  certainly  exists,  for  we  can  point 
out  at  least  one  member  of  it,  namely,  w  itself,  for  w  is  similar  to 
w.  On  the  other  hand,  we  have  reason  to  deny  that  there  is  such 
a  class  as  the  cardinal  number  of  w,  and  most  mathematicians  ex- 
press this  by  saying  that  the  cardinal  number  in  question  does  not 
"exist."  Of  course,  the  solution  of  this  apparent  contradiction  is 
that  "the  cardinal  number  of  w"  is  a  phrase  denoting  nothing — 
there  is  no  such  entity  as  the  cardinal  number  of  w.  If  it  did 
denote  a  class,  that  class  would  be  existent. 

So,  in  my  above-quoted  paper,  I  distinguished  between  the 
existence  of  a  class  u  from  the  entity  of  a  thing  v.  The  symbol 
"3.u"  was  used,  following  Peano,  to  denote  that  u  exists,  and  the 
symbol  "Ez>"  was  used  to  denote  the  proposition  that  v  is  an  entity. 
The  symbol  "Ez/"  was  defined  by  the  definition  of  "not-Ez/"  as 
"v  is  a  member  of  the  null-class."  Since  the  null-class  has  no 
members,  and  is  defined  as  the  x's  satisfying  a  prepositional  func- 
tion, such  as  x  is  not  identical  with  x,  which  is  always  false,  this 
is  a  most  paradoxical  way  of  stating  the  case  about  non-entity,22 
and  the  paradox  results  from  the  assumption  that,  in  some  sense, 
there  is  a  v, — that,  as  MacColl  would  have  said,  v  has  a  "symbolical 

"On  printing  the  above  article,  Professor  Peano  wrote  to  me,  on  Jan.  1, 
1906,  as  follows:  " I  see  the  new  symbol  E,  which  you  do  not  define  sym- 
bolically, but  the  importance  of  which  I  believe  I  have  understood It  would 

be  necessary  to 'introduce  many  kinds  of  null-class  (A):  AO  =  that  of  the 
Formulaire ;  Aj  =  the  class  of  classes,  which  has  no  classes ;  A,  for  the  classes 


I5O  THE  MONIST. 

existence."  But,  as  Dr.  Whitehead  and  Mr.  Russell23  pertinently 
remark:  "We  cannot  first  assume  that  there  is  a  certain  object,  and 
then  proceed  to  deny  that  there  is  such  an  object."  Russell's  solu- 
tion of  the  difficulty  about  propositions  asserting  that  "the  so-and-so 
is  not  an  entity"  is  to  reduce  all  such  propositions  to  a  form  not 
involving  the  assumption  that  "the  so-and-so"  is  a  grammatical 
subject.  "The  so-and-so,"  whether  it  appears  to  denote  something  or 
not,  is  an  incomplete  symbol,  like  the  d/dx  of  mathematics. 

*       *       * 

It  has,  I  trust,  been  not  quite  without  interest  to  see  how  the 
important  distinction  of  existence  and  entity  in  mathematics  strug- 
gled into  clearness.  We  have  seen  before24  that  the  discussions  on 
"existence"  of  MM.  Poincare  and  Couturat  were  conducted  in  ob- 
scurity. This  obscurity  was  produced  by  the  confusion  of  the  two 
notions  of  existence  and  entity,  and  the  consequent  use  of  one  word 
to  denote  both. 

When,  in  a  paper  published  in  1904,  I  used  the  badly  chosen 
term  "inconsistent"  for  an  aggregate  whose  cardinal  number  is  a 
non-entity — "does  not  exist,"  I  said  then — Mr.  Russell  rightly  ob- 
jected that,  given  a  class  u,  its  cardinal  number  must  exist,  since  u 
is  a  member  of  the  class  called  the  cardinal  number  of  u.  And  yet 
there  was  an  undoubted  difficulty  about  what  I  called  "inconsistent" 
classes.  We  know  now  that — at  any  rate  when  the  number  of  a 
class  is  defined  logically — it  is  a  delusion  that  there  are  such  "in- 
consistent" classes, — they  are  non-entities.  If  they  were  entities, 
their  cardinal  numbers  would  "exist." 

There  is  one  more  thing  to  be  noticed:  it  is  the  entity  of 
a  number  that  is  most  important,  the  proof  of  its  existence  is 
less  so.  In  his  Principles  of  1903,  Mr.  Russell  laid  great  stress  on 
the  existence-proofs  of  numbers  and  classes  of  spaces.  Let  us  con- 
sider the  case  of  real  numbers.  A  real  number  is,  according  to  Mr. 

of  classes;    A,   for   the   classes   of   classes   of   classes;....    An,. . .  .A-^ 

There  is  the  generation  of  the  transfinite  numbers,  in  the  principles  of  logic. 
There  results  this  rather  laughable  consequence,  that  the  new  philosophers 
have  decomposed  nothing  into  a  transfinite  number  of  classes!" 

*  Op.  cit.,  p.  69.  We  may  remark  here,  as  I  have  done  in  a  review  of 
Whitehead  and  Russell's  Principia  in  the  Cambridge  Review  for  1911,  that  the 
authors  (cf.  pp.  32,  69,  182,  229)  use  the  word  "existence"  ambiguously; 
though,  of  course,  there  is  no  ambiguity  when  the  proper  technical  symbols 
(a  and  E;  E  only  occurring  in  a  phrase  involving  incomplete  symbols)  are 
used. 

"Monist,  Jan.  1910,  Vol.  XX,  pp.  113-116. 


CRITICISMS  AND  DISCUSSIONS. 

Russell,  a  certain  class  of  rational  numbers ;  its  existence  can  be 
proved,  and  one  feels  satisfied.  But  a  rational  number  or  a  negative 
number,  being  a  relation,  does  not  "exist,"  and  yet  one  would  have 
thought  existence  quite  as  important  in  these  cases  as  in  the  case 
of  real  numbers.25  I  hope  to  go  more  fully  into  this  question  on 
another  occasion. 

PHILIP  E.  B.  JOURDAIN. 
CAMBRIDGE,  ENGLAND. 


IDEALISM  AS  A  FORCE. 

A    MECHANICAL  ANALOGY. 

In  the  present  state  of  knowledge  the  man  of  intelligence  has 
much  difficulty  in  deciding  what  course  of  conduct  he  should  adopt 
in  regard  to  beliefs  and  social  and  religious  practice  without  at  the 
same  time  violating  these  principles  which  he  has  obtained  from 
science  and  critical  philosophy.  Before  venturing  to  suggest  exactly 
what  position  he  should  (and  eventually  must)  take  up,  a  little  con- 
sideration of  the  importance  of  the  older  ideas  and  their  relation  to 
new  ones  would  be  advisable.  I  propose  to  introduce  various  me- 
chanical analogies  in  this  sketch,  for  two  reasons.  First,  because  I 
think  they  show  forth  more  clearly  the  nature  of  the  phenomena 
described,  and  second,  a  training  in  scientific  thought  soon  shows  one 
that  mechanical  laws  pervade  the  whole  universe,  mental,  moral  and 
physical.  I  do  not  use  the  word  "mechanical"  in  at  all  a  derogatory 
sense.  As  a  matter  of  fact,  although  it  seems  at  first  contrary  to 
our  ideas  of  perfection  realized  by  a  continuous  process  of  adjust- 
ment, the  really  perfect  state  is  the  mechanical  one,  where  each  part 
has  a  definite  and  unchanging  relation  to  all  the  other  parts,  so  that 
a  change  in  its  condition  is  accompanied  by  a  change  in  all  other 
parts  in  accordance  with  the  nature  of  that  mutual  relation.  Surely 
this  is  what  is  meant  by  "correspondence  with  environment,"  if 
there  is  the  proviso  of  stability.  All  moral  philosophers  have  more 
or  less  directly  stated  that  the  key  to  morality  is  the  Golden  Rule, 
"Do  as  you  would  be  done  by,"  or  as  K'ung-f  u-tze  puts  it,  in  one  word, 
"Reciprocity,"  i.  e.,  mutual  bearing  upon  one  another.  This  condi- 
tion of  mutual  bearing  is  essentially,  when  complete,  a  mechanical 

*  Frege  (Grundlagen  dcr  Arithmetik,  Breslau,  1884,  pp.  114-115)  indicated 
such  definitions  of  all  the  numbers  of  analysis  as  would  enable  him  to  prove 
the  existence  in  every  case. 


152  THE  MONIST. 

state.  Similarly  in  matters  of  thought  consistency  is  the  great  prin- 
ciple, and  what  is  consistency  but  a  mechanically  perfect  state  of 
balance?  As  to  the  mechanical  character  of  physical  conditions 
there  can  be  no  question,  provided  we  do  not  necessarily  limit  the 
concept  to  the  Newtonian  exposition. 

I  wish  to  use  frequently  the  idea  of  force.  In  natural  philos- 
ophy a  force  is  that  which  tends  to  produce  or  hinder  motion,  and 
it  is  the  characteristic  of  all  natural  phenomena  that  the  forces 
acting  on  them  shall  be  in  a  state  of  balance.  Whether  they  are  still 
or  moving,  this  balance  exists  either  in  the  form  of  opposed  pulls, 
pushes,  stresses  or  accelerations  of  mass.  It  is  the  criterion  in  the 
light  of  which  all  mechanical  problems  may  be  attacked.  I  wish  to 
extend  this  idea  of  force  to  matters  of  thought  and  ideal,  by  a  defini- 
tion such  as  the  following:  A  mental  force  is  that  which  produces 
or  tends  to  produce  change  of  thought. 

The  ever-famous  Newton,  in  studying  natural  forces,  announced 
three  laws  of  motion.  There  is  no  definite  proof  of  these,  but  we 
have  no  experience  which  contradicts  them. 

With  the  suggested  psychical  analogues  these  laws  are  as  fol- 
lows: 

1.  Any  body  tends  to  remain  in  its  condition  of  rest  or  motion 
until  acted  on  by  some  force. 

To  extend  this  to  matters  of  thought  we  can  say: 

Any  idea  (group  of  concepts)  tends  to  remain  in  its  state  of 

rest  or  change  along  certain  lines  until  acted  upon  by  some  mental 

force. 

2.  Change  of  motion  is  proportional  to  the  magnitude  of  the 
applied  force. 

This  becomes: 

Change  of  thought  is  greater  or  less  according  to  the  effective 
importance  of  the  mental  force. 

3.  To  every  action  there  is  a  reaction,  i.  e.,  whenever  a  force 
acts  upon  a  body  there  is  called  out  in  that  body  a  force  opposed 
to  (and  equal  to)  the  first  force  which  manifests  itself  as  internal 
stress  or  acceleration  of  mass. 

In  mental  matters  this  notion  is  expressed  by  the  change  in 
thought  which  takes  place  as  the  result  of  applying  mental  force, 
appearing  either  as  a  new  formation  of  ideas  or  a  reaction  of  old 
ideas  on  the  new  mental  force. 

It  must  be  understood  at  this  point  that  I  do  not  mean  anything 


CRITICISMS  AND  DISCUSSIONS.  1 53 

extremely  mystical  or  undiscovered  by  this  term  "mental  force." 
I  simply  give  this  name  to  a  set  of  ideas,  in  the  first  place  external 
to  the  mind  in  question,  then  received  through  the  ordinary  chan- 
nels of  sense,  and  acting  upon  the  ideas  already  existing  there, 
either  producing  resistance  or  modifying  those  ideas.  The  tech- 
nical word  "suggestion"  is  almost  identical  in  meaning. 

The  engineer,  in  the  spirit  of  Newton,  takes  our  above-described 
three  laws  into  one  equivalent,  as  follows : 

Force  is  the  rate  of  change  of  motion  attached  to  matter  (tech- 
nically "momentum"). 

This  simply  means  that  wherever  and  whenever  a  force  acts 
upon  a  body  it  produces  a  change  in  its  motion,  or,  vice  versa,  a 
change  in  motion  is  caused  by  a  force. 

This  can  be  made  the  basis  of  a  more  sweeping  statement  which 
describes  mental  force  thus: 

Mental  force  is  the  rate  of  change  of  thought  attached  to  mind. 
(Brain-matter  is  perhaps  not  to  be  regarded  as  the  absolute  medium 
of  thought,  since  psychologists  regard  the  latter  as  contemporaneous 
with,  but  not  necessarily  the  same  as,  change  in  cerebral  substance) . 

Idealism  I  wish  to  describe  as  a  particular  type  of  mental 
force  proceeding  in  the  first  place  from  some  external  source,  and 
then  by  its  action  on  different  minds  in  accordance  with  the  above 
laws  and  by  the  reactions  of  such  minds  on  physical  and  moral 
actions,  producing  an  effect  tending  to  the  realization  of  certain 
progressive  states  which  are  for  the  time  being  regarded  as  perfect. 

In  the  light  of  this  conception  all  religions  are  forms  of  ideal- 
ism. 

If  we  examine  any  religion  from  its  commencement  we  usually 
find  some  such  development  as  this: 

1.  Absorption  by  a  master  mind  (the  founder)  of  certain  older 
ideals,  the  mutual  reactions  of  which  together  with  the  mental  con- 
tition  induced  in  him  by  his  surroundings  (physical  and  social)  pro- 
duce a  new  system  with  one  central  ideal. 

2.  This  result  in  many  cases  is  accompanied  by  very  severe 
mental  strain,  and  in  some  cases  by  nervous  disease  (cf.  Mohammed 
who  is  believed  to  have  suffered  from  epilepsy)  after  which  this 
ideal  takes  the  leading  part  in  his  thought  and  life  (monoideism). 

3.  The  ideal  now  works  through  him  to  the  minds  of  certain 
followers  or  disciples  who  receive  it  according  to  their  previous 


154  THE  MONIST. 

training  and  heredity,  and  so  is  formed  a  circle  of  minds  in  which 
the  ideal  circulates  for  a  time,  gaining  an  ever  increasing  potential. 

4.  The  widening  of  the  circle  and  frequently  the  loss  by  decease 
of  the  founder,  causes  the  ideal  to  cease  its  original  evolution  and 
take  on  certain  new  features  according  to  the  reactions  in  the  minds 
of  its  various  adherents.    Hence  we  have  lesser  circles  forming,  to 
which  certain  new  phases  have  more  and  more  relation,  until  there 
is  a  schism  of  the  original  community  and  the  most  energetic  minds 
found  sects. 

5.  These  sections  expand  or  not  according  as  the  ideal  is  re- 
sisted or  absorbed  by  the  further  minds  upon  which  it  acts,  and  we 
may  finally  have  a  large  community  with  the  ideal  (usually  much 
modified  by  reaction)   controlling  and  connecting  the  units.     This 
arrangement  persists  until  external  ideas  of  a  different  kind  or  in- 
ternal resistances  destroy  its  energy  and  it  is  replaced  by  other 
ideals  or  a  great  modification  of  the  old  one. 

The  mechanical  analogy  to  the  action  of  external  forces  on 
matter  already  possessing  kinetic  energy  is  so  obvious  if  the  lines 
previously  indicated  are  followed,  that  I  will  not  trace  out  each 
link  of  the  chain,  but  merely  point  out  the  steps  in  which  we  draw 
a  comparison. 

1.  Composition   (i.  e.,  combining  together)   of  various  forces 
(ideals)  in  one  point  (mind)  which  possesses  considerable  freedom 
(enthusiasm). 

2.  Acceleration  in  this  point  (mind)  under  the  resultant  force 
(new  ideal)  finally  acting  on  other  bodies  (minds)  in  a  greater  or 
less  degree  according  to  their  condition  of  stability  (environment). 

3.  Composition  of  the  forces  in  these  individual  bodies  (minds) 
resulting  in  a  balanced  but  unstable  system  (idealist  community). 

4.  Splitting  up  of  systems  into  smaller  systems  (sects)  balanced 
in  themselves  with  moderately  high  stability  (sects)  and  balanced 
as  a  whole  (unstably)  as  a  general  system  (national  religion). 

5.  Modification  of  system  by  new  forces   (ideals)   finally  re- 
sulting in  a  new  system  (religion). 

At  this  point  it  is  necessary  to  discuss  the  importance  of  ideal- 
ism in  its  effect  on  the  social  life.  Once  a  definite  ideal  or  system 
of  ideals  has  become  established  among  a  set  of  minds  it  acts  as  a 
"superhuman"  power  (not  in  the  accepted  sense  of  "supernatural" 
but  as  the  simple  result  of  evolution)  whose  magnitude  is  the  re- 
sultant of  the  various  forces  which  it  has  impressed  on  individual 


CRITICISMS  AND  DISCUSSIONS.  1 55 

minds  and  whose  direction  (i.  e.,  tendency  to  progress  or  degen- 
erate) is  determined  by  the  manner  in  which  it  has  combined  with 
the  mental  forces  previously  impressed  on  these  minds. 

We  see  then  that  it  has  a  definite  (but  fluctuating)  value,  a 
more  or  less  constant  direction  (for  the  time)  and  it  is  attached 
to  a  certain  number  of  unit  minds. 

It  may  be  compared  with  the  constitution  of  the  atom  in  which 
there  are  a  number  of  electrons  each  possessing  a  peculiar  resultant 
motion  of  its  own  but  at  the  same  time  coordinating  with  other 
electrons  to  confer  on  the  atom  as  a  whole  certain  dynamic  properties 
which  manifest  themselves  as  polarity  or  chemical  attraction,  which, 
although  the  equivalent  of  the  electronic  energy,  are  different  in 
kind. 

Similarly  our  ideal  may  be  attached  to  a  large  number  of  minds 
of  varying  caliber,  force  and  direction,  but  as  a  whole  organism  the 
system  will  be  possessed  of  properties  differing  from  those  of  its 
units. 

Such  a  force  as  this  centered  in  a  community  constitutes  a 
divine  being  controlling  and  working  through  its  members,  just  as 
according  to  modern  psychology,  the  soul  is  a  centering  of  nervous 
energy.  The  Christian  church  in  which  the  members  are  said  to 
belong  to  the  mystical  body  of  Christ  exemplifies  this.  The  whole 
of  the  church  is,  so  long  as  homogeneity  prevails,  a  force  whose  mag- 
nitude is  the  resultant  of  the  mental  and  moral  efforts  of  the  units. 
These  efforts  may  be  distinct  in  kind,  amount  and  object,  but  never- 
theless on  the  whole  they  are  cumulative  and  there  is  a  resultant 
which  may  be  well  called  the  living  Christ,  for  it  is  an  intelligent 
force  realizing  within  itself  to  some  extent  the  ideal  which  the 
master-mind  of  Jesus  impressed  on  his  disciples  to  such  a  degree 
as  their  capacities  permitted. 

In  this  way  the  doctrines  of  salvation  (i.  e.,  separation  from 
anti-Christian  community  and  ideals)  and  grace  (impression  of 
idealism  according  to  capacity  for  receiving  it)  become  explicable 
and  even  reasonable.  Of  this  more  later. 

I  am  of  course  aware  that  I  at  once  lay  myself  open  to  severe 
criticism  from  the  adherents  of  all  faiths  who  conceive  their  deity 
to  be  omnipotent  and  omniscient.  To  this  notion  I  would  say  that 
such  a  force  as  described  above  has  within  itself  the  means  of  doing 
and  knowing  all  those  things  which  come  within  the  ken  of  the 
units,  and  that  further  it  combines  with  the  resultant  forces  of  the 


156  THE  MONIST. 

universe,  being  either  decreased  or  increased  in  effect  according  as  it 
is  opposed  to  or  in  line  with  such  world  forces.  So  long  as  a  religion 
progresses  (apart  from  the  consideration  of  certain  artificial  condi- 
tions such  as  politics)  it  must  be  to  some  extent  in  conformity  with 
the  laws  of  the  universe,  known  and  unknown.  So  soon  as  it 
directly  opposes  those  laws  (still  subject  however  to  certain  socio- 
logical factors)  it  must  degenerate.  The  gods  of  a  religion  live  and 
die  with  it,  their  energy  appearing  in  other  faiths  after  reaction 
has  taken  place  in  the  minds  of  the  interregnum.  The  only  case  in 
which  they  (or  he)  are  immortal  is  when  they  are  definitely  identi- 
fied with  some  permanent  force  in  the  universe  so  that  the  mental 
force  runs  contemporaneously  with  a  natural  one,  each  producing 
proportionate  effects  on  mind  and  matter.  It  is  from  this  cause  that 
Judaism  has  ensured  its  immortality.  About  the  time  of  the  Cap- 
tivity it  definitely  connected  its  tribal  deity  Yahweh  not  only  with 
the  ideal  of  tsedek  (righteousness)  but  with  that  unitary  world- 
power  which  under  various  names  (such  as  "the  eternal  energy") 
all  philosophers  and  scientists  recognize,  with  or  without  moral 
attributes.  This  element  of  permanence  has  been  transmitted  to 
Christianity  and  Islam  so  that  these  three  are  probably  the  most 
stable  of  all  faiths.  It  does  not  however  necessarily  follow  that 
because  the  force  survives,  the  attachment  of  the  community  to  the 
ideal  force  will  also  survive.  Its  energy  may  be  transferred  to  other 
minds,  possibly  in  other  forms,  but  practically  never  losing  all  con- 
nection with  the  primal  natural  force  with  which  it  has  been  asso- 
ciated. 

In  order  that  the  idealism  of  a  community  shall  have  a  perma- 
nent effect  it  is  necessary: 

1.  That  there  should  be  a  continual  supply  of  mental  energy 
on  the  part  of  unit  minds ; 

2.  That  the  individual  energies  shall  be  so  directed  generally 
and  of  such  amount  that  there  always  is  an  external  resultant  pro- 
ducing progress  by  its  reaction  on  the  minds  of  both  the  units  of 
the  community  and  those  outside  of  the  community. 

In  order  to  assure  the  first  condition  some  definite  "cult"  is 
required,  which  by  the  repetition  of  various  practices  concentrates 
the  mind  on  the  ideal  tending  to  develop  its  realization  in  that  mind 
and  directing  the  energy  of  the  mind  to  that  end,  both  within  and 
without. 

In  the  second  condition  it  is  essential  that  certain  agreements 


CRITICISMS  AND  DISCUSSIONS.  157 

concerning  the  ideal  shall  be  established,  so  that  the  energies  put 
forth  are  not  contrary  in  tendency.  This  is  the  foundation  of  dogma, 
which  states  as  far  as  possible  the  ideal  in  words  and  symbols,  which 
produce  in  the  various  minds  a  more  or  less  homogeneous  concep- 
tion of  the  ideal. 

Further,  it  is  necessary  in  order  that  the  mental  forces  shall  not 
equilibriate,  that  all  the  members  of  the  community  shall,  as  far  as 
practicable  within  the  limits  of  the  competition  necessitated  by  the 
law  of  selection  and  survival,  support  one  another,  so  that  the 
mutual  stress  between  them  is  minimized  and  the  external  resultant 
increased. 

To  return  to  our  electron  analogy,  if  electrons  move  at  right 
angles  to  the  general  path,  collisions  will  occur  which  reduce  the 
external  force  exerted  by  the  atom,  and  if  sufficiently  numerous 
may  be  conceived  quite  to  destroy  that  force  and  even  disintegrate 
the  atom.  (Cf.  "The  house  divided  against  itself.") 

This  necessity  for  internal  balance  gives  rise  to  ethics,  which 
is  summarized  by  the  Golden  Rule. 

HERBERT  CHATLEY. 
CHINESE  GOVERNMENT  ENGINEERING  COLLEGE. 

TANG  SHAN,  CHIH-LI. 

CLASSICAL  CONFUCIANISM. 

Sinology  has  so  far  not  yet  passed  the  stage  of  crude  and 
amateurish  translation.  No  interpretative  work  worthy  of  serious 
consideration  has  yet  appeared.  Mr.  Miles  Menander  Dawson's 
recently  published  book,  The  Ethics  of  Confucius:  The  Sayings  of 
the  Master  and  his  Disciples  upon  the  Conduct  of  the  "Superior 
Man/'1  is  an  attempt  in  the  direction  of  interpreting  Confucianism 
to  the  West.  We  congratulate  him  on  his  highly  successful  exposi- 
tion of  one  of  the  greatest  ethical  systems  of  the  world.  His  work 
has  at  least  met  a  need  which  has  long  been  felt  by  all  who  desire 
to  bring  about  a  better  understanding  of  Chinese  civilization  in  the 
occidental  world.  For  ever  since  the  days  of  Marshman  and  Legge 
the  true  meaning  of  Confucianism  has  been  lying  hidden  in  those 
painstaking  but  unfortunately  too  expensive  and  out-of-print  trans- 
lations ;  and  the  general  public  have  long  had  to  swallow  what  super- 
ficial and  biased  writers  are  pleased  to  call  "Confucianism."  Mr. 
Dawson's  book  is  based  entirely  on  Legge's  translation  of  The 

i  New  York,  G.  P.  Putnam's  Sons.    Pp.  xviii,  305.    Price,  $1.50  net. 


158  THE  MONIST. 

Chinese  Classics,  and  he  has  so  classified  and  arranged  his  material 
that  the  reader  can  easily  comprehend  what  Confucius  and  the 
early  Confucians  actually  said  on  the  various  fundamental  prob- 
lems of  life. 

This  book  has  many  notable  merits.  First,  the  handling  of  the 
immense  quantity  of  material  is  excellent.  The  work  is  divided 
into  seven  chapters :  I.  What  Constitutes  the  Superior  Man ;  II. 
Self -Development ;  III.  General  Human  Relations ;  IV.  The  family ; 
V.  The  State;  VI.  Cultivation  of  the  Fine  Arts;  VII.  Universal 
Relations.  Mr.  Dawson  has  seized  upon  a  very  important  point  in  Con- 
fucianism when  he  arranges  his  book  in  accordance  with  the  scheme 
of  The  Great  Learning.  For  the  Confucian  ethics  is  essentially  a 
system  of  human  relations:  all  extension  of  knowledge  contributes 
to  the  cultivation  of  individual  conduct,  and  from  the  individual 
there  radiate  the  relationships  of  the  family,  the  state  and  the  world. 

Secondly,  the  illustrative  quotations  from  the  Confucian  classics 
are,  with  a  few  exceptions,  very  well  chosen.  The  quotations  are 
all  accompanied  by  the  name  of  the  book,  the  number  of  chapter, 
paragraph  and  verse.  The  carefulness  and  patience  with  which  the 
numerous  passages  are  selected  and  classified,  certainly  commands 
our  admiration.  The  index  appended  to  the  book  also  enhances  its 
usefulness. 

Thirdly,  the  first  two  chapters  in  particular  constitute  the  best 
portion  of  the  book.  In  these  chapters  Mr.  Dawson  sets  forth  the 
Confucian  ideal  man,  "the  Superior  Man,"  which  forms  the  sub- 
title of  the  book.  The  Superior  Man,  which  can  be  more  literally 
translated  as  "the  lordly  man"  or  better  still  as  "the  gentleman," 
is  quite  different  from  the  dianoetic  man  of  the  Greeks;  neither 
does  he  aspire  to  the  Nirvanic  life  of  Buddhism,  nor  aim  at  the 
attainment  of  a  union  with  God,  which  forms  the  ideal  of  Chris- 
tianity. The  Confucian  ideal  is  simply  a  life  made  ever  nobler  and 
richer  by  individual  reticence  and  by  a  conscious  adoption  as  one's 
own  of  the  social  moral  institutions  which  constitute  the  li  (trans- 
lated "rules  of  propriety")  or  what  the  Hegelians  call  Sittlichkeit. 
In  expounding  these  basic  elements  of  Confucianism  Mr.  Dawson 
has  exhibited  a  high  degree  of  clarity  of  exposition  and  richness 
of  illustration. 

Lastly,  we  believe  that  the  greatest  merit  of  the  book  lies  in 
its  objectivity,  by  which  is  meant  the  impartiality  and  disinterested- 
ness with  which  the  author  expounds  the  Confucian  doctrines. 


CRITICISMS  AND  DISCUSSIONS.  1 59 

Mr.  Dawson  has  no  desire  to  prove  that  Confucianism  is  inferior 
to  any  particular  ethical  or  religious  system,  nor  does  he  wish  to 
proselyte  his  readers  into  Confucianism.  He  simply  presents  to  us 
what  the  great  Confucians  thought  and  taught  concerning  the 
multifarious  complexities  of  life  and  conduct.  He  speaks  of  con- 
cubinage with  the  same  calmness  with  which  he  discusses  the  Con- 
fucian conception  of  the  state. 

It  is  natural  that  an  undertaking  of  this  kind  by  one  who  has 
no  access  to  the  original  texts  cannot  be  entirely  free  from  occa- 
sional errors.  Numerous  unimportant  mistakes  may  be  pointed  out 
at  random.  For  example :  ( 1 )  on  page  xiii,  the  name  of  Confucius 
appears  twice  as  Kung  Chin,  which  should  be  Kung  Chiu;  (2)  on 
page  xiv,  Chun  Chin  should  read  Chun  Chiu ;  (3)  on  page  xvi,  it 
is  wrong  to  include  the  Hsiao  King  instead  of  the  Chun  Chiu  in  the 
Five  Classics ;  and  (4)  on  the  same  page  "Pan  Ku"  and  The  His- 
tory of  Han  Dynasty  are  mentioned  as  two  separate  works ;  whereas, 
as  a  matter  of  fact,  Pan  Ku  is  the  author  of  The  History  of  Han 
Dynasty. 

Of  errors  of  a  more  serious  nature  we  find  at  least  three. 
In  the  first  place,  the  title,  "The  Ethics  of  Confucius,"  is  not  correct. 
It  is  as  if  a  compilation  of  the  ethical  theories  contained  in  the 
works  of  Plato,  Aristotle  and  Theophrastus  were  to  be  called  "The 
Ethics  of  Socrates."  Mr.  Dawson's  book  deals  with  the  ethics, 
not  of  Confucius  alone,  but  of  what  we  may  call  classical  Con- 
fucianism. For  it  is  almost  needless  to  point  out  that  many  of  the 
Confucian  classics,  like  the  Shu  King  and  the  Shi  King,  deal  with 
historical  periods  long  before  Confucius;  while  others,  like  the  Book 
of  Mencius  and  the  Li  Ki,  came  long  after  the  death  of  Confucius. 
Book  III  of  the  Li  Ki,  for  example,  was  compiled  in  the  second 
century  B.  C. 

In  the  second  place,  Mr.  Dawson  has  at  times  misinterpreted 
the  meaning  of  certain  passages.  Take  this  illustration: 

"The  scholar  keeps  himself  free  from  all  stain"  (Li  Ki, 
xxxviii,  15).  The  Master  said,  "Refusing  to  surrender  their 
wills  or  to  submit  to  any  taint  to  their  persons ;  such,  I  think, 
were  Pih-E  and  Shuh-Tse"  (Analects,  xviii,  8). 

"These  two  passages,"  says  Mr.  Dawson,  "illustrate  the  sage's 
insistence  upon  sexual  continence,  among  other  virtues."  Now  the 
word  "stain"  in  the  first  quotation  has  no  reference  to  sexual  rela- 
tions. Nor  does  the  phrase  "taint  to  their  persons"  in  the  second  quo- 


l6o  THE  MONIST. 

tation  mean  sexual  immorality.  The  story  of  Pih-E  and  Shuh-Tse  (or 
Po-I  and  Shu-Chi),  who  abandoned  their  hereditary  kingdom  and 
retired  into  obscurity,  and  who,  when  the  Chou  Dynasty  was 
founded,  died  of  hunger  rather  than  live  under  the  new  dynasty, — 
this  story  is  well  known  to  every  Chinese,  and  is  given  in  a  note  in 
Legge's  translation  (v.  22). 

In  the  third  place,  Mr.  Dawson  has  on  several  occasions  taken 
a  passage  quite  apart  from  its  immediate  and  inseparable  context, 
thus  losing  the  meaning  that  was  intended.  An  example  of  this 
kind  is  found  on  page  248: 

"When  good  government  prevails  in  the  empire,  cere- 
monies, music  and  punitive  military  expeditions  proceed 
from  the  emperor"  (Analects,  xvi,  2). 

This  passage  Mr.  Dawson  takes  as  "suggesting  that  wise  pa- 
tronage and  encouragement  of  art  by  the  government  which  has 
distinguished  the  most  enlightened  governments  of  ancient  and 
modern  times."  Now  this  passage  cannot  be  taken  apart  from  its 
context.  Here  is  the  context: 

"When  good  government  prevails  in  the  empire,  cere- 
monies, music,  and  punitive  military  expeditions  proceed 
from  the  emperor.  When  bad  government  prevails,  these 
things  proceed  from  the  princes.  When  these  things  pro- 
ceed from  the  princes,  rarely  can  the  empire  maintain  itself 
more  than  ten  generations."2 

Here  we  can  easily  see  that  the  point  of  emphasis  in  this 
passage  is  from  what  source  these  institutions  should  derive  their 
authority.  The  passage  no  more  illustrates  the  wise  patronage  of 
art  than  it  illustrates  the  encouragement  of  punitive  expeditions. 
It  must  be  pointed  out,  however,  that  such  errors  are  very 
rare  in  the  entire  work.  On  the  whole,  Mr.  Dawson's  book  may 
be  recommended  to  all  students  of  Chinese  philosophy  and  religion 
as  an  excellent  exposition  of  classical  Confucianism. 

SUH  Hu. 

COLUMBIA  UNIVERSITY,  New  York. 
2  This  is  my  translation.    Legge's  rendering  is  not  correct. 


VOL.  XXVII  APRIL,  1917  NO.  2 


THE  MONIST 


THE  TEXT  OF  THE  RESURRECTION  IN  MARK, 

AND  ITS  TESTIMONY  TO  THE  APPA- 

RIT1ONAL  THEORY. 

WITH  A  PREFACE  ON  LUKE'S  MUTILATION  OF  MARK. 


~^HE  greatest  literary  problem  in  the  New  Testament 
A  is:  What  is  the  matter  with  the  Gospel  of  Mark? 
Something  happened  to  the  end  of  it  in  the  first  or  second 
century,  and  for  ages  thereafter  it  was  left  truncated  in 
the  middle  of  a  sentence  or  else  supplied  with  a  shorter 
conclusion  than  the  present  one,  which  scholars  long  kept 
to  themselves.  Edwin  A.  Abbott,  however,  gave  it  in  his 
forgotten  Gospel  analysis  of  1884,  and  the  Nonconformist 
translators  of  The  Twentieth  Century  New  Testament 
have  also  given  it;  but  it  does  not  appear  in  any  official 
translation,  though  the  Revised  Version  mentions  it  in  a 
note  at  Mark  xvi.  8.  This  is  the  note  : 

"The  two  oldest  Greek  manuscripts,  and  some  other 
authorities,  omit  from  verse  9  to  the  end.  Some  other 
authorities  have  a  different  ending  to  the  Gospel." 

Here  is  the  "different  ending,"  translated  from  a  ninth- 
century  manuscript  in  the  National  Library  of  France, 
Codex  L,  which  gives  both  conclusions,  but  puts  this  one 
first.  (We  prefix  to  it  the  connecting  words  of  Mark)  : 

233 

II.  And  they  went  out  and  fted  from  the  sepulcher,  for 


l62  THE  MONIST. 

trembling  and  astonishment  had  come  upon  them;  and  they 
said  nothing  to  any  one,  for  they  were  afraid  of. 

£  .      i|i      •  $         4*  ••      j|c         j|c     •'  ^ji    •••  ;|t        £     •    4t.        4*     £    4*        4( 
[Thirteen  ornamental  marks.] 

Where  also  you  must  give  currency  to  this : 
Now,  all  things  that  were  commanded,  they  showed 
forth  in  few  words  unto  those  about  Peter.  And  after 
these  things  Jesus  himself,  also,  from  the  East  even  unto 
the  West,  sent  forth  through  them  the  holy  and  incor- 
ruptible preaching  of  eternal  salvation. 

But  there  is  also  current  the  following,  after  the  words : 

FOR  THEY  WERE  AFRAID  OF: 

Now,  when  he  was  risen  early  etc.  (as  in  our  common 
versions,  Mark  xvi.  9-20). 

In  their  Introduction  to  the  New  Testament  (Cam- 
bridge, 1 88 1,  pp.  298,  299)  Westcott  and  Hort  remark  on 
the  above  "less  known  alternative  supplement"  to  Mark: 
"In  style  it  is  unlike  the  ordinary  narrative  of  the  Evan- 
gelists, but  comparable  to  the  four  introductory  verses  of 
St.  Luke's  Gospel."  Conybeare,  in  his  great  book,  Myth, 
Magic  and  Morals,  throws  out  the  suggestion  that  Luke 
mutilated  the  first  edition  of  Mark  because  he  disagreed 
with  its  contents :  viz.,  an  account  of  apparitions  in  Galilee, 
whereas  he  expressly  limits  all  these  phenomena  to  Judea, 
by  making  Jesus  order  the  apostles  to  stay  in  Jerusalem 
until  Pentecost.  (Luke  xxiv.  49;  Acts  i.  4).  If  Luke 
mutilated  Mark,  then  why  not  go  further  and  say  that  he 
wrote  this  smooth-flowing  supplement  to  round  him  out? 
The  word  ODVTO^ICO^,  "in  few  words,"  is  never  found  in  the 
New  Testament  except  in  this  shorter  Mark  Appendix  and 
in  Luke's  Acts  of  the  Apostles  (xxiv.  4). 

'E^cr/YE^ro,  "to  show  forth,"  also  occurs  only  in  the 
Pauline  or  Lucan  (for  Luke  was  Paul's  secretary)  Epistle 
of  Peter  (i  Pet.  ii.  9).  'E|ajioateXXco,  "to  send  forth,"  is 
used  seven  times  in  Acts,  thrice  in  Luke's  Gospel,  and  once 


THE  TEXT  OF  THE  RESURRECTION  IN  MARK.  163 

by  his  master  Paul.  "Incorruptible"  occurs  only  in  Paul 
and  the  Pauline  i  Peter. 

Luke  represents  the  aristocratic  tradition  of  the  capital, 
which  said:  "It  all  happened  here!"  Mark  represents  the 
rural  tradition  of  Galilee,  which  said:  "Our  poor  parish 
was  the  scene  of  these  wonderful  things  !"So  the  young  man 
in  white,  in  Mark,  says  at  the  sepulcher:  Go,  tell  his  dis- 
ciples and  Peter:  Behold,  I  am  going  to  Galilee  ahead  of 
you.  There  shall  ye  see  me.  (Thus  read  some  of  our  best 
manuscripts,  in  the  first  person.) 

Another  thing:  Luke  and  John  both  make  the  appari- 
tions real.  In  these  later  Gospels  Jesus  is  objective  after  the 
Resurrection:  he  eats  broiled  fish  in  Luke,  while  in  John 
the  wounds  in  his  hands  and  side  are  felt  by  Thomas. 
Now  as  the  earliest  account  of  the  Resurrection  in  Paul 
(i  Cor.  xv.  4-8)  makes  the  event  a  series  of  apparitions, 
it  is  probable  that  the  second  earliest  account,  Mark's,  did 
the  same.  Indeed  in  Matthew  xxviii.  17  (under  suspicion 
of  being  taken  from  the  lost  ending  of  Mark),  "some 
doubted."  This  was  because  some  saw  the  figure  and 
others  did  not.  Luke  and  John  leave  no  room  for  doubt: 
the  evidence  is  sensuous,  not  subjective. 

The  first  Christian  heresy  was  Docetism,  the  belief  that 
Jesus  even  in  life  was  a  phantom.  His  flesh  and  blood  were 
unreal ;  he  did  not  really  suffer ;  his  bodily  functions  were 
different  from  human  ones  or  even  non-existent.  To  fight 
this  heresy  the  First  Epistle  of  John  was  written,  and  a 
curse  pronounced  upon  those  who  doubted  that  Jesus  had 
been  actual  flesh  and  blood  ( i  John  iv.  2,  3).  Consequently 
if  Mark  repeated  Paul's  impression  that  the  Galilean  ap- 
paritions were  the  same  in  kind  as  the  one  to  himself  on 
the  Damascus  road,  then  Mark  must  go.  Who  was  the 
likeliest  one  to  do  this  work  of  excision?  Answer:  Luke. 
He  was  the  most  literary  of  all  the  Evangelists.  He  is 


164  THE  MONIST. 

the  only  one  of  them  who  says  "I."  Moreover,  as  Harnack 
has  pointed  out,  he  betrays  an  animus  against  Mark,  ani- 
madverting upon  his  conduct  in  Acts  xv.  36-41.  In  his 
own  Gospel  Prologue,  Luke  is  undoubtedly  thinking  of 
him  as  one  of  the  "many"  who  have  "undertaken"  to  write 
the  life  of  Jesus,  but  who  have  not  begun  "accurately  from 
the  first"  nor  set  forth  "in  order"  the  sayings  and  events. 
Add  to  this  the  Jerusalem  tradition  of  the  Resurrection 
against  the  Galilean,  and  the  flesh-and-blood  appearances 
against  the  phantom  who  is  only  to  be  "seen"  ("there  shall 
ye  see  me,"  in  Mark),  and  we  have  motive  enough  for 
Luke's  high-handed  act. 

Indeed,  we  can  even  surmise  the  reason  why  he  made 
the  excision  in  the  middle  of  a  sentence.  He  would  hardly 
do  this  except  to  get  rid  of  an  offensive  word.  If  Mark 
had  read: 

They  said  nothing  to  any  one,  for  they  were  afraid  of 
the  apparition, 

this  last  word  would  have  been  the  red  rag.  There  must 
be  no  apparition:  there  must  be  objective  forms.  The 
young  man  in  white,  who,  in  several  MSS.,  speaks  in  the 
person  of  Jesus,  was  indeed  he  himself  in  his  glorified 
being.  Thus  do  I  read  the  texts.  Luke  too  had  read  some- 
thing of  this  kind,  which  he  reproduces  thus : 

But  they  were  terrified  and  affrighted  and  supposed 
that  they  beheld  a  spirit.  ( Luke  xxiv.  37.  The  Cambridge 
MS.  and  Marcion's  edition  of  Luke  both  read  "apparition" 
instead  of  "spirit") 

Let  it  be  understood  that  I  do  not  deny  the  possibility 
of  ectoplastic  phantoms,  which  Myers  himself  believed  in, 
though  he  said  he  would  not  press  them  upon  the  credence 
of  the  reader,  because  of  the  difficulty  of  correct  observa- 
tion and  the  chances  of  fraud.  Dr.  Reichel  of  Germany 
has  testified  to  their  occurrence  here  in  America.  The 
difficulty  in  the  New  Testament  is  that  they  only  appear  in 


THE  TEXT  OF  THE  RESURRECTION  IN  MARK.  1 65 

the  later  accounts.  Paul  and  (I  shall  show  presently) 
Mark,  our  earliest  witnesses,  know  of  apparitions  alone, 
not  of  materialized  forms. 

For  the  fullest  account  in  English  of  all  the  problems 
the  reader  should  consult  The  Resurrection  in  the  New 
Testament,  by  Clayton  R.  Bowen,  of  Meadville,  Pennsyl- 
vania (New  York,  1911).  Professor  Bowen  is  one  of  a 
long  series  of  laymen  and  liberals,  like  Griesbach,  Lach- 
mann,  Tischendorf  and  Tregelles,  who  have  taken  the  New 
Testament  out  of  clerical  hands.  The  three  German  lay 
professors  and  Tregelles,  the  English  Quaker,  were  the 
ones  whose  work  led  directly  to  the  Revised  Version  of 
1 88 1 ;  but  the  task  of  revision  is  by  no  means  ended  yet. 

Bowen  was  a  Unitarian  minister,  but  is  now  professor 
at  Meadville.  Before  reading  him,  a  shorter  and  clearer 
book  by  Kirsopp  Lake  should  first  be  mastered. 


Kirsopp  Lake,  of  Harvard  University,  published  in 
1907  The  Historical  Evidence  for  the  Resurrection  of 
Jesus  Christ  (London,  Williams  and  Norgate).  Professor 
Lake  at  that  time  held  the  chair  of  New  Testament  Exege- 
sis in  the  University  of  Leiden,  to  which  Rendel  Harris 
was  elected  in  1903,  but  did  not  serve.  The  book  appeared 
in  the  Crown  Theological  Library  and  has  been  widely 
read.  It  contains  a  masterly  analysis  of  the  Resurrection 
narratives  in  i  Corinthians,  the  Synoptical  Gospels,  the 
Acts  of  the  Apostles,  the  Mark  appendices,  the  Fourth 
Gospel  and  the  apocryphal  ones  of  Peter  and  the  Hebrews. 
The  conclusion  reached  is  that  Paul  and  Mark's  accounts 
are  historical,  and  the  later  ones  exaggerated.  Babylonian 
and  other  resurrection  theories  are  reviewed,  and  the  book 
ends  with  an  allusion  to  Myers  and  psychical  research. 
F.  C.  Burkitt,  of  Cambridge,  in  placing  the  essay  in  a 


l66  THE  MONIST. 

bibliography,  says :  "I  introduce  this  book  here  as  the  first 
example  in  original  English  work  of  the  doctrine  of  the 
priority  of  Mark  being  consistently  applied  throughout  an 
historical  investigation."  (The  Earliest  Sources  for  the 
Life  of  Jesus,  Boston,  1910,  p.  129). 

The  method  is  that  of  the  Lower  Criticism,  though  the 
Higher  is  also  freely  used.  What  I  especially  wish  to 
criticize  is  the  following  passage  (pp.  61-65)  which  here 
we  must  read  in  full: 

"The  young  man  at  the  tomb. — The  account  of  what 
the  women  saw  at  the  tomb  is  contained  in  Mark  xvi.  5. 
Dependent  narratives  are  found  in  Matthew  xxviii.  2-5 
and  in  Luke  xxiv.  3-5. 

"And  entering  into  the  tomb,  they  saw  a  young  man 
sitting  on  the  right  side,  clothed  in  a  white  garment;  and 
they  were  astonished. 

"As  it  stands  in  Mark,  this  account  gives  rise  at  once 
to  two  questions:  Did  they  see  for  themselves  that  the 
grave  was  empty?  and  who  was  the  young  man  who  ap- 
peared to  them?  Neither  question  is  answered  in  Mark, 
but  before  considering  the  bearing  of  this  fact,  it  is  first 
necessary  to  ask  whether  the  version  given  above  repre- 
sents the  original  text.  According  to  it,  the  women  entered 
the  tomb  and  found  a  young  man  seated  within  on  the 
right  hand.  No  other  meaning  can  be  extracted  from  it, 
or  ever  could  have  been,  in  the  presence  of  the  word  eiaeX- 
ftoijaai,  'entering  into,'  in  verse  5  and  the  reference  con- 
tained in  the  corresponding  e|eA$oi)aai,  'going  out/  in 
verse  8.  But  in  case  of  neither  of  these  words  is  the  text 
perfectly  certain.  The  former  is  in  the  Vatican  MS.  weak- 
ened to  eAftovaai,  'coming/  while  the  latter  is  not  repre- 
sented in  the  Arabic  Diatessaron,  and  in  some  MSS.  is 
altered  to  axowavrec;,  'having  heard/  The  weight  of  tex- 
tual evidence  is  against  these  alterations,  but  on  the  other 
hand  transcriptional  probability  is  in  their  favor.  It  is 


THE  TEXT  OF  THE  RESURRECTION  IN  MARK.  167 

unlikely  that  later  scribes  would  have  introduced  changes 
in  the  text  which  were  calculated  to  weaken  the  evidence 
for  the  belief  that  the  women  had  made  a  complete  exam- 
ination of  the  tomb,  and  if  these  changes  be  made,  the  text 
of  Mark  would  leave  it  doubtful  whether  the  women  saw 
the  young  man  on  the  right  hand  of  the  inside  or  of  the 
outside  of  the  tomb;  for  eA-ftouaai  eig  TO  [wjneiov  need  not 
mean  more  than  'when  they  came  to  the  tomb.'  Is  it  pos- 
sible that  this  represents  the  original  form  of  the  narrative  ? 
In  the  absence  of  other  evidence,  it  may  not  be  ill-advised 
to  consider  the  evidence  of  a  comparison  with  the  two  other 
gospels,  Matthew  and  Luke,  which  are  closely  based  on  the 
Marcan  narrative,  and  of  the  Fourth  Gospel  and  the  Gos- 
pel of  Peter,  which  follow  it  with  greater  freedom.  It  has 
already  been  seen,  in  cases  in  which  the  Marcan  document 
is  undoubtedly  ambiguous  or  difficult,  that  the  dependent 
narratives  adopted  divergent  methods  of  elucidating  the 
points  at  issue.  It  may  therefore  be  allowed  to  reverse 
this  argument  and  see  whether  the  dependent  narratives 
in  the  present  case  support  the  suggestion  that  the  ground 
document  was  ambiguous.  They  certainly  seem  to  do  so. 
Matthew  represents  the  angel,  who  is  in  his  narrative  the 
equivalent  of  the  young  man  of  Mark,  as  seated  on  the 
stone  which  he  had  just  rolled  away;  he  was  therefore 
regarded  by  Matthew  as  outside  the  tomb.  It  is  equally 
plain  that  Luke  regards  the  two  men,  who  in  his  narrative 
represent  the  Marcan  young  man,  as  appearing  within  the 
tomb.  Furthermore,  the  Fourth  Gospel  and  the  Gospel  of 
Peter  narrate  that  the  women  did  not  enter  the  tomb,  but 
stooped  down  and  saw  an  angel  or  angels  sitting  within. 
These  two  last  accounts  may  quite  well  represent  an  attempt 
at  conflation  between  two  traditions  which  differed,  or 
were  not  explicit,  as  to  the  position  of  the  women  and  the 
angel  with  regard  to  the  tomb,  and  so  far  they  support  the 
suggestion,  which  is  rather  strongly  made  by  Matthew  and 


1 68  THE  MONIST. 

Luke,  that  the  ground  document  was  ambiguous  on  this 
point.  The  weak  point  in  this  argument  is  that  it  does 
not  take  account  of  the  possibility  that  Matthew  altered 
the  Marcan  document  owing  to  the  influence  of  the  story 
of  the  watchers.  It  could  be  argued  that  the  angel  had 
to  be  kept  in  the  presence  of  the  watchers  and  of  the  women, 
and  that  the  word  cbreXftovacci,  'going  from,'  in  verse  8  is 
a  proof  that  the  ground  document  of  Matthew  contained 
an  account  of  an  actual  entry  into  the  tomb.  This  is  per- 
haps not  a  convincing  argument,  but  it  may  be  taken  as 
practically  balancing  the  previous  one.  It  is  impossible 
finally  to  decide  between  the  two.  I  think  that  the  balance 
of  probability  remains  slightly  in  favor  of  the  view  that 
the  original  Marcan  document  narrated  the  story  of  the 
vision  at  the  tomb  in  such  a  way,  as  not  to  state  plainly 
that  the  women  entered  the  tomb,  but  I  should  not  be  pre- 
pared to  put  emphasis  on  the  argument." 

I  hope  to  show  that  there  is  every  reason  for  Professor 
Lake  to  emphasize  the  argument  that  the  original  text  of 
Mark  did  actually  keep  the  women  outside  the  tomb.  We 
may  say  does  actually,  for  the  original  text  of  Mark  can  be 
reconstructed  from  extant  manuscripts  and  versions,  with- 
out any  appeal  to  the  Higher  Criticism.  In  one  case  only  do 
we  have  to  appeal  to  a  lost  source,  but  even  this  is  supported 
by  a  patristic  quotation,  and  therefore  belongs  to  the  Lower 
Criticism. 

Let  us  begin  with  this  lost  source.  Eusebius,  in  his 
Questions  of  Marinus,  Question  i,  which  deals  with  the 
absence  of  the  Mark  Appendix  (Mark  xvi.  9-20)  from  the 
oldest  manuscripts,  says  this: 

"He  who  rejects  the  passage  itself  might  say  that  the 
story  does  not  exist  in  all  the  copies  of  the  Gospel  according 
to  Mark;  at  least,  the  accurate  ones  among  the  copies  de- 
scribe the  end  of  the  story  according  to  Mark  in  the  words 


THE  TEXT  OF  THE  RESURRECTION  IN  MARK.  169 

of  the  youth  who  appears  to  the  women,  saying  to  them: 
'Be  not  astonished;  ye  seek  Jesus  the  Nazarene,'  and  so 
forth.  'And  when  they  heard,  they  fled,  and  said  nothing 
to  anyone,  for  they  were  afraid  of.  . .  .'  For  herein  the 
end  is  described  in  nearly  all  the  copies  of  the  Gospel  ac- 
cording to  Mark,  and  what  follows  is  seldom  found  in 
any,  but  would  not  be  superfluous  in  all,  and  especially  if 
they  should  contain  a  contradiction  to  the  witness  of  the 
rest  of  the  Evangelists." 

We  may  remark  that  "afraid  of"  is  Kirsopp  Lake's 
own  translation  of  the  concluding  words  of  the  genuine 
text  of  Mark,  and  it  has  been  adopted  by  James  Moffatt  in 
his  splendid  translation  of  the  New  Testament  (London, 
1913).  But  the  words  for  which  we  have  copied  this 
famous  passage  of  Eusebius  are :  "when  they  heard"  (dxoi)- 
aaaai).  Now  it  is  known  that  Eusebius  had  access  to  the 
library  collected  by  Origen  in  the  third  century  and  ex- 
tended by  Pamphilus.  Indeed  Conybeare  has  made  use  of 
this  fact  to  delete  the  trinitarian  formula  and  the  baptismal 
charge  at  the  end  of  the  Gospel  of  Matthew,  in  the  teeth 
of  all  existing  manuscripts.  He  shows  that  Eusebius  read 
Matthew  xxviii.  18-20  without  these  theological  additions, 
and  places  over  against  three  thousand  extant  MSS.,  all 
later  than  the  fourth  century,  that  other  thousand,  now 
lost,  which  went  back  to  the  third  and  the  second. 

Applying  this  principle  we  can  put  in  the  forefront  of 
our  textual  evidence  for  dxovaccaai  instead  of  eleXftovaai 
the  whole  weight  of  the  earliest  Christian  manuscripts. 
The  ungrammatical  dxo-uaavreg  quoted  by  Lake  is  from  a 
medieval  manuscript  in  Russia,  numbered  565  by  Caspar 
Rene  Gregory  in  his  Prolegomena  to  Tischendorf's  Greek 
Testament.  Of  course  Eusebius  gives  the  right  reading, 
dxovaaacu  (feminine).  Rallying  to  the  support  of  this 
ancient  Greek  original  are  the  Washington  manuscript 
and  the  Old  Syriac  and  Old  Armenian  versions,  overlooked 


I/O  THE  MONIST. 

by  Lake.  Their  testimony  is  very  important;  especially 
the  Armenian,  for  the  Old  Syriac  and  the  Washington 
Greek  betray  a  transition  stage  which  was  tautological. 
The  "went  out"  was  evidently  interpolated  before  the  de- 
letion of  the  "having  heard." 

The  following  table  will  give  a  view  of  the  process  of 
corruption.  As  Eusebius  expressly  tells  us  that  the  most 
accurate  MSS.  omitted  the  Mark  appendix,  we  need  only 
deal  with  those  that  do  so.  This  gives  us  a  sure  criterion. 
Six  MSS.  that  omit  this  can  therefore  be  pitted  against 
6000  that  add  it.  To  the  trustworthy  ones  we  may  add 
those  which  contain  the  spurious  matter  with  a  caveat,  also 
those  which  have  a  different  ending  from  the  current  ap- 
pendix. To  these  also  must  be  added  a  few  MSS.  that 
contain  attestations  of  careful  copying  from  Jerusalem 
copies,  such  as  No.  565. 

Lost  MSS.  of  the  Early  Centuries  quoted  by  Eusebius. 


[First  clause  not  traced.] 

And  when  they  heard  they  fled  and  said  nothing  to  any 

one,  for  they  were  afraid  of 

(End  of  Mark.) 


Armenian  Version. 

And  entering  into  the  sepulcher. 

*  *       * 

And  when  they  heard,  they  fled  from  the  sepulcher, 
because  they  were  terrified;  and  they  said  nothing  to  any 
one  for  they  were  afraid. 

Gospel  according  to  Mark.1 

*  *        # 
Introduction  to  Luke. 

1  The  colophons  here  printed  in  bold-faced  type  are  rubricated  in  the 
original. 


THE  TEXT  OF  THE  RESURRECTION  IN  MARK. 

Frank  Normart,  of  Glenolden,  Pennsylvania,  but  a 
native  of  Erzerum,  has  translated  for  me  the  passage  from 
the  Old  Armenian,  as  found  in  his  own  printed  edition  ( Con- 
stantinople, 1895)  and  in  a  valuable  manuscript  owned  by 
John  P.  Peters  (Bedrosian)  of  Philadelphia.  (The  colo- 
phon is  from  the  manuscript,  for  the  Bible  Society  has 
printed  the  Appendix,  as  in  the  King  James  version,  with 
a  note  accusing  the  Greeks  for  omitting  it,  but  carefully 
suppressing  the  fact  that  nearly  all  Armenian  MSS.  before 
A.  D.  noo  omit  it  also.) 

Both  in  the  Syriac  and  the  Armenian  this  colophon  is 
rubricated. 


Washington  MS. 
And  entering  into  the  sepulcher 

*       *       * 

And  when  they  heard,  they  went  out  and  fted  from  the 
sepulcher,  for  fear  and  astonishment  had  come  upon  them, 
and  they  said  nothing  to  any  one,  for  they  were  afraid  of. 
Now  when  he  was  risen  early,  on  the  first  day  of  the  week, 
he  appeared  to  Mary  Magdalene  etc. 

[This  is  the  earliest  MS.  that  contains  the  Appen- 
dix, which  it  has  in  an  unusual  form,  hitherto  only 
partially  known  from  a  fragment  in  Jerome.] 


Old  Syriac. 
And  they  entered  into  the  sepulcher 

And  when  they  heard,  they  came  forth  and  went  away 
and  said  nothing  to  any  one,  for  they  had  been  afraid. 

ENDETH  GOSPEL  OF  MARK. 


172  THE  MONIST. 

The  South  Coptic 
(Sahidic  or  Thebaic.) 

[First  clause  wanting.] 

*       *       * 

And  when  they  had  heard,  they  came  out  of  the  sepulcher, 
and  they  ran,  for  a  trembling  was  laying  hold  on  them, 
and  a  confusion;  and  they  said  not  any  word  to  any  one, 
for  they  were  fearing.  But  all  the  things  which  were 
ordered  them,  to  those  who  followed  Peter  they  said  them 
openly.  After  these  things  also  again  Jesus  was  manifested 
to  them  from  the  place  of  rising  of  the  sun  unto  the  place 
of  setting.  He  sent  through  them  the  preaching  which  is 
holy  and  incorruptible  of  the  eternal  salvation.  Amen. 

But  these  also  belong  to  them. 

[Then  follows  the  Longer  Appendix,  after  a  repetition 
of  the  words  at  the  juncture.] 


The  Vatican  MS. 

And  coming  unto  the  sepulcher 

*       *       * 

And  they  went  out  and  fled  from  the  sepulcher,  for  trem- 
bling and  astonishment  had  come  upon  them ;  and  they  said 
nothing  to  any  one,  for  they  were  afraid  of 

ACCORDING  TO  MARK. 


The  Sinaitic  MS. 

And  entering  into  the  sepulcher 

*       *       * 

And  they  went  out  and  fled  from  the  sepulcher,  for  trem- 
bling and  astonishment  had  come  upon  them ;  and  they  said 

nothing  to  any  one,  for  they  were  afraid  of 

Gospel  according  to  Mark. 


THE  TEXT  OF  THE  RESURRECTION  IN  MARK.  173 

The  Old  Latin  at  Turin. 

And  when  they  had  entered,  they  saw  a  youth  etc. 

*       #       * 

But  when  they  went  out  from  the  sepulcher,  they  fled;  for 
trembling  held  them,  and  awe  by  reason  of  fear. 

But  all  things  whatsoever  that  were  commanded,  those 
also  who  were  with  the  boy  briefly  explained.  And  after 
these  things  Jesu  himself  appeared,  and  from  the  East 
even  unto  the  East  (sic)  he  sent  through  them  the  holy 
and  uncorrupted  [preaching]  of  eternal  salvation.  Amen. 

Endeth  Gospel  according  to  Mark.  Beginneth  happily 
according  to  Matthew. 


The  Ethiopic  version  also  omits  the  Mark  appendix, 
while  medieval  MSS.  L,  and  Nos.  I  and  209  show  the  doubt 
about  it;  L  gives  both  endings,  like  the  South  Coptic, 
putting  the  shorter  appendix  first  as  we  have  seen  already. 
Nos.  i  and  209  say  at  xvi.  8 : 

"In  some  copies,  the  Evangelist  ends  here,  as  far  as 
Eusebius  the  [friend]  of  Pamphilus,  has  placed  his  canons; 
in  others,  there  are  found  also  these  [words] : 

"NOW  WHEN  HE  WAS  RISEN/'  etc. 


Several  Fathers  support  the  testimony  of  Eusebius, 
so  that  the  proof  is  overwhelming. 

By  restoring  "when  they  heard"  at  xvi.  8,  we  get  rid 
of  the  eleAftooxrai,  but  this  was  introduced  as  correlative 
to  eiaeTiftovaoci :  the  two  stand  or  fall  together.  If  we  had 
not  a  single  manuscript  that  read  eMtowai  at  xvi.  5,  the 
Higher  Criticism  would  bid  us  read  it.  But  our  very 
oldest  Greek  authority  reads  it,  plus  an  eleventh-century 
MS  numbered  127,  together  with  the  fourth-century  Gothic 


174  THE  MONIST. 

version.  This  reads  atgaggandeins,  "coming  at,"  or.  "going 
unto."  Lower  Criticism  therefore  permits  the  restoration. 

So  we  have  unimpeachable  ancient  testimony  that  there 
was  no  e|eA,dxyuaat  at  Mark  xvi.  8 :  the  women  did  not  flee 
out  of  the  tomb,  because  they  had  never  been  in  it. 

It  is  vain  to  protest  that  we  cannot  put  three  authorities 
against  three  thousand  that  read  eiaeA.fto'uaai,  "entering 
into";  for,  by  the  laws  of  the  Lower  Criticism,  we  can, 
not  only  on  the  grounds  already  given,  but  by  a  well-known 
law  of  textual  criticism.  Dean  Alford,  in  the  critical  ap- 
paratus to  his  Greek  Testament,  gives  us  the  reason: 

"Received  text,  eioEA,ftovaai,  from  the  parallel  in  Luke." 
(Henry  Alford,  Greek  Testament:  New  York,  1859,  Vol. 

I,  P-  391-) 

Nay,  more :  the  Dean  of  Canterbury  Cathedral  did  not 
hesitate  to  put  eMhwaai  into  his  Greek  text  and  to  translate 
it  in  his  New  Testament  for  English  Readers  (London, 
1868,  Vol.  I): 

"And  when  they  came  to  the  sepulcher,"  etc. 

The  principle  upon  which  Alford  did  this  is  perfectly 
sound.  It  is  thus  expressed  by  Jerome  in  his  letter  to  Pope 
Damasus,  introducing  his  novel  Vulgate  edition  of  the 
Gospels  in  the  year  384 : 

"Great  error,  if  indeed  it  be  (so),  has  grown  up  in  our 
codices,  so  long  as  what  one  Evangelist  has  said  further 
on  the  same  thing,  (the  scribes),  have  added  in  another 
because  they  thought  it  too  little.  Or,  so  long  as  another 
expressed  otherwise  the  same  meaning,  he  who  had  read 
first  any  one  of  the  Four,  considered  that  the  rest  ought 
also  to  be  amended  to  the  pattern  of  that  one.  Whence  it 
happens  that  among  us  everything  is  mixed,  and  there  are 
found  in  Mark  more  things  of  Luke  and  of  Matthew,  and 
again  in  Matthew  more  things  of  John  and  of  Mark,  and  in 
others  of  the  rest  things  which  are  peculiar  to  others." 

Here  we  have  the  Protestant  reason,  stated    by  the 


THE  TEXT  OF  THE  RESURRECTION  IN  MARK.  175 

prince  of  Catholics,  for  modern  revision  of  the  text.  Al- 
though we  have  not  access  to  so  many  ancient  manuscripts 
as  Jerome  had,  yet  we  have  ancient  versions  neglected  by 
him,  as  well  as  two  Greek  codices  of  his  own  time  and 
several  Latin  ones.  We  are  more  justified  in  ceasing  to 
regard  his  work  as  final  because  he  tells  us  himself  that 
he  did  not  do  it  thoroughly: 

"This  short  preface  offers  only  the  four  gospels,  the 
order  of  which  is  as  follows :  Mathew,  Mark,  Luke,  John, 
amended  by  a  collation  of  the  Greek  codices,  but  (only)  of 
old  ones.  Lest,  however,  they  should  differ  much  from  the 
accustomed  form  of  the  Latin  reading,  we  have  so  re- 
strained the  pen  that,  when  such  things  only  were  corrected 
as  seemed  to  change  the  sense,  we  suffered  the  rest  to 
remain  as  they  were." 

We  are  now  in  a  position  to  reconstruct  the  end  of  the 
Gospel  of  Mark,  and  to  show  that  this  most  historic  of  all 
the  Evangelists  never  told  a  story  about  a  corpse  that  got 
up  and  walked  off,  but  simply  of  some  women  who  came 
to  a  tomb  and  saw  a  strange  young  man.  When  they  saw 
him  they  were  astonished,  but  when  he  addressed  them  they 
were  terrified  and  ran  away.  At  this  point  the  Gospel  ends, 
as  we  now  have  it.  The  reason  for  this  abrupt  ending  requires 
a  separate  discussion,  such  as  briefly  outlined  in  our  pro- 
logue or  such  as  I  attempted  in  The  Open  Court  at  Easter, 
1910.  Unfortunately  I  had  not  then  read  Kirsopp  Lake,  or 
my  attempt  would  have  been  better.  (At  the  top  of  page 
133  I  made  a  blunder;  line  i  should  read:  "Now  this  note 
of  doubt  is  Marcan,  not  Matthaean,"  etc.) 

In  giving  the  following  text,  several  readings  differing 
from  the  common  ones  have  also  been  given  in  addition 
to  those  noted.  Thus  the  phrase  "on  their  right"  in  verse 
5  is  from  the  Sinai  Syriac,  and  is  all  part  of  the  reformed 
readings,  for  it  no  longer  smacks  of  the  inside  of  the  grave 
where  a  youth  or  an  angel  was  sitting  on  the  right  side  of 


176  THE   MONIST. 

the  corpse.  The  word  "daughter"  is  also  from  the  Old 
Syriac.  The  Greek  has  a  blank  here,  which  our  common 
translations  supply  by  "mother."  But  a  second-century 
version  in  the  language  which  Jesus  spoke  may  be  pre- 
sumed to  go  back  to  authentic  tradition. 

Our  reconstruction  is  guided  by  the  Lower  Criticism 
alone.  If  we  were  to  venture  upon  the  dangerous  ground 
of  the  Higher,  I  should  strike  out  the  words :  "He  is  risen ; 
he  is  not  here."  These  are  lacking  in  important  MSS.  of 
Luke,  and  as  Luke  used  the  first  edition  of  Mark,  he  prob- 
ably did  not  find  them  therein.  But,  as  they  appear  in  all 
extant  MSS.  of  Mark,  the  principles  of  the  Lower  Criti- 
cism require  that  they  should  be  retained.  Higher  Criti- 
cism would  also  query  the  historicity  of  the  spices  and  oint- 
ment. Matthew  says  nothing  about  them,  but  tells  us  that 
the  women  came  "to  see  the  tomb,"  and  had  no  errand 
inside.  John  expressly  rules  out  the  proceeding  by  an 
elaborate  embalming  before  burial.  If  Luke's  eiaeMhwaai 
could  be  copied  into  Mark,  as  Alford  following  the  state- 
ment of  Jerome  and  the  abundant  witness  of  the  manu- 
scripts would  have  us  admit,  why  should  not  Luke's  oint- 
ment and  spices  also  have  found  their  way  into  Mark  in 
very  early  times?  But  here  again  we  are  faithful  to  the 
Lower  Chriticism  and  insert  the  spices  and  ointment. 

Another  point.  When  Dr.  Lake  says:  "eTiftouaai  els 
TO  |j,VTi(i8iov  need  not  mean  more  than  'when  they  came  to 
the  tomb,'  "  does  he  not  understate  the  case?  Can  we  not 
confidently  say  that  they  do  not  mean  more?  Thayer,  in 
his  lexicon,  long  since  pointed  out  that  elg  to  |ivr]|ieiov  in 
John  xx,  the  parallel  passage  to  the  one  in  Mark,  means 
simply  "unto  the  tomb."  In  verse  I,  the  Revised  Version 
renders  the  phrase :  "unto  the  tomb" ;  in  verse  3,  "toward 
the  tomb" ;  in  verses  4  and  8  "to  the  tomb."  So  too  in  John 
xi.  31  and  38.  In  New  Testament  Greek,  therefore,  els  TO 
means  "unto  the  tomb,"  and  in  order  to  introduce 


THE  TEXT  OF  THE  RESURRECTION  IN  MARK.  177 

the  idea  of  entrance,  Luke  and  the  copyists  of  Mark  had 
to  alter  eXftouaai  to  eiaeXfto'uaai.  Then,  having  gotten  the 
women  into  the  tomb,  they  must  be  gotten  out  again ;  hence 
the  correlative  corruption  of  axovaaacti  ecpvyov  mto  £§eX- 
dovaai  eqwyov,  which  in  the  Sinai  Syriac  is  even  more 
tautological : 

they  went  out  and  went. 

This  text  is  a  conflation,  for  it  has  already  given  the 
original  reading  known  to  Eusebius  in  his  Csesarean  manu- 
scripts : 

And  when  they  heard. 

In  fact,  a  close  study  of  the  documents  reveals  the  fact  that 
the  whole  passage  has  been  systematically  tampered  with. 

Mark  xvi  entire,  as  in  the  Oldest  Manuscripts. 
Revised  Text. 


230  And  when  the  sabbath  was  past,  Mary  Magda- 
VIII  lene  and  Mary  the  daughter  of  James,  and  Salome, 

brought  spices,  that  they  might  come  and  anoint  him. 

231  And  very  early  in  the  morning,  the  first  day  of 
I    the  week,  they  came  unto  the  sepulcher  at  the  rising 

of  the  sun.  And  they  said  among  themselves:  Who 
shall  roll  us  away  the  sepulcher  stone?  (for  it  was 
exceeding  great).  And  when  they  looked  they  saw 
that  the  stone  was  rolled  away.  And  coming  unto 
the  sepulcher,  they  saw  a  young  man  sitting  on  their 
right,  clothed  in  a  white  robe;  and  they  were  bewil- 
dered. 

232  And  he  saith  unto  them:  Be  not  bewildered;  ye 
II    seek  Jesus  the  Nazarene,  who  was  crucified.     [He  is 

risen;  he  is  not  here.]  Behold,  there  is  his  place  where 
they  have  laid  him.  But  go  your  way,  tell  his  dis- 


178  THE  MONIST. 

ciples  and  Peter :  I  am  going  to  Galilee  ahead  of  you. 
There  shall  ye  see  me,  as  I  have  said  unto  you. 

233           And  when  they  heard,  they  fled,  and  said  nothing 
II    to  any  one,  for  they  were  afraid  of 

HERE  ENDETH  THE  GOSPEL  OF  MARK. 


The  numbered  paragraphs  are  known  as  the  Ammonian 
Sections  and  appear  in  the  Sinaitic  manuscript  and  down 
through  the  Middle  Ages.  Underneath  the  Hindu  numeral 
is  the  canon  ascribed  to  Eusebius,  which  is  numbered  in 
Roman.  These  canons  represent  an  ancient  Gospel  anal- 
ysis of  no  mean  ability:  Canon  I  means  that  the  section  is 
common  to  all  four  Gospels;  II,  to  the  three  Synoptists; 
VIII,  to  Mark  and  Luke;  X,  peculiar  to  one;  the  rest  to 
different  pairs. 

RECAPITULATION. 

From  early  manuscripts,  versions  and  Fathers  we  can 
reconstruct  the  text  of  the  Resurrection  in  Mark,  so  as  to 
read,  at  xvi.  5,  "coming  unto  the  sepulcher,"  instead  of 
"entering  into"  it.  Comparison  of  these  authorities  reveals 
the  fact  that  the  text  has  been  systematically  corrupted  in 
early  times  by  altering  "coming  unto"  to  "entering  into," 
and  "when  they  heard,"  in  verse  8,  to  "going  out  of."  This 
has  been  done  so  as  to  make  it  appear  that  the  women 
found  the  grave  empty,  implying  the  doctrine  of  a  fleshly 
resurrection.  By  the  Lower  Criticism  alone  we  are  en- 
abled to  correct  these  corruptions  and  to  show  that  Mark 
originally  contained  no  account  of  a  physical  resurrection. 
The  familiar  problem  of  the  lost  ending  of  Mark  is  inci- 
dentally dealt  with. 

ALBERT  J.  EDMUNDS. 

PHILADELPHIA,  PA. 


BERGSONISM  IN  ENGLAND. 

MBERGSON  has  told  us  that  on  the  arena  of  Europe 
•  to-day  we  have  a  spectacle  of  "life"  in  arms  against 
"matter."  He  takes  the  two  terms  to  express  respectively 
the  spirit  of  his  own  nation  and  that  of  the  enemy.  We 
may  take  them  as  expressing  at  least  the  alternatives  be- 
tween which  his  own  philosophy  moves.  On  the  arena  of 
the  universe  as  a  whole  life  and  matter  are  in  conflict  and 
his  philosophy  seeks  to  decide  between  the  two.  Which  is 
the  ultimate  reality?  Which  is  to  be  explained  by  the 
other?  Is  life  a  product  of  matter  or  is  the  truth  the  other 
way,  has  matter  itself  been  created  by  life?  Bergson's 
works  may  be  considered  to  constitute  an  elaborate  answer 
to  this  question,  and  to  decide  it  in  favor  of  the  latter  alter- 
native. He  thus  takes  rank  as  a  champion  of  the  living 
against  the  dead.  But  the  main  position  against  which  he 
argues — what  one  could  call  roughly  scientific  naturalism 
— is  one  with  which  philosophy  in  this  country  has  long 
been  accustomed  to  deal,  and  with  a  like  result.  Now  that 
Bergson's  philosophy  may  be  said  to  have  struck  its  first 
roots  in  this  country,  the  time  seems  opportune  for  a  com- 
parison. We  might  with  profit  compare  what  he  has  to  say 
with  what  the  philosophy  of  the  past  two  generations  has 
been  teaching.  From  some  points  of  view  it  seems  as 
though  his  special  followers  did  not  realize  sufficiently  the 
likeness  of  the  two  ways  of  thinking.  Nor  do  they  seem 
to  have  seen  as  yet  how  much  some  of  Bergson's  most 


ISO  THE  MONIST. 

valued  positions  seem  to  invite  and  indeed  to  demand 
strengthening  by  reference  to  matters  which  have  been 
developed  at  least  partially  by  the  older  idealism,  and  which 
are  absent  from  him. 

The  opportunity  for  making  the  comparison  is  ready 
to  our  hand.  As  is  well  known,  the  head  and  front  of 
Bergson's  English  following  has  come  to  be  pretty  much 
identified  with  Dr.  H.  Wildon  Carr.  While  his  work,  The 
Philosophy  of  Change,1  shows  the  influence  of  Bergson  at 
every  turn,  and  while  Dr.  Carr  does  not  conceal  his  convic- 
tion that  Bergson's  is  the  most  successful  attempt  yet  made 
to  deal  with  the  questions  of  metaphysics,  still  it  would  be 
wrong  to  say  that  in  this  work  he  has  merely  been  expound- 
ing another  man's  thought.  He  makes  the  endeavor  to 
restate  Bergson's  central  principle  and  to  substantiate  it 
by  fresh  applications ;  and  a  glance  at  the  main  topics  will 
serve,  we  believe,  to  give  a  fairly  true  view  of  one  important 
feature  of  Bergson's  thinking  which  it  is  necessary  to  take 
account  of,  if  we  would  bring  its  relation  to  English 
thought  properly  into  focus.  It  will  show  us  the  kind  of 
questions  which  figure  in  Bergson's  pages. 

The  recurring  topic  is  matter  and  spirit  and  the  prob- 
lems which  arise  out  of  their  relation.  For  instance,  is 
mind  produced  by  the  brain?  If  so,  how  could  mind  ever 
get  at  what  is  outside  of  this  brain?  for  apparently  mind 
can. reach  not  only  what  is  outside  of  it  but  what  is  even 
separated  from  it  by  immeasurable  gulfs  of  space  and  time. 
Or,  what  are  we  to  make  of  the  question  as  to  the  consti- 
tution of  matter,  now  that  recent  discoveries  seem  to  re- 
solve the  material  atom  into  something  that  is  not  a  sub- 
stance at  all  but  only  so  much  electricity  ?  Then  there  are 
the  problems  connected  with  the  relation  of  conscious  mind 
to  movement.  Why  is  it  that  a  higher  form  of  conscious- 
ness, in  the  animal  world,  is  so  constantly  accompanied  by 

1  Macmillan  &  Co.,  1914. 


BERGSONISM  IN  ENGLAND.  l8l 

an  increased  capability  on  the  animal's  part  of  choosing  its 
movements?  There  is  further  the  nature  of  life  itself. 
How  are  we  to  construe  the  fact  that  the  whole  past  of 
every  living  being  appears  to  be  recorded  in  its  present 
structure?  And  again  there  is  the  law  of  evolution  and 
its  ways  of  working.  How  is  it  that  in  a  material  world 
supposed  to  be  governed  by  mechanical  law,  evolution  can 
bend  and  govern  the  most  dissimilar  series  of  conditions 
so  as  to  produce  a  like  result — as  when  (to  cull  an  illustra- 
tion from  Bergson  himself)  the  series  of  conditions  which 
produces  a  mollusk  and  the  very  different  series  which 
produces  a  vertebrate  animal  should  both  alike  end  in  the 
one  result,  the  endowment  of  the  creature  with  an  eye? 

An  outstanding  characteristic  of  the  whole  way  of 
thought  will  be  apparent  from  this  cursory  survey.     Its 
leading  questions  are  such  as  would  arise  in  the  course  of 
the  study  of  natural  science.     They  are  questions  which 
would  occur  to  a  scientist  with  philosophical  interests.    No 
doubt  the  motive  which  impels  the  mind  to  raise  them  lies 
far  back  in  the  perennial  human  sources  of  the  philoso- 
phizing habit.  But  the  actual  questions  raised  come  straight 
out  of  modern  physics,  mathematics,  biology  and  psychol- 
ogy.   They  don't  arise  out  of  a  study  of  the  history  of  phil- 
osophical disputes.    This  is  characteristic  of  Bergson.    He 
is  preeminently"  one  of  the  writers  who  attack  problems, 
not  other  people's  solutions  of  problems.    Hence  his  philo- 
sophical freshness.    He  does  indeed  deal  with  the  history 
of  thought.    He  deals  also  with  some  of  the  standing  con- 
troversies of  current  philosophy.     But  these  are  not  the 
center  of  his  interest.    He  does  not  begin  with  these.    His 
speculation  thus  acquires  an  interest  for  scientific  minds 
which  most  philosophy  does  not  possess.    And  it  is  perhaps 
not  altogether  fanciful  to  say  that  this  feature  gives  his 
thought  from  the  outset  a  certain  advantage  with  the  Eng- 
lish mind.    Moreover,  in  Dr.  Carr  Bergson  seems  to  have 


l82  THE  MONIST. 

found  a  follower  whose  interests  also  are  preeminently  in 
the  concrete  present.  The  central  matter  with  him  seems 
to  be,  not  what  can  we  make  of  the  systems  of  the  past,  but 
rather  what  can  we  make  of  the  report  which  the  sciences 
give  us  now  of  the  world  we  are  in;  what  is  important  is 
that  report  and  the  right  interpretation  of  it. 

Fortunately,  however,  what  we  have  just  referred  to  as 
the  perennial  sources  of  all  philosophy — the  great  hypoth- 
eses of  our  emotional  nature — are  touched  upon  by  Dr. 
Carr  much  less  lightly  and  less  fleetingly  than  by  his  mas- 
ter. He  even  makes  bold  to  adopt  as  a  heading  for  one  of 
his  later  chapters  the  well-known  words  in  which  Kant 
summed  up  the  whole  demands  of  our  higher  emotional 
nature,  the  phrase  "God,  Freedom  and  Immortality."  It 
is  a  fortunate  circumstance  for  the  comparison  which  we 
have  in  hand.  In  Kant  we  have  the  original  source  of  that 
general  idealistic  view  in  philosophy  which  has  practically 
held  the  field  in  English  teaching  until  recently  and  with 
which  we  wish  to  compare  Bergson's.  These  postulates 
will  therefore  give  us  a  starting-point.  We  can  compare 
what  light  upon  them  has  been  derived  from  the  older 
sources,  with  what  has  been  given  to  us  by  Bergson.  And 
we  can  compare  the  two  philosophies  further  as  regards 
their  ability  to  justify  what  they  have  to  say  on  these  things, 
and  give  a  reason  for  the  hope  that  is  in  them. 

Before  we  can  consider  what  light  Bergson  throws 
upon  ultimate  questions,  or  compare  it  with  any  derived 
from  elsewhere,  we  must  first  try  to  gain  some  rough  idea 
of  his  general  view.  It  is  clearly  impossible  to  go  into 
detail.  We  cannot  indicate  Bergson's  actual  answers  to 
the  questions  we  have  cited  above,  still  less  his  answers  to 
all  the  questions  of  which  these  are  only  a  few  taken  at 
random.  But  it  may  be  possible  to  indicate  his  principle. 
It  may  be  possible  to  point  out  what  in  the  universe  Berg- 


BERGSONISM  IN  ENGLAND.  183 

son  especially  sees  and  values ;  what  it  is  which  he  believes 
to  be  capable  of  providing  a  solution  to  these  and  all  the 
various  problems  with  which  he  deals.  It  is  in  point  of 
fact  nothing  else  than  that  elan  vital  which  has  figured  so 
often  in  the  reviews  of  the  new  French  philosophy,  that 
vital  impulse  which  we  behold  forcing  itself  along  the 
whole  course  of  evolution  and  of  which  in  the  long  run 
Bergson  holds  the  universe  itself  to  be  the  creation. 

With  this  mere  hint  of  the  view,  let  us  turn  at  once  to 
the  question  how  it  stands  to  the  idealism  which  has  been 
taught  in  England  and  which  finds  its  classical  English  ex- 
pression in  the  writings  of  T.  H.  Green, — how  it  stands, 
especially,  as  regards  an  attitude  to  problems  which  are 
of  the  last  importance  for  the  human  mind. 

In  the  first  place  there  are  striking  general  resem- 
blances. In  Green's  doctrine  as  in  Bergson's  the  funda- 
mental reality  of  the  universe  is  not  matter  but  conscious- 
ness. Like  the  former  view,  Bergsonism  professes  never- 
theless to  be  neither  "idealism"  nor  "realism"  simpliciter. 
And  up  to  a  point  it  adduces  the  same  reason  for  repu- 
diating the  former  of  these  titles — for  refusing,  i.  e.,  to 
identify  its  teaching  with  any  such  "idealism"  as  is  usually 
associated  with  the  name  of  Berkeley.  The  reason  in  both 
cases  is  this.  The  reality  of  the  world,  though  for  both 
views  it  is  consciousness,  is  not  for  either  of  them  anything 
constituted  simply  by  our  private  minds.  No  thesis  is  put 
forward  by  either  theory  to  the  effect  that  you  and  I  and 
other  minds  like  us  are  all  that  exist.  The  consciousness 
referred  to  is  in  the  literal  sense  universal.  It  is  over  all 
the  universe,  a  feature  of  the  whole  of  it.  No  such  doctrine 
is  put  forward  from  either  quarter  as  that  the  universe 
which  we  usually  see  and  know  does  not  exist.  What  is 
said  is  that  it  is  conscious  and  is  the  product  of  its  con- 
sciousness. But,  for  Bergsonism,  that  self-creative  con- 
sciousness which  the  universe  is  differs  from  our  private 


184  THE  MONIST. 

minds  also  in  another  way,  a  way  reminiscent  of  Schopen- 
hauer as  well  as  of  the  neo-Kantian  idealism  of  Green. 
That  consciousness  is  not  preeminently  representative  or 
pictorial.  It  is  active.  Not  the  static  picture  of  knowledge 
is  its  characteristic  expression,  but  the  energy  of  will. 

In  consciousness  so  conceived,  then,  Bergsonism  finds 
the  key  to  the  broad  facts  of  life  and  evolution  as  science 
has  revealed  them  to  us.  In  the  evolutionary  history  of 
life  on  this  planet — in  the  genesis  and  progress  of  vegetable 
world,  animal  kingdom  and  man — what  we  have  is  this 
active  consciousness  in  the  form  of  life,  pushing  itself,  as 
it  were,  through  the  surface  of  matter  and  seeking  free 
way.  Man's  physical  organism  is  the  one  configuration  of 
matter  through  which  it  finds  the  free  course  which  it  seeks. 
The  human  body  is  organized  for  giving  outlet  to  this  ac- 
tivity. The  brain  and  nervous  system  are  but  its  cutting- 
edge  by  means  of  which  it  thrusts  itself  forward.  The 
story  of  evolution  is  the  story  of  how  the  main  current  of 
this  vital  impulse  has  worked  its  devious  way  through 
matter.  The  different  forms  of  life  which  we  see  are  the 
different  channels  into  which  the  central  stream  has  split 
itself  up  in  process  of  thrusting  itself  into  matter.  The 
central  stream  has  not  quite  dissipated  itself  as  yet  into 
these  branches.  The  main  current  is  still  traceable.  It  is 
found  in  the  life  and  consciousness  of  man.  It  is  for  this 
reason  that  man  is  at  the  head  of  creation.  His  life  and 
mind  contain  the  most  complete  concentration  to  be  found 
anywhere  of  what  was  in  the  original  world-impulse.  The 
fundamental  reality  of  the  universe,  then,  is  life;  but  it  is 
a  life  which  comes  to  view  best,  not  in  the  plant  or  the 
animal  but  in  the  conscious  life  of  man.  We  must  note 
further  that  the  "matter"  through  which  the  stream  of  life 
thrusts  itself  is  in  the  last  resort  its  own  creation,  though 
we  need  not  go  into  Bergson's  proof  of  that  here.  The 
vital  impulse  is  thus  creative  of  matter  and  of  all  the  forms 


BERGSONISM  IN  ENGLAND.  l8$ 

of  life  in  which  it  finds  an  outlet,  and  the  whole  process 
of  its  advance  is  named  by  Bergson  "creative  evolution." 

Some  such  view  is  the  only  one  capable,  in  Bergson's 
opinion,  of  meeting  the  necessities  of  the  case  which  nat- 
ural science  has  presented  to  us.  Dr.  Carr  has  endeavored 
to  go  further  and  show  that  as  a  general  view  it  is  specially 
in  harmony  with  some  quite  recent  scientific  discoveries. 
The  "vital  impulse"  is  nothing  if  it  is  not  movement.  It  is, 
in  fact,  pure  movement.  If  it  be  creative  of  "things"  then 
somehow  things  must  be  generated  out  of  movement.  And 
this,  Dr.  Carr  points  out,  is  precisely  what  science  is  now 
finding. 

"The  essential  principle  of  the  philosophy  of  change," 
he  says,  "is  that  movement  is  original.  Things  are  derived 
from  movement,  and  movement  is  not  a  quality  or  character 
that  things  have  added  to  themselves."2  "A  very  few  years 
ago,"  he  says  again,  "such  a  doctrine  would  have  sounded 
paradoxical  and  absurd.  But  now  compare  the  philosoph- 
ical doctrine  of  original  movement  with  the  new  theories 
of  science.  Let  us  take  first  the  structure  of  the  atom. 
The  electrical  theory  of  matter  teaches  that  the  atom  is 
composed  of  a  central  mass  or  core,  which  is  far  the  larger 
part  of  its  substance,  and  an  envelope  small  in  comparison. 
The  central  core  is  positive  electricity,  and  the  outer  en- 
velope consists  of  negatively  electrified  particles  held  in 
position  by  the  electrical  relation  to  the  central  core.  The 
atom,  in  fact,  is  a  solar  system  in  which  the  positive  element 
is  the  sun  and  the  negative  element  the  planets.  And  all 
the  qualities  of  atoms  depend  upon  the  arrangement  of 
these  outer  negative  elements.  But  what  is  the  ultimate 
reality  of  this  atom — something  or  other  that  is  electrified  ? 
No,  it  is  electricity,  not  something  electrified,  and  electricity 
is  a  form  of  energy,  and  energy  degrades  and  disperses. 

2  Philosophy  of  Change,  p.  11. 


l86  THE  MONIST. 

Reduced  to  simple  everyday  concepts  it  is  this,  that  what 
we  call  matter  is  a  form  of  movement."8 

But  it  is  not  merely  in  the  case  of  the  atom  that  recent 
discoveries  have  tended  to  resolve  into  terms  of  movement 
what  we  had  been  accustomed  to  regard  under  quite  other 
terms;  elsewhere  also  they  have  begun  to  transform  the 
static  into  the  changing,  the  resting  into  the  active. 

"But  now  turn  to  the  other  side,"  Dr.  Carr  continues 
(pp.  17-18).  "In  the  last  few  years  it  has  been  possible  to 
demonstrate  that  our  solar  system  is  not,  as  was  supposed, 
at  rest  in  an  absolute  space  or  else  moving,  if  it  be  moving, 
without  regard  to  forces  outside  itself.  It  belongs  to  a 
larger  system,  all  the  parts  of  which  are  in  movement  in 
relation  to  one  another.  The  fifty  million  stars  that  our 
telescopes  reveal  are  not  scattered  at  random  over  the 
firmament,  but  are  moving  along  regular  courses  coordi- 
nated to  one  another.  The  members  of  this  stellar  system 
are  not,  like  the  planets,  revolving  round  a  central  mass, 
but  millions  of  suns  are  streaming  across  an  unoccupied 
center.  The  speed  of  our  sun  (now  about  12%  miles  a  sec- 
ond) has  been  calculated,  and  its  direction  and  the  acceler- 
ation it  will  undergo  as  it  travels  across  the  center  and 
passes  outward  again  to  the  periphery.  This,  however,  is 
not  all.  A  discovery  has  been  announced  that  seems  likely 
to  extend  indefinitely  further  than  astronomers  have  yet 
imagined  the  vastness  of  the  spatial  universe.  Observa- 
tions which  have  been  made  on  the  great  spiral  nebula  in 
Andromeda  show  that  its  spectrum  is  inconsistent  with 
the  hitherto  generally  held  supposition  that  it  consists  of 
gaseous  matter  in  a  state  of  extreme  tenuity.  It  is  now  said 
to  be  a  spectrum  that  is  given  out  by  solid  glowing  masses, 
and  thus  seems  to  confirm  an  old  view  that  the  nebulae  are 
star  groups  immensely  distant.  This  nebula  is  apparently 
not  within  our  stellar  system,  but  itself  a  vast  stellar  system 

8  Ibid.,  pp.  16-17. 


BERGSONISM  IN  ENGLAND.  187 

lying  outside  the  latter  and  at  an  enormous  distance  away 
from  it.  What  other  systems  lie  outside  these  we  do  not 
know,  but  all  that  we  discover  suggests  universal  move- 
ment. There  is  no  absolute  rest.  If  we  conceive  an  ob- 
server placed  anywhere  in  this  great  universe  that  we  look 
out  upon  from  our  position  on  an  insignificant  planet  of  an 
insignificant  sun,  whether  we  suppose  him  to  gather  into 
one  embrace  what  to  us  are  vast  stellar  systems  or  to  be 
confined  to  the  negatively  charged  ion  of  an  hydrogen 
atom,  there  will  stretch  out  for  him  on  either  side  an  unlim- 
ited expanse  of  reality  of  which  the  ultimate  essence  is 
movement." 

And  Dr.  Carr  finds  a  suggestion  of  the  same  point, 
viz.,  that  things  are  not  more  original  than  movement  but 
that  movement  is  more  original  than  things,  in  the  way  in 
which  the  recent  "principle  of  relativity"  in  physics  threat- 
ens to  transform  our  conceptions  of  space  and  time  and  re- 
move the  ether  from  its  place  as  a  scientific  hypothesis. 
All  this  seems  to  him  to  confirm  the  view  expressed  by 
Bergson  in  La  preception  de  changement:  "Movement 
is  the  reality  itself,  and  what  we  call  rest  (immobilite) 
is  a  certain  state  of  things  identical  with  or  analogous  to 
that  which  is  produced  when  two  trains  are  moving  with 
the  same  velocity  in  the  same  direction  on  parallel  rails; 
each  train  appears  then  to  be  stationary  to  the  travelers 
seated  in  the  other."  And  again :  "There  are  changes,  but 
there  are  not  things  that  change;  change  does  not  need  a 
support.  There  are  movements,  but  there  are  not  neces- 
sarily constant  objects  which  are  moved;  movement  does 
not  imply  something  that  is  movable." 

The  discovery  that  the  whole  universe  is  movement  is, 
however,  very  little,  if  we  know  nothing  about  this  move- 
ment except  simply  that  it  moves.  Even  when  we  have 
brought  it  so  far  that  we  can  regard  this  movement  as  life, 


1 88  THE  MONIST. 

as  creative,  and  as  able  at  last  to  burst  into  consciousness, 
we  have  not  even  then  got  very  far  philosophically  unless 
we  learn  something  of  the  inner  character  of  this  vast 
spiritual  force.  On  its  inner  character  too  must  rest  our 
judgment  upon  Dr.  Carr's  bold  claim  for  the  "new  method," 
that  it  is  nothing  less  than  a  revolution  and  that  it  has 
reversed  the  direction  that  philosophy  has  followed 
throughout  its  history  of  2500  years.4  It  is  on  the  subject 
of  the  inner  character  of  this  movement  that  Bergson's 
teaching  most  directly  challenges  comparison  with  that  of 
Green ;  and,  we  may  add,  most  clearly  demands  to  be  sup- 
plemented by  it. 

For  Bergson  is  by  no  means  the  only  teacher  who  has 
conceived  of  a  universal  spiritual  energy  as  sustaining  the 
universe.  Green  teaches  the  same.  In  the  words  of  by  far 
the  ablest  existing  short  exposition  of  it,  the  central  concep- 
tion of  his  philosophy  "is  that  the  universe  is  a  single, 
eternal,  activity  or  energy  of  which  it  is  the  essence  to  be 
self-conscious."5  Nor  can  we  get  a  distinction  for  Bergson 
out  of  his  repeated  claim  that  the  spiritual  activity  of 
which  he  is  thinking  is  not  purely  conceptual,  because 
Green  in  essentials  makes  the  same  claim.  A  great  deal 
is  made  by  Bergson  of  the  non-conceptual  character  of  true 
philosophical  apprehension.  You  cannot  apprehend  that 
ultimate  essence  or  spiritual  force  whereby  the  universe 
exists,  in  the  ordinary  way;  that  is,  by  the  intellect.  The 
reason  why,  is  that  the  intellect  can  only  apprehend  what 
is  dead,  static,  given.  It  cannot  grasp  living  movement. 
Now  Green  has  quite  as  little  use  as  Bergson  has  for  what 
can  only  grasp  the  given  or  static.  Natural  science,  for 
Bergson,  is  based  on  the  intellect  and  that  is  why  it  cannot 
conduct  us  into  the  presence  of  what  verily  explains  things. 

4  See  Philosophy  of  Change,  p.  20. 

6  R.  L.  Nettleship  in  his  biography  of  Green :  Green's  Works,  Vol.  Ill,  p. 
Ixxv. 


BERGSONISM  IN  ENGLAND.  189 

Science  only  sees  in  the  universe  what  is  dead,  and  there- 
fore it  cannot  exhibit  its  ultimate  spiritual  essence.  This 
is  Green's  complaint  too.  Green,  indeed,  does  not  say  that 
we  must  appeal  to  something  else  than  the  intellect  (this 
is  Bergson's  way  of  putting  the  matter),  but  he  does  say 
that  we  must  understand  the  intellect.  We  must  not  be 
content  to  use  it  uncritically,  as  natural  science  and  naive 
common  sense  do.  We  must  lay  hold  of  life  and  activity 
with  it.  But  Green  is  clear  that  the  life  and  activity  of 
which  the  intellect  must  lay  hold,  is  its  own.  His  philo- 
sophic creed,  shortly  stated,  is  that  this  is  possible — that 
intellect,  itself  an  energy,  will  reveal  a  spiritual  energy  at 
the  heart  of  the  universe,  if  it  be  persevered  with  and 
rightly  used.  Green  does  not  say  so  in  these  words.  But 
his  philosophy  says  so  to  any  one  who  has  entered  into  it. 
He  insists  on  the  one  hand  that  reason  and  will  are  one, 
in  the  sense  that  they  are  alike  expressions  of  one  prin- 
ciple;6 and  he  speaks  constantly  of  that  principle  as  active 
and  as  self-active.  His  phrase  is  "self-realizing."  On 
the  other  hand,  his  whole  contention  against  the  empirical 
school  in  philosophy  was  that  this  self-activity,  the  essence 
of  men's  minds,  was  not  in  men's  minds  alone.  It  was  the 
essence  of  the  universe.  The  spiritual  principle  was  "in 
knowledge,"  and  also  it  was  "in  nature,"  as  the  most  ele- 
mentary student  of  his  chief  work  soon  learns  to  know. 
Practically  all  he  had  to  say  about  nature  in  fact  was  just 
this :  that  it  was  not  inert,  dead,  merely  given ;  that  it  was 
a  spiritual  life,  of  which  our  individual  minds  were  the 
highest  finite  manifestation.  So  far  Green  and  Bergson 
are  on  common  grounds. 

There  is,  however,  a  real  divergence  between  Berg- 
son's  and  the  older  teaching.  They  differ  in  their  doctrine 
of  time.  Both  agree  that  what  we  can  see  around  us  with 

6  See,  inter  alia,  Works,  II,  p.  329. 


THE   MONIST. 

the  bodily  eye  is  not  the  ultimate  spiritual  energy  but  its 
manifestations  only.  Even  with  the  eye  of  the  mind,  they 
both  hold,  the  ultimate  spirit  itself  cannot  be  apprehended 
in  its  whole  nature,  for  only  part  of  its  original  totality 
is  contained  in  the  mind  of  man.  But  they  differ  as  to  what 
it  is  in  its  own  whole  nature.  With  Green  it  is  in  itself 
already  perfect,  whereas  with  Bergson  it  is  still  developing 
and  changing  with  the  course  of  time  and  has  an  immense 
and  entirely  uncertain  future  ahead  of  it.  With  Green  the 
ultimate  spirit  is  complete,  with  Bergson  it  is  incomplete. 
With  Green  it  is  a  consciousness,  morally  and  intellectually 
all  that  we  could  conceive  ourselves  becoming.  With  Berg- 
son it  is  a  consciousness  still  always  turning  into  something 
different  and  turning  always  into  something  which  could 
not  have  been  predicted. 

At  this  point  also  occurs  the  most  marked  difference  of 
the  two  doctrines  in  regard  to  the  light  which  they  cast 
upon  the  assumptions  of  the  moral  and  religious  conscious- 
ness. And  as  regards  religion,  it  is  not  hard  to  judge 
which  is  in  the  stronger  position.  So  far  as  the  religious 
mind  has  entertained  the  belief  that  behind  the  phenomena 
of  the  universe  and  acting  as  their  source,  there  exists  a 
mind  which  is  eternal,  one  who  is  above  time  and  vicissi- 
tude, who  is  perfect  and  is  not  subjected  to  change,  a  God 
"who  was  and  is  and  ever  will  be,"  in  so  far  it  will  find  its 
faith  countenanced  in  Green's  teaching  but  discountenanced 
in  Bergson's.  Dr.  Carr  himself  is  clear  that  the  change 
Bergson's  theory  invites  us  to  make  in  our  religious  con- 
ceptions is  profound,  though  he  thinks  that  it  has  com- 
pensations. 

"How  is  the  conception  of  God  affected  by  the  principle 
of  this  new  philosophy  ?  One  attribute  that  has  seemed  to 
attach  to  this  conception  can  certainly  not  belong  to  it — 
eternity,  in  the  sense  of  timelessness.  Reality  is  essentially 
movement,  movement  is  duration,  duration  is  change.  If 


BERGSONISM  IN  ENGLAND. 

we  call  the  original  impulse  of  life  God,  then  God  is  not 
a  unity  that  merely  resumes  in  itself  the  multiplicity  of  time 
existence,  a  unity  that  sums  up  the  given.  God  has  nothing 
of  the  already  made.  He  is  not  perfect  in  the  sense  that  He 
is  eternally  complete,  that  He  endures  without  changing. 
He  is  unceasing  life,  action,  freedom. 

"No  more  profound  change  can  be  imagined  in  the  con- 
ception of  the  universe,  in  the  conception  of  human  nature, 
in  the  whole  outlook  of  life,  than  is  involved  in  this  new 
conception  of  God.  The  conception  of  God  to  which  we 
have  been  accustomed  in  philosophy, — the  most  perfect  be- 
ing, the  ens  realissimum,  the  first  cause,  the  causa  sui,  the 
end  or  final  cause, — is  the  conception  of  a  reality  which 
time  does  not  affect.  Hence  the  continual  attempt  both  in 
ancient  and  modern  philosophy  to  conceive  two  orders  or 
kinds  of  existence,  the  temporal  and  the  eternal,  and  the 
whole  problem  of  philosophy  has  been  to  conceive  the  rela- 
tion of  these  two  orders  to  one  another.  Time  and  the 
whole  order  of  changing  reality  must,  it  has  seemed,  be  of 
the  nature  of  an  emanation  from  God,  or  a  manifestation 
of  God.  But  however  conceived,  the  time  order  is  regarded 
as  essentially  unreal,  appearance  and  not  reality;  change 
and  movement  are  relative  to  us."7 

Connected  with  the  same  difference  in  regard  to  time, 
there  is  again  a  difference  of  the  two  theories  as  regards 
the  light  they  throw  upon  another  of  the  Kantian  postu- 
lates. So  far  as  the  religious  consciousness  has  fixed  its 
hope  on  immortality  in  the  sense  of  a  life  out  of  time  Berg- 
sonism  can  offer  no  corroboration,  for  time  and  the  change 
which  constitutes  it  are  to  this  philosophy  reality  itself, 
and  to  be  out  of  time  is  ipso  facto  to  be  out  of  existence. 
Here  again  the  older  view  is  considerably  different.  For 
it  the  idea  of  a  life  beyond  time  is  at  least  not  contradictory. 
Nay,  any  completing  or  perfecting  of  our  best  life  here 

7  Philosophy  of  Change,  pp.  187-188. 


IQ2  THE  MONIST. 

would  inevitably  have  this  character,  since  for  this  view 
we  are  already  above  time  in  so  far  as  we  think  what  is 
true  and  do  what  is  unselfish. 

As  regards  the  postulates  of  God  and  immortality,  then, 
the  effect  of  Bergsonism  is  of  a  negative  character.  But 
these  are  preeminently  religious  postulates.  The  point 
upon  which  Bergsonism  claims  most  confidently  to  have 
substantiated  our  higher  emotional  demands  is  in  regard 
to  the  moral  postulate,  that  of  freedom.  In  its  clearness 
upon  this  question,  indeed,  Dr.  Carr  finds  the  chief  com- 
pensation for  its  attitude  upon  the  others. 

"The  philosophy  of  change  does  not  sound  any  clear 
and  confident  note  as  to  what  lies  beyond  us  in  the  unseen 
world.  It  does  not  present  to  us  God  as  the  loving  father 
of  the  human  race,  whom  He  has  begotten  or  created  that 
intelligent  beings  may  recognize  Him  and  find  happiness 
in  communion  with  Him.  There  may  be  truth  in  this  ideal, 
but  it  is  no  part  of  philosophy.  Neither  does  it  teach  us  the 
brotherhood  of  the  human  race — on  the  contrary  it  seems 
to  insist  that  strife  and  conflict  are  the  essential  conditions 
of  activity.  Life  is  a  struggle,  and  the  opposing  elements 
are  the  nature  of  life  itself,  the  very  principle  of  it.  The 
evolution  of  life  is  the  making  explicit  of  what  lies  implicit 
in  the  original  impulse.  Philosophy  reveals  no  ground  for 
the  belief  in  personal  survival,  and  it  shows  us  that  'how- 
ever highly  we  prize  our  individuality  we  are  the  realiza- 
tion of  the  life-impulse  which  in  producing  us  has  produced 
also  myriad  other  forms.  What  then  is  the  attraction  that 
this  philosophy  exercises  ?  What  is  there  of  supreme  value 
that  it  assures  to  us?  The  answer  is  freedom."8 

Here  at  length  we  reach  the  philosophically  important 
matter.  For  here  we  can  interrogate  the  two  views,  not 
merely  as  to  whether  they  can  corroborate  our  religious 

8  Philosophy  of  Change,  pp.  195-186. 


BERGSONISM  IN  ENGLAND.  193 

sense,  but  as  to  their  grounds  for  doing  so.  The  whole 
question  for  the  critical  evaluation  of  the  philosophy  of 
Bergson,  it  may  be  said,  is  that  of  the  nature  of  and  the 
evidence  for  the  freedom  which  he  says  characterizes  that 
ultimate  spiritual  force  of  which  we  are  the  offspring,  and 
which  by  its  vast  uprush  through  the  universe  and  through 
us  creates  us  and  the  universe  as  it  goes. 

For  Green  too,  as  everybody  knows,  there  is  freedom. 
And  he  puts  the  rationale  of  it  thus.  Man  is  free,  for  him, 
both  in  his  knowing  and  his  acting,  because  in  both  of  these 
functions  the  past  is  gathered  up  in  the  present  which  is 
now  before  him.  Except  this  were  so,  says  Green,  we 
could  not  know.  To  know,  is  to  know  succession.  Now  if 
there  were  only  succession  itself — that  is,  if  the  past  were 
not  thus  gathered  up — there  could  be  no  consciousness 
thereof.  This  is  straightforward  reasoning,  and  at  bottom 
quite  simple.  If  I  am  gathering  a  bunch  of  flowers,  I  must 
hold  the  first  ones  in  my  hand  while  I  gather  the  rest.  If 
I  did  not  do  this  but  dropped  each  one  as  I  picked  it,  I 
should  never  have  a  bunch.  Quite  similarly,  if  I  hear  or 
see  a  succession,  say,  of  strokes  upon  a  knocker,  and  if  I 
know  that  it  is  a  succession  of  five  knocks,  my  knowing 
is  evidence  sufficient  that  the  earlier  strokes  have  not  es- 
caped me  but  have  been  gathered  up  in  my  mind  and  pre- 
sented along  with  the  last  one.  If  each  had  disappeared 
as  it  occurred  there  would  have  been  no  succession  of  five 
for  me.  Each  one  would  have  been  number  one;  and 
when  it  was  over  would  have  been  nothing.  To  perceive 
time  at  all  I  must  not  merely  have  the  present  before  me. 
I  must  have  the  past  along  with  the  present.  In  Green's 
phrase,  the  various  members  of  the  series  must  be  "co- 
present"  to  consciousness. 

Bergson  has  made  an  analysis  of  this  same  experience, 
and  has  given  the  matter  profound  attention.  He  too  sees, 
that  to  know  succession  in  the  ordinary  sense  of  knowl- 


194  THE  MONIST. 

edge  the  members  of  the  succession  must  be  somehow  co- 
present,  but  he  gives  the  whole  matter  another  turn.  He 
cannot  feel,  apparently,  that  in  knowing  the  successive  as 
thus  co-present  we  are  really  knowing  the  successive  at 
all.  His  refrain  therefore  is,  we  try  to  know  a  time-suc- 
cession by  the  ordinary  use  of  our  intellect,  but  cannot. 
We  do  not,  in  this  fashion,  know  a  time-succession.  We 
only  succeed  in  knowing  space.  In  counting  the  strokes 
we  set  out  the  series  of  events  in  a  row,  along  a  line,  in  a 
kind  of  mental  space.  This  we  call  perceiving  their  tem- 
poral succession.  And  if  one  asks,  "Why  does  the  intellect 
fail?  how  are  we  to  apprehend  time,  or  what  would  it  be 
like  if  we  would  apprehend  it?"  the  whole  argument  of 
Bergson's  Time  and  Free  Will  converges  in  effect  upon 
this  answer:  that  the  intellect  which  fails  to  apprehend 
time-succession  fails  because  it  can  only  set  out  the  events 
separately  along  an  imaginary  spatial  line,  whereas  for 
the  "intuition"  which  really  apprehends  time  these  events 
are  not  separate,  they  interpenetrate.  This  interpenetra- 
tion  is  time.  It  is  fairly  easy  to  see  further  how,  out  of 
the  apprehension  of  such  time,  he  gets  free  will.  We  have 
to  pull  ourselves  together  in  order  to  grasp  this  interpene- 
tration;  and  in  this  attitude,  in  this  tense  summoning  of 
ourselves  together,  we  are  free. 

We  have  here  the  fundamental  impeachment  of  reason 
to  which  Bergson's  philosophy  seems  compelled  to  have  re- 
course. To  reject  the  intellect  as  a  means  of  attaining  to 
the  truth  is  an  obvious  weakness,  as  compared  with  the 
other  view,  thus  far — that  it  is  a  species  of  self-subversion 
which  the  view  with  which  we  are  contrasting  it  does  not 
commit.  Both  Bergson  and  Green  in  philosophizing  at  all 
are  endeavoring  to  settle  their  account  with  the  problems 
of  life  by  thinking  them  out.  Both,  in  other  words,  are 
making  use  of  the  intellect.  The  difference  between  them 
in  regard  to  the  matter  before  us  is  that  Green  trusts  the 


BERGSONISM  IN  ENGLAND. 

instrument  he  is  using.  Having  found  what  the  intellect 
perceives  time  and  succession  to  be,  he  says  frankly  that 
that  is  what  they  are.  But  Bergson,  unable  to  accept  the 
verdict,  will  rather  make  bold  to  say  that  our  rational  mind 
is  incompetent,  that  it  is  incapable  of  seeing  things  as  they 
are,  and  so  has  no  authority  in  the  case.  This  is  a  serious 
matter.  One  cannot  feel,  after  this,  that  the  intellect  can 
be  a  very  safe  instrument  to  philosophize  with.  This  is 
perhaps  the  rock  on  which  all  philosophies  eventually  split 
which  attempt  to  reason  the  reader  into  preferring  some 
supra-rational  or  sub-rational  power  before  reason  itself. 
Mere  reason  may  not  be  fit  to  see  what  reality  is;  but  if 
not,  is  it  fit  to  attack  itself  either  ?  We  cannot  endorse  this 
intellectual  abuse  of  the  intellect.  If  the  intellect  cannot 
justify  itself  it  cannot  justify  anything.  We  must  accept 
the  intellect,  or  our  whole  attitude  is  sceptical. 

"But  the  intellect  can't  allow  you  free-will,"  it  will  be 
at  once  objected.  This  is  an  ancient  objection,  of  which, 
as  we  shall  see,  Bergson  himself  shows  us  how  to  get  the 
better.  What,  we  have  to  ask, — what  precisely  is  the  free- 
dom that  Bergson's  argument  itself  will  bring  us  if  it  is 
true?  It  is  easy  for  Dr.  Carr  to  speak  as  if  Bergson  pre- 
served for  us  the  privilege  of  a  wide  choice  in  an  open 
universe.  All  defenders  of  freedom  have  used  such  lan- 
guage. The  question  is,  what  evidence  has  he?  What  is 
there  in  our  own  experience  that  we  can  fall  back  upon  and 
see  that  the  universe  is  open  before  us  ?  What  reveals  our 
identity  with  a  universal  principle  of  freedom  which  creates 
the  universe  itself,  and  in  whose  life  we  are  free? 

Whatever  answer  can  be  got  out  of  Bergson  to  the 
question  must  come  from  the  "interpenetration"  just  men- 
tioned. And  on  inquiry  we  find  that  it  is  a  solid  answer 
enough.  We  do  get  evidence  of  freedom.  And  it  is  from 
the  "interpenetration"  that  we  get  it.  Bergson  is  one  of 
the  few  people  who  see  where  the  freedom  issue  really  lies. 


196  THE  MONIST. 

In  Time  and  Free  Will  he  insists  that  freedom  is  to  be 
looked  for  in  the  character  of  an  act  itself.  It  is  the  ques- 
tion "what  was  the  act?"  that  is  essential;  not  the  question 
"what  might  it  have  been?"  or  "could  it  have  been  differ- 
ent ?"  What  we  have  to  ask  about  two  alternative  courses 
of  conduct  ahead  of  us,  when  we  want  to  know  whether  we 
are  free  agents,  is  not  "is  either  equally  possible  to  me 
now?"  but  "what  is  the  inner  character  of  the  one  chosen 
when  it  does  eventuate?"  And  he  indicates,  in  language 
which  might  have  been  copied  from  Green,  that  our  charac- 
ter must  be  in  our  act.  "We  are  free,"  he  says  in  Time 
and  Free  Will,9  "when  our  acts  spring  from  our  whole  per- 
sonality, when  they  express  it,  when  they  have  that  in- 
definable resemblance  to  it  which  one  sometimes  finds  be- 
tween the  artist  and  his  work.  It  is  no  use  asserting  that 
we  are  then  yielding  to  the  all-powerful  influence  of  our 
character.  Our  character  is  still  ourselves;"  etc.,  etc.  And 
what  we  learn  from  his  lengthy  subsequent  discussion  of 
the  matter  is  simply  this:  that  where  "interpenetration" 
occurs,  there  our  character  is;  where  the  multiplicity  con- 
sciously present  in  us  is  made  up  of  items  which  interpene- 
trate, there  our  personality  has  its  seat.  And  where  the 
multiplicity  of  interpenetrating  states  is  at  its  maximum 
in  the  great,  critical  decisions  of  our  life,  there  our  free- 
dom is  at  its  maximum  because  our  personality  is  so.  "It  is 
the  whole  soul ....  which  gives  rise  to  the  free  decision ; 
and  the  act  will  be  so  much  the  freer,  the  more  the  dynamic 
series  with  which  it  is  connected  tends  to  be  the  fundamen- 
tal self."10 

It  takes  a  great  effort,  often,  to  draw  the  scattered 
multiplicity  of  our  conscious  states  into  this  interpene- 
trating unity.  And  in  his  later  work,  Creative  Evolution, 
Bergson  tries  to  show  that  when  this  concentration  of  spirit 

9  English  Translation,  p.  172. 

10  Time  and  Free  Will,  p.  167. 


BERGSONISM  IN  ENGLAND.  197 

is  relaxed  an  order  of  freedom  transforms  itself  into  an 
external  order  of  necessity.  There  is  no  disorder  in  spirit, 
but  only  these  two  opposite  kinds  of  order.  That  is  how 
he  accounts  for  matter.  It  is  the  de-tension  of  the  universal 
life-impulse.  But  the  present  point  is,  that  an  act  is  free 
when  our  personality  is  in  it,  and  that  happens  when  it  is 
one  such  as  gives  outlet  or  utterance  to  a  multiplicity  of 
states  held  in  an  intense  interpenetrating  unity. 

So  far,  Bergson  conducts  us  along  safe  and  solid 
ground.  But  let  us  not  make  a  mystery  of  this  interpene- 
tration.  The  highest  examples  of  it  are  to  be  found  only 
rarely,  no  doubt.  We  find  them  in  moments  when  the  en- 
tire being  of  a  richly  endowed  mind,  all  its  desires,  fears, 
hopes,  knowledge,  emotions,  converge  in  one  direction, 
meditate  one  high  and  hard  decision,  and  that  decision  is 
taken.  There  you  have  that  contracting  together  of  the 
entire  soul  for  the  effort,  of  which  Bergson  speaks  under 
so  many  similes,  and  which  is  perhaps  the  highest  act  of 
a  life.  But  there  are  simpler  examples.  The  simplest  is 
the  common  experience  we  have  already  referred  to — the 
mere  watching  a  series  of  events  go  by.  The  vague  im- 
pression left  by  the  last  "click"  of  a  series  to  which  we  have 
not  been  attending  will  tell  us,  says  Bergson,  (if  we  start 
up  afterwards  and  try  to  count  how  many  we  have  missed) 
when  we  have  counted  enough.  In  such  a  case  the  objects 
we  consciously  count  are  set  out  in  a  sort  of  mental  row. 
Not  so  the  vague  impression  which  acts  as  our  standard 
and  says  to  us  when  we  have  counted  up,  say,  four,  "that 
is  enough."  This  vague  impression  does  itself  contain 
four.  It  is  an  impression  of  four.  But  it  contains  them 
in  a  different  way.  In  it  they  are  not  set  out  in  a  row,  but 
interpenetrate.  Its  "four"  character,  its  quadruplicity  if 
you  will,  is  a  unique  quality. 

"Whilst  I  am  writing  these  lines,  the  hour  begins  to 
strike  upon  a  neighboring  clock,  but  my  inattentive  ear 


198  THE  MONIST. 

does  not  perceive  it  until  several  strokes  have  made  them- 
selves heard.  Hence  I  have  not  counted  them.  Yet  I  only 
have  to  turn  my  attention  backwards  to  count  up  the  four 
strokes  which  have  already  sounded  and  add  them  to  those 
which  I  hear.  If,  then,  I  question  myself  carefully  on  what 
has  just  taken  place,  I  perceive  that  the  first  four  sounds 
had  struck  my  ear  and  even  affected  my  consciousness,  but 
that  the  sensations  produced  by  each  one  of  them,  instead 
of  being  set  side  by  side,  had  melted  into  one  another  in 
such  a  way  as  to  give  the  whole  a  peculiar  quality,  to  make 
a  kind  of  musical  phrase  out  of  it.  In  order,  then,  to  esti- 
mate retrospectively  the  number  of  strokes  sounded,  I  tried 
to  reconstruct  this  phrase  in  thought :  my  imagination  made 
one  stroke,  then  two,  then  three,  and  so  long  as  it  did  not 
reach  the  exact  number  four,  my  feeling,  when  consulted, 
answered  that  the  total  effect  was  qualitatively  different. 
It  had  thus  ascertained  in  its  own  way  the  succession  of 
four  strokes,  but  quite  otherwise  than  by  a  process  of  addi- 
tion, and  without  bringing  in  the  image  of  a  juxtaposition 
of  distinct  terms.  In  a  word,  the  number  of  strokes  was 
perceived  as  a  quality  and  not  as  a  quantity;  it  is  thus  that 
duration  is  presented  to  immediate  consciousness,  and  it 
retains  this  form  so  long  as  it  does  not  give  place  to  a  sym- 
bolical representation  derived  from  extensity."1 

Now  the  freedom  which  Bergson  secures,  and  which 
he  says  cannot  be  apprehended  by  the  intellect  but  only  by 
what  he  calls  "intuition,"  is  this  interpenetration.  The  in- 
tellect, he  holds,  cannot  grasp  it.  But  if  we  put  aside  his 
statement  that  the  intellect  cannot  grasp  this  unity  of  inter- 
penetrating items,  and  attend  solely  to  his  description  of 
what  the  intellect  is  alleged  not  to  be  able  to  grasp,  we  find 
that  his  statement  is  quite  wrong.  The  intellect  can  grasp 
it,  and  Green's  doctrine  is  precisely  that  it  can.  True, 
"interpenetration"  is  not  a  favorite  word  of  Green's.  He 

11  Time  and  Free  Will,  Eng.  Trans.,  pp.  127-128. 


BERGSONISM  IN  ENGLAND.  199 

speaks  of  relation.  He  holds  that  the  members  of  a  suc- 
cession in  order  to  be  known  to  our  minds  as  a  succession 
must  be  related;  so  related  that  they  are  co-present.  But 
this  interrelation  which  Bergson  says  is  a  misreading  of 
time  and  a  translation  of  it  into  mere  "space  symbolism" 
because  the  members  don't  interpenetrate,  this  intellectual 
apprehension  of  a  succession,  is  already  to  Green  precisely 
a  complex  of  interpenetrating  elements.  True,  the  items 
are  connected  by  relation,  but  relations  are  internal  for 
Green.  They  are  constitutive  of  the  thing's  character.  The 
relations  in  which  each  thing  stands  to  the  others  are  what 
make  its  nature.  The  nature  of  all  the  others,  therefore, 
enters  into  each,  and  that  of  each  into  all  the  others.  They 
must  interpenetrate;  their  natures  do  so  as  truly  and  lit- 
erally as  two  brushes  which  have  been  stuck  together.  The 
fact  is,  it  is  altogether  the  same  whether  we  say  of  certain 
elements  that  their  mutual  relations  are  internal  to  each 
of  them,  or  that  they  penetrate  one  another. 

"But  this  is  not  the  interpretation  that  Bergson  means," 
it  will  be  replied  at  once.  "This  interrelation  of  Green's 
would  never  yield  anything  like  freedom.  What  Bergson 
means  is  a  vital  interpenetration,  not  any  dead  static  thing 
such  as  could  be  illustrated  by  the  mere  material  inter- 
penetration  of  the  bristles  of  two  brushes."  Entirely  so. 
The  metaphor  does  not  do  justice  to  Bergson's  position, 
and  neither  does  it  to  Green's.  With  Bergson  the  inter- 
penetration  of  the  elements  seen  by  "intuition"  is  vital,  it 
is  an  intense  living  movement,  and  he  strains  language  to 
express  how  the  elements  fuse  together,  melt  into  each 
other,  inter-work  and  support  a  real  life.  But  neither, 
with  Green,  are  the  objects  of  the  intellect  in  a  dead  rela- 
tion. A  relation,  with  him,  is  a  relating — a  living  activity, 
therefore.  He  has  nothing  to  teach  if  he  does  not  teach 
this.  He  has  nothing  to  urge  if  he  does  not  urge  that  a 
system  of  relations  "implies  a  relating  mind."  And  surely 


2OO  THE  MONIST. 

no  one  ever  took  him  to  mean  by  that,  that  the  implied 
"mind"  merely  made  the  system,  set  it  down,  and  left  it  for 
ever  alone  to  stand  there,  dead,  cold  and  finished.  The  re- 
lations are  alive.  They  are  being  kept  up.  They  are  a 
deed;  and  not  a  deed  done  but  a  deed  ever  a-doing.  A 
relation  of  two  things,  with  Green,  is  a  supporting  of  them 
in  an  energy  of  ceaseless  spiritual  movement,  in  precisely 
the  Bergsonian  sense. 

"But  this  movement  constitutes  the  things,  with  Berg- 
son;  it  is  their  source,  the  very  stuff  of  which  they  are 
made."  Even  so  with  Green,  and  much  he  has  been  made 
to  suffer  for  it !  It  is  not,  says  Bergson,  things  which  are 
first  and  which  come  to  interpenetrate  afterward.  It  is 
the  movement  or  interpenetration  which  constitutes  the 
things.  It  is  not,  says  Green,  things  which  are  first  and 
which  come  to  be  related  or  interpenetrated  afterward.  It 
is  their  relation  or  interpenetration  which  constitutes  them. 
A  thing  is  nothing  apart  from  its  relations. 

So  far  as  regards  the  tracing  of  reality  to  a  spiritual 
source  Bergson  indeed  uses  a  language  which  is  different 
from  that  of  the  older  idealists.  But  in  this  general  matter 
his  fundamental  thought  is  accurately  the  same.  The  only 
difference  is  that  the  older  teaching  does  not  fall  back  on 
any  special  intuition  in  order  to  be  assured  that  reality  has 
a  spiritual  source.  It  relies  on  the  more  thorough  applica- 
tion and  the  critical  use  of  the  intellect  itself.  It  holds  that 
this  most  important  of  truths  still  is  truth,  and  that  by 
those  who  persevere  it  may  be  reached  by  the  same  meth- 
ods through  which  other  convincing  truth  is  reached, 
namely,  by  the  exercise  of  reason. 

But  this  one  difference  is  a  difference  as  of  heaven  and 
earth.  By  disparaging  intellect  it  puts  Bergson  in  the  un- 
happy position  of  constantly  needing  to  discredit  that  very 
faculty  of  "reasoning"  upon  which  as  a  philosopher  he 


BERGSONISM  IN  ENGLAND.  2OI 

must  stake  his  own  results;  and  that  is  not  the  whole 
of  the  trouble.  It  also  gives  a  false  cast  on  the  moral  side 
to  the  entire  physiognomy  of  his  teaching.  And  with  a 
glance  at  this  we  may  close  our  review. 

The  significant  point  is  that  Bergson  does  not  believe 
in  the  intellect,  or  in  the  typical  object  of  the  intellect, 
namely,  space.  By  not  believing  in  them  we  mean  that  he 
does  not  believe  in  their  spirituality.  Green  does.  Green 
finds  in  space  itself  that  very  interpenetration  or  spiritual 
movement  which  Bergson  insists  cannot  be  found  there.  He 
finds,  that  is  to  say,  in  the  (spatial)  object  of  the  intellect 
something  which  fully  answers  the  essentials  of  Bergson's 
description  of  the  interpenetrating,  while  Bergson  con- 
stantly speaks  of  this  character  in  things  as  though  it  could 
not  be  seen  at  all  intellectually,  but  only  in  glimpses,  by  the 
special  power  of  apprehension  which  he  calls  intuition. 
Green,  in  a  word,  finds  in  the  spatial-intellectual  that  reality 
and  truth  which  Bergson  can  only  find  when  all  "space- 
symbolism"  has  been  done  away  with.  This  is  a  serious 
difference.  For  this  "space-symbolism,"  in  the  wide  mean- 
ing which  Bergson  gives  to  it,  is  the  very  stuff  and  fiber  of 
the  moral  life.  His  teaching  therefore  means  that  to  be 
at  the  moral  point  of  view  is  to  be  out  of  touch  with  the  real 
truth  of  the  world. 

And  unfortunately  his  actual  ethical  teaching  bears  out 
the  suggestion.  It  is  quite  a  mistake,  we  may  note  in  pass- 
ing, to  say  that  Bergson  has  not  written  on  ethics.  It  is 
true  he  has  not  written  any  book  with  that  name.  But  he 
has  a  work  the  real  burden  of  which  is  an  interpretation 
of  the  moral  and  social  life.  This  is  his  little  treatise  On 
Laughter.  His  thesis  in  that  work  is  that  laughter  is  a 
species  of  social  castigation.  It  is  designed  to  rid  society 
of  the  conduct  that  provokes  it.  And  the  question  for  the 
moral  implications  of  Bergson's  teaching  is,  what  is  it  whose 
destiny  is  thus  to  be  socially  castigated?  Startling  as  the 


2O2  THE  MONIST. 

answer  may  seem,  it  is  the  moral.  It  is  called  the  mechan- 
ical. In  the  wide  sense  in  which  Bergson  eventually  uses 
the  term,  it  is  the  intellectual-spatial.  But  in  the  concrete 
what  is  it?  It  is  simply  faithfulness  to  principle  where 
such  faithfulness  is  awkward.  In  other  words  it  is  the  very 
soul  of  the  moral  life,  if  that  is  anything  at  all  distinct  from 
the  "esthetic"  life.  This  disbelief  in  space  and  the  spatial, 
this  disbelief  in  the  negation  which  is  at  the  root  of  these, 
is  what  the  present  writer  has  ventured  to  call  the  pessi- 
mism of  Bergson.12 

Without  repeating  here  what  has  been  worked  out  else- 
where,13 reference  may  be  permitted  to  one  little  point  in 
elucidation  of  this  view.  It  concerns  Bergson's  first  illus- 
tration in  Laughter,  his  picture  of  the  runner  who  stumbles 
and  falls.  It  is  a  small  matter,  of  course,  but  it  has  always 
struck  the  present  writer  as  a  peculiarly  significant  accident 
that  Bergson  should  have  opened  an  essay  On  Laughter 
by  taking  as  his  first  example  of  the  ridiculous  precisely 
that  figure  which  has  served  so  many  moralists  for  their 
type  of  the  moral  life.  The  runner  of  Bergson's  illustra- 
tion, as  Bergson  describes  him,  with  his  eagerness  and  his 
"rigidness,"  with  his  omitting  to  look  where  he  is  going, 
his  stumbling  over  obstacles  and  his  abundant  inability  to 
adapt  his  conduct  as  circumstances  require,  and  follow  the 
sinuosities  of  his  crooked  path,  is  indeed  ridiculous.  But  it 
is  only  Bergson's  light  vein  that  makes  him  so.  There  is 
nothing  essentially  ludicrous  about  such  a  man.  In  essen- 
tials, he  might  be  Bunyan's  pilgrim  fleeing  toward  the 
wicket-gate  or  St.  Paul's  runner,  who  also  heeds  nothing 

12  See  articles  in  The  Hibbert  Journal  for  October  1912,  The  International 
Journal  of  Ethics  for  January  1914  and  Mind  for  July  1913.  Compare  an  article 
on  "Bergson,  Pragmatism  and  Schopenhauer"  by  Giinther  Jacobi  in  The 
Monist,  Vol.  XXII,  pp.  593ff.  The  latter  article,  however,  should  be  read  with 
caution.  The  present  writer  has  the  best  of  reasons  to  believe  that  the  mar- 
velous correspondence  in  detail  which  exists  between  Bergson  and  the  prince 
of  pessimists  is  largely  accidental.  Bergson  himself  learned  about  it  only  after 
his  own  principles  had  been  evolved  into  practically  their  mature  shape. 

18  In  the  article  in  The  International  Journal  of  Ethics  referred  to. 


BERGSONISM  IN  ENGLAND.  2O3 

either  right  or  left,  but  simply  "presses  toward  the  mark." 
Of  course  there  would  be  nothing  in  a  mere  illustration 
as  such,  but  this  one  is  so  absolutely  well  chosen.  This  is 
the  type  of  man — this  steadfast  man,  this  man  who  just  is 
not  sinuous  and  yielding  and  pliable  and  graceful  and  free, 
this  straight-going  individual  who  cannot  do  anything  but 
go  straight — this  is  the  type  whose  proper  destiny,  accord- 
ing to  the  whole  tenor  of  the  essay,  is  to  be  laughed  out  of 
society ;  this  is  the  man  for  whom  society  has  no  use.  "Since 
when?"  some  may  feel  inclined  to  ask,  not  without  a  tinge 
of  indignation.  We  confess  that  to  us,  hitherto,  society 
has  seemed  to  have  considerable  need  for  him ;  nay,  to  have 
had,  perhaps,  prodigiously  little  use  for  the  other  sort  in 
comparison. 

Moreover  it  is  the  discovery  of  precisely  what  this  social 
theory  neglects,  namely  the  spirituality  in  spatiality  itself, 
that  enables  the  idealist  to  endorse  the  religious  conscious- 
ness of  God  as  eternal  and  perfect,  without  losing  the  other 
point,  equally  important,  that  the  divine  nature  must  also 
be  movement,  activity,  freedom.  To  science  the  natural- 
spatial  world  is  a  completed  order.  If  such  order  implies 
spirit,  then,  there  must  be  a  completed  mind.  As  for  the 
compatibility  of  such  completeness  with  freedom,  the  very 
reasons  which  make  Bergson  to  see  real,  active,  free  spirit- 
ual life  except  in  a  present  which  has  the  past  in  it,  make  it 
impossible  for  the  idealist  to  see  the  perfection  of  such  free- 
dom except  in  a  living  present  charged  not  only  with  the 
whole  past,  but  with  the  whole  future  as  well.  The  whole 
of  reality  must  interpenetrate  as  Bergson  makes  the  reality 
which  has  so  far  elapsed  do.  That  interpenetration,  with 
its  inner  activity,  movement  and  freedom,  makes  up  the 
content  of  what  the  religious  consciousness  has  conceived 
as  the  perfect  mind  of  God.  Its  inward  intensity  is  God's 
perfect  life,  which  is  also  ours  so  far  as  we  are  both  good 
and  great. 


2O4  THE  MONIST. 

With  the  claim  then,  which  is  put  forward  by  most  of 
Bergson's  following  here  and  elsewhere  that  his  philosophy 
is  both  true  and  "new,"  we  cannot  agree.  So  far  as  we 
have  been  able  to  examine  it,  it  differs  from  other  idealism 
in  an  essentially  philosophical  way  only  when  it  has  some- 
thing to  say  which  is  indefensible.  Bergson  has  done  im- 
portant work  in  matters  which  in  this  paper  we  have  had 
to  pass  over  because  they  are  extra-philosophical.  He  has 
done  great  work  in  psychology ;  and  he  has  also  done  great 
work  in  the  interpretation  of  the  actual  story  of  evolution, 
by  bringing  out  new  facts  there  which  could  easily  be 
shown  to  be  as  compatible  with  the  classical  idealistic  de- 
fense of  spirit  as  with  his  own.  That  kind  of  work  is  the 
limit,  it  seems  to  us,  of  his  service;  except  indeed  it  be  a 
service  to  have  presented  a  great  deal  of  the  substance  of 
idealism  from  an  angle  so  entirely  fresh  as  almost  to  trans- 
port the  reader  into  the  idealistic  center  of  vision,  without 
his  suspecting  that  he  is  there.  We  are  not  convinced  that 
this  is  a  small  service.  Nay,  rightly  understood,  there  is 
perhaps  no  greater. 

J.  W.  SCOTT. 

UNIVERSITY  OF  GLASGOW. 


THE  PRESENT  STATUS  OF  THE  UNCONSCIOUS. 


^HE  unconscious  is  a  topic  with  which  some  writers 
JL  have  tried  to  coquet  freely,  which  others  have  shunned 
scrupulously,  and  which  still  others  have  approached  in  a 
true  scientific  spirit  in  the  endeavor  to  find  out  precisely 
what  it  is  and  how  well  it  can  explain  the  phenomena  that 
usually  come  under  its  name.  The  motives  that  have  led 
men  to  write  about  the  unconscious  have  differed  so  widely 
that  it  is  not  surprising  to  find  the  works  of  some  fantas- 
tical, and  those  of  others  useful  for  practical  purposes  only 
but  devoid  of  scientific  information.  The  interests  of  the 
former  type  have  been  purely  metaphysical.  Their  object 
being  to  discover  unity  and  continuity  in  the  universe,  they 
have  postulated  the  unconscious  as  the  absolute  principle. 
The  interests  of  the  latter,  who  are  chiefly  physicians,  have 
been  entirely  practical.  Naturally  they  have  considered 
and  still  continue  to  consider  the  subconscious  from  the 
functional  point  of  view.  What  its  real  nature  is  and  how 
it  is  related  to  consciousness  as  such,  is  a  problem  that  does 
not  fall  within  their  sphere.  The  third  group  comprises 
the  few  psychologists  who,  motivated  by  a  true  scientific 
and  progressive  spirit,  are  seeking  to  discover  the  "what" 
and  the  "how"  of  subconscious  activities.  As  a  result  of 
these  diverse  attitudes  we  find  that  one  writer  thought 
the  unconscious  a  topic  sufficiently  great  and  all-embracing 


2O6  THE  MONIST. 

to  deserve  three  large  volumes,  while  another  laconically 
dismisses  it  in  three  monosyllabic  words.  But  the  fact 
that  we  still  have  the  problem  on  our  hands  shows  that 
neither  the  three  volumes  of  von  Hartmann  nor  the  three 
words  of  Miinsterberg1  have  either  brought  us  any  nearer 
to  its  solution  or  diminished  in  the  least  our  unavoidable 
duty  as  scientific  psychologists  to  search  out  the  cause  of 
unconscious  activities  and,  if  possible,  to  bridge  the  gap 
between  the  conscious  and  the  unconscious. 

As  an  example  of  the  psychologist  who  avoids  all  dis- 
cussion of  the  subconscious,  I  need  only  mention  Titchener, 
who,  after  defining  the  subconscious  as  "an  extension  of 
the  conscious  beyond  the  limits  of  observation,"2  goes  on  to 
say  that  it  is  always  >a  matter  of  inference  and  therefore 
"it  can  not  be  a  part  of  the  subject  matter  of  psychology." 
It  is  merely  employed  as  an  explanatory  concept,  he  de- 
clares, but  there  are  two  reasons  against  its  use  in  psy- 
chology: First,  that  the  scientific  psychologist,  like  the 
scientist  in  general,  is  not  called  upon  to  explain  anything ; 
and  secondly,  the  introduction  of  this  inferential  concept 
may  lead  to  danger  inasmuch  as  it  is  "impossible  to  draw 
the  line  between  legitimate  and  illegitimate  inference." 

As  an  example  of  the  thinker  who  interprets  all  mental 
phenomena  in  terms  of  the  subconscious,  I  may  mention 
von  Hartmann  and  Schopenhauer.  The  former  endows 
the  entire  universe  with  an  unconscious  mind,  declaring 
it  to  be  the  absolute  principle  which  operates  in  all  things 
organic  and  inorganic.  But  as  James  says,  "his  logic  is 
so  lax  and  his  failure  to  consider  the  most  obvious  alter- 
native so  complete,  that  it  would ....  be  a  waste  of  time 
to  look  at  his  arguments  in  detail."  Nor  are  the  views  of 
Schopenhauer  much  more  reasonable.  According  to  him 
every  sense  organ  unconsciously  infers  its  impinging  stim- 

1  Psychotherapy,  p.  125. 

2  A  Beginner's  Psychology,  p.  327. 


THE  PRESENT  STATUS  OF  THE  UNCONSCIOUS.          2O7 

ulus  "as  the  only  possible  cause  of  some  sensation  which  it 
unconsciously  feels."3 

But  the  theory  of  the  subconscious  dates  back  farther 
than  von  Hartmann  and  Schopenhauer.  Although  Wein- 
gartner  traces  it  to  Plato  and  Plotinus,  we  may  say  that  it 
received  its  first  definite  formulation  at  the  hands  of  Leib- 
niz in  his  conception  of  the  petites  perceptions  which  play 
the  main  role  in  psychic  activity.  These  subliminal  per- 
ceptions are  individually  too  faint  to  arouse  consciousness, 
according  to  Leibniz,  but  in  their  totality  they  come  to  a 
high  degree  of  consciousness.  To  use  his  own  words,  "the 
belief  that  there  are  no  other  perceptions  in  the  soul  than 
those  of  which  it  is  conscious,  is  a  great  source  of  error." 

Kant's  view  of  the  subconscious  is  somewhat  analogous 
to  that  of  some  modern  authors,  particularly  Lipps.  He 
declares:  "To  have  sensations  and  not  to  be  conscious  of 
them  is  a  contradiction,  for  how  can  we  know  that  we  have 
them,  when  we  are  not  conscious  of  them?  But  we  may 
infer  we  have  had  a  sensation  or  a  perception,  although 
we  were  not  immediately  aware  of  it."4  Such  perceptions 
Kant  calls  "vague,"  and  their  field  he  declares  is  much 
broader  than  that  of  the  clear  and  definite  ones. 

Turning  to  the  English  school,  we  find  Sir  William  Ham- 
ilton asserting  that  although  he  does  not  wish  to  maintain 
that  all  consciousness  is  the  product  of  unconscious  percep- 
tions, and  that  knowledge  as  such  is  the  product  of  the  un- 
known and  the  unknowable,  still  we  must  confess,  he  says, 
"that  there  are  things  which  we  neither  know  nor  can  know 
directly,  but  which  manifest  their  existence  indirectly 
through  the  medium  of  their  effects."6  Hence,  since  the 
mind  in  its  behavior  manifests  processes  of  which  it  is  un- 
conscious, these  processes  must  have  come  about  through 

3  James,  Psychology,  I,  p.  170. 

4  Soewenfeld,  Bewusstsein  und  psychisches  Geschehen,  p.  2. 

5  Carpenter,  Principles  of  Physiology,  p.  518. 


2O8  THE  MONIST. 

some  modification  of  mind,  and  this  may  be  called  the  un- 
conscious. Or  as  Carpenter  believes,  Hamilton  meant  by 
this  "unconscious  cerebration." 

Maudsley  undertakes  to  interpret  the  greater  part  of 
conscious  behavior  in  terms  of  unconscious  psychophysical 
processes.  He  observes  that  almost  from  the  moment  of 
birth  the  sensorium  receives  multifarious  impressions  which 
it  assimilates  unconsciously,  and  makes  use  of  them  in  a 
purely  mechanical  manner  even  in  so-called  intelligent  ac- 
tivity. Even  our  general  and  abstract  concepts  are  de- 
veloped unconsciously;  in  short,  "the  process  upon  which 
our  thinking  depends,"  he  says,  "goes  on  of  its  own  accord, 
without  our  awareness." 

Carpenter  devotes  a  whole  chapter  to  unconscious  cere- 
bration and  declares  that  since  there  is  reason  to  believe 
that  the  greater  part  "of  our  intellectual  activity" — whether 
it  be  reasoning  or  imagination — is  essentially  automatic, 
it  is  not  unlikely  that  "the  cerebrum  may  act  upon  impres- 
sions transmitted  to  it,  and  may  elaborate  intellectual  re- 
sults, such  as  might  have  been  attained  by  the  intentional 
direction  of  our  minds  to  the  subject,  without  any  con- 
sciousness on  our  own  part"  (p.  515). 

This  view  was  subsequently  taken  up  by  Huxley  and 
made  to  explain  all  intellectual  activity.  Noticing  that 
epileptics  can  execute  complex  actions  without  having  any 
memory  of  them  upon  recovery,  and  also  that  somnam- 
bulists can  write  letters  and  compose  original  verse  while 
in  their  so-called  sleeping  state,  he  concluded  that  since 
"these  cases  are  examples  of  purposive  and  intelligently 
controlled  action  taking  place  without  consciousness,  it 
would  seem  to  follow  that  the  mere  mechanism  of  the 
nervous  system"  is  all  that  is  needed  for  the  execution  of 
such  actions,  independently  of  all  consciousness  and  con- 
scious guidance;  and,  therefore,  we  are  compelled  to  as- 
sume that  when  similar  actions  are  accompanied  by  con- 


THE  PRESENT  STATUS  OF  THE  UNCONSCIOUS.          2OO, 

sciousness,  the  nervous  mechanisms  are  the  only  essential 
conditions,  and  "consciousness  is  a  superfluous  accom- 
paniment, so  far  as  the  causal  sequence  is  concerned."8 

It  is  obvious  from  the  above  that  not  only  do  earlier 
writers  disagree  on  how  the  subconscious  functions  but 
they  even  differ  with  respect  to  its  essential  nature. 

Turning  to  modern  authors  we  find  that  among  them, 
too,  there  are  almost  as  many  views  as  writers  on  the  sub- 
ject. On  the  one  hand,  men  like  Freud  declare  that  sub- 
conscious phenomena  are  due  to  dissociated  and  suppressed 
ideas;  that  these  unconscious  ideas  are  active,  though  the 
individual  may  not  be  aware  of  them  while  going  through 
the  bodily  actions  of  which  those  ideas  are  the  prime 
causes.7  On  the  other  hand,  Sidis  retorts  that  the  existence 
of  unconscious  ideas  is  inconceivable,  for  "ideas  are  essen- 
tially of  a  conscious  nature"  f  hence  their  introduction  into 
psychology  is  a  self-contradictory  concept.  The  subcon- 
scious, according  to  him,  is  rather  "a  diffused  consciousness 
below  the  margin  of  personal  consciousness."9  Again, 
writers  like  Prince  and  James  conceive  of  the  subconscious 
as  the  outerlying  fringe  of  consciousness,  as  dim  conscious- 
ness, or  better  still,  as  the  base  of  a  cone,  the  apex  of  which 
is  attentive  consciousness.  While  Irving  King  refutes  this 
view,  declaring  that  consciousness  either  exists  or  does  not 
exist,  that  "it  may  be  more  intense  at  one  moment  than  at 
another ....  But  at  any  one  moment  it  is ....  a  unitary 
existence  without  parts  which  might  be  thought  of  as  clus- 
tering about  a  center  with  different  degrees  of  intensity 
and  adhesion."1  Finally,  while  Sidis  mocks  unconscious 
cerebration,  characterizing  nerve  currents,  nerve-paths  and 

"McDougal,  Body  and  Mind,  pp.  109-110. 

7  Freud,  "A  Note  on  the  Unconscious,  "Proc.  Soc.  Psy.  Res.,  XXVI,  1912, 
p.  314. 

8  Sidis,  "The  Theory  of  the  Unconscious,"  Proc.  Soc.  Psy.  Res.,  XXVI, 
1912,  p.  337. 

9  Ibid.,  p.  319. 

10  "The  Problem  of  the  Subconscious,"  Psychol.  Rev.,  XIII,  1906,  p.  43. 


2IO  THE  MONIST. 

neurograms  as  "figments  of  imagination/'11  Ribot  main- 
tains that  the  psychological  aspects  of  the  subconscious 
play  but  a  secondary  role,  that  they  are  a  result,  an  effect 
of  physiological  or  neural  processes.12 

No  small  part  of  the  above  controversy  and  disagree- 
ment is  due  to  the  fact  that  the  term  subconscious  has  been 
used  in  widely  different  senses.  Prince  gives  no  less  than 
six  different  meanings  in  which  the  term  has  been  em- 
ployed. 

1.  The  word  subconscious  has  been  employed  to  describe 
that  portion  of  our  field  of  consciousness  which  at  any 
moment  is  outside  the  focus  of  attention.    In  this  sense  it 
is  equivalent  to  James's  fringe  of  consciousness. 

2.  The  second  meaning  asserts  that  the  subconscious  is 
composed  of  ideas  that  are  dissociated  or  split  off  from 
the  personal  consciousness,  i.  e.,  the  focus  of  attention; 
that  though  the  subject  is  unaware  of  their  existence  they 
are  none  the  less  active,  and  that  "they  form  a  conscious- 
ness coexisting  with  the  primary  consciousness,  and  thereby 
a  doubling  of  conscious  results." 

3.  According  to  the  third  meaning  of  the  term,  "sub- 
conscious states  are  conceived  of  as  becoming  synthesized 
among  themselves,  forming  a  larger  self-conscious  personal- 
ity, to  which  the  term  self  is  given."    These  subconscious 
states  are  personified  by  the  people  who  hold  this  view, 
and  referred  to  as  the  "subconscious  self"  or  "the  hidden 
self." 

4.  The  fourth  view  conceives  the  subconscious  as  "in- 
cluding all  those  past  conscious  states  which  are  either  for- 
gotten and  cannot  be  recalled,  or  which  may  be  recalled  as 
memories,"  their  non-existence  being  due  to  the  fact  that 
they  are  crowded  out  of  consciousness  by  the  bulk  of  pres- 
ent experience. 

11  Op.  cit.,  p.  325. 

12  "A  Symposium  on  the  Subconscious,"  Journ.  Abnorm.  Psychol.,  II,  1907, 
p.  37. 


THE  PRESENT  STATUS  OF  THE  UNCONSCIOUS.          211 

5.  The  fifth  view,  which  is  that  of  Frederick  Myers, 
"declares  that  subconscious  ideas,  instead  of  being  mental 
states  dissociated  from  the  main  personality,  are  the  main 
reservoir  of  consciousness,  and  the  personal  consciousness 
is  a  subordinate  stream  flowing  out  of  this  great  storage." 
In  short,  we  have  within  us  a  great  tank  of  con- 
sciousness, but  are  aware  of  only  a  small  portion  of  it. 

6.  The  sixth  and  final  view  asserts  that  there  are  no 
psychical  elements  in  subconscious  phenomena  at  all,  that 
automatic  writing  and  speech,  the  solution  of  mathematical 
problems  in  sleep,  and  the  carrying  out  of  post-hypnotic 
suggestion  "are  the  result  of  pure  neural  processes/'  un- 
accompanied by  any  mentation  whatever.13 

Before  presenting  the  detailed  arguments  in  support  of 
the  above  theories,  let  us  hastily  review  some  of  the  more 
common  subconscious  phenomena  in  order  that  we  may 
have  freshly  before  our  minds  the  facts  which  these  theories 
endeavor  to  explain. 

The  main  test  that  a  by-gone  experience  was  accom- 
panied by  consciousness  is  memory.14  The  ability  to  recall 
an  experience  without  the  artificial  aid  of  suggestion  or 
abstraction,  shows  that  the  individual  was  conscious  of 
that  experience  at  the  time  he  underwent  it.  But  memory 
is  composed  of  three  factors :  registration,  conservation  and 
reproduction.  Something  must  be  impressed  on  the  sen- 
sorium  in  order  to  be  recalled,  and  it  must  also  be  conserved 
in  some  form.  The  question  therefore  arises:  Does  every 
impression,  however  faint  it  may  be,  stir  up  a  pulse  of  con- 
sciousness which  is  immediately  forgotten  because  of  its 
brevity  or  faintness,  or  can  reproducible  impressions  be 
made  without  the  least  awareness  at  the  time  being?  And 
if  so,  how  are  they  conserved  ?  Daily  observation  and  lab- 
oratory experiments  demonstrate  that  perceptions  of  the 

18  "A  Symposium  on  the  Subconscious,"  Jour.  Abnorm.  Psychol.,  p.  22. 
"McDougal,  Of.  tit.,  p.  109. 


212  THE  MONIST. 

environment  of  which  the  individual  did  not  have  the  least 
awareness,  may  be  conserved.  You  may  pass  an  acquain- 
tance on  the  street  without  being  aware  of  him  at  the  time, 
but  two  or  three  minutes  later  it  will  suddenly  dawn  on 
you  that  you  had  seen  your  friend  so  and  so.  Again  in 
hypnosis,  by  means  of  automatic  writing  or  abstraction, 
people  have  been  able  to  recall  paragraphs  in  the  news- 
papers read  through  casual  glances  without  awareness 
thereof.  Or  the  experiment  may  be  put  under  controlled 
conditions,  by  having  the  subject  take  a  brief  survey  of 
the  room,  and  then  while  blindfolded  dictate  as  detailed  a 
description  of  it  as  he  can.  Thereafter  if  he  is  hypnotized 
and  asked  to  describe  the  room  once  more,  "it  is  often  quite 
surprising,"  says  Morton  Prince,  "to  note  with  what  detail 
the  objects  which  almost  entirely  escaped  conscious  ob- 
servation are  subconsciously  perceived  and  remembered."1 
Another  method  of  proving  the  conservation  of  uncon- 
scious experiences  is  to  have  a  person  concentrate  his  atten- 
tion by  giving  him  something  to  read  or  an  arithmetical 
problem  to  perform,  and  while  he  is  so  engaged  to  place 
cautiously  and  surreptitiously  objects  within  his  peripheral 
field  of  vision.  After  their  removal  he  is  asked  to  state 
in  detail  what  he  has  seen.  Invariably  he  is  unable  to  men- 
tion any  of  these  surreptitiously  introduced  objects.  On 
being  hypnotized,  however,  he  mentions  them  with  con- 
siderable accuracy  and  readiness.16 

Automatic  writing  furnishes  another  group  of  facts 
which  presuppose  subconscious  processes.  If  into  the  an- 
esthetic hand  of  an  hysterical  person  a  pencil  be  put  the 
hand  will  commence  to  write  mechanically,  and  the  subject 
will  observe  the  movements  of  the  hand  as  if  that  member 
belonged  to  some  other  person.  Nor  will  the  patient  rec- 
ognize the  written  ideas  as  his,  but  if  he  is  hypnotized  he 

15  Prince,  The  Unconscious,  p.  53. 
« Ibid. 


THE  PRESENT  STATUS  OF  THE  UNCONSCIOUS.          213 

will  claim  them  immediately  and  explain  what  he  meant 
by  them.  Sometimes  the  two  hands  of  the  same  subject 
may  be  made  to  give  written  expression  to  two  different 
kinds  of  mental  content. 

Perhaps  the  most  interesting  and  common  source  of 
subconscious  phenomena  is  somnambulism.  People  in  this 
state  have  been  known  to  perform  the  most  delicate  feats 
of  physical  skill,  such  as  walking  across  roofs  on  narrow 
planks.  Others  have  been  known  to  perform-  events  that 
in  waking  life  require  a  great  deal  of  intelligence, — such 
as  writing  letters  or  verse.  Yet  they  have  no  memory  for 
these  events.  The  question  arises :  Are  these  highly  com- 
plex mental  activities  performed  mechanically,  without  any 
mentation,  or  are  they  consciously  performed,  but  forgotten 
in  waking  life  because  dissociated  from  the  personal  con- 
sciousness ? 

Post-hypnotic  suggestion  is  no  less  a  mystery  than  com- 
plete change  of  personality.  An  individual  is  hypnotized 
and  is  told  that  at  a  fixed  time  after  he  awakens — be  it  sev- 
eral minutes,  an  hour  or  a  day  later — he  is  to  do  a  certain 
deed.  He  is  awakened  and  asked  if  he  remembers  any- 
thing that  had  been  said  to  him  during  the  hypnosis.  He 
does  not.  He  is  permitted  to  depart  and  goes  about  his 
business  in  his  customary  manner.  But  precisely  at  the 
fixed  time  he  will  carry  out  the  post-hypnotic  suggestion, 
whether  it  be  to  ask  for  a  pail  of  coal  in  a  jewelry  store  or 
to  purchase  an  overcoat  in  summer.  When  he  is  asked  why 
he  did  this  he  can  only  reply  that  something  within 
prompted  him  to  it,  that  he  felt  it  was  a  voluntary  deed. 

Whether  such  a  case  as  that  of  the  Rev.  Ansel  Bourne 
would  fall  into  the  group  of  epileptic  phenomena  or  not 
matters  little.  In  both  instances  we  know  that  the  subject 
will  go  through  many  complicated  activities,  denoting  a 
high  degree  of  consciousness  or  the  presence  of  the  cus- 
tomary kind  of  intelligence  as  judged  by  the  adaptation  of 


214  THE  MONIST. 

the  subject  to  his  environment,  yet  in  neither  case  does  the 
normal  personality  have  a  memory  for  these  experiences. 

How  are  these  phenomena  to  be  explained  in  the  light 
of  modern  psychology? 

Two  general  theories  are  proposed:  the  psychical  and 
the  physiological.  And  it  is  to  these  two  that  the  six  fore- 
going views  can  be  reduced  after  we  eliminate  Myers's 
metaphysical  notion  which  conceives  of  the  subconscious  as 
the  reservoir  of  all  consciousness,  and  that  other  view  which 
interprets  the  subconscious  as  the  larger  self-conscious  per- 
sonality. 

Freud,  Sidis  and  Janet  may  be  taken  as  the  chief  ex- 
ponents of  the  psychological  theory  of  the  subconscious, 
while  Pierce,  Jastrow  and  Ribot,  not  to  mention  a  host  of 
others,  hold  to  the  physiological  view.  The  former  trio  in 
one  form  or  another  declare  that  the  subconscious  is  dis- 
sociated consciousness,  or  awareness  that  is  dissociated 
from  the  synthesizing  personality,  and  that  this  aware- 
ness exists  in  consciousness  in  a  latent  form  all  the  time. 
The  latter  maintain  that  not  only  is  it  unscientific  to  speak 
of  latent  ideas  and  latent  feelings,  but  that  there  is  no 
causal  relation  among  psychic  elements  at  all,  that  the 
explanation  of  unconscious  phenomena  must  be  sought  in 
neural  processes. 

Let  us  examine  their  views  individually. 

Freud  suggests  that  the  term  "conscious"  should  be 
applied  to  the  perception  which  is  present  to  our  conscious- 
ness and  of  which  we  are  aware, — while  the  latent  per- 
ceptions should  be  denoted  by  the  term  "unconscious." 
"Hence  an  unconscious  idea  is  one  of  which  we  are  not 
aware,  but  the  existence  of  which  we  are  nevertheless  ready 
to  admit  because  of  other  proofs  or  signs."17  This  un- 
conscious idea,  though  latent  in  the  sense  that  it  does  not 
attain  awareness,  is  by  no  means  inactive  while  in  the 

"  Op.  tit.,  p.  3i3ff. 


THE  PRESENT  STATUS  OF  THE  UNCONSCIOUS. 

mind.  That  unconscious  ideas  are  active,  undergoing  com- 
bination and  recombination  among  themselves,  is  demon- 
strated by  the  hysterical  patient.  "If  she  is  executing  the 
jerks  and  movements  constituting  her  fit,"  says  Freud,  "she 
does  not  consciously  represent  to  herself  the  intended  ac- 
tions, she  may  perceive  those  actions  with  the  detached 
feelings  of  an  onlooker.  Nevertheless,  analysis  will  show 
that  she  is  acting  her  part  in  the  characteristic  reproduc- 
tion of  some  incident  in  her  life,  the  memory  of  which  was 
unconsciously  present  during  the  attack." 

Freud  distinguishes  two  kinds  of  latent  ideas:  those 
which  enter  consciousness  with  no  difficulty  whatever,  and 
those  which  do  not  penetrate  into  consciousness  however 
strong  they  may  be.  The  first  type  constitute  the  fore- 
conscious,  the  second  type  the  unconscious.  "The  term 
unconscious,"  he  says,  "now  designates  not  only  latent 
ideas  in  general,  but  especially  ideas  with  a  certain  dynamic 
character,  ideas  keeping  apart  from  consciousness,  in  spite 
of  their  intensity  and  activity."  In  explaining  the  phenom- 
enon of  double  personality  Freud  would  say  that  it  is  a 
shifting  of  consciousness,  an  oscillation  between  two  dif- 
ferent psychical  complexes  which  become  conscious  and 
unconscious  alternately.18 

But  the  question  still  remains :  Why  does  foreconscious 
activity  pass  into  consciousness  with  no  difficulty,  while  an 
unconscious  activity  is  cut  off  from  consciousness?  (It  is 
to  be  noticed  here  that  he  no  longer  speaks  of  foreconscious 
and  unconscious  ideas,  but  replaces  the  word  idea  by  the 
term  activity.)  In  answering  this  question  he  says  that 
frequently  when  we  try  to  represent  an  idea  or  a  situation 
to  ourselves  we  become  aware  of  a  distinct  feeling  of  re- 
pulsion which  must  be  overcome ;  and  when  we  try  to  inject 
such  an  idea  into  a  patient,  we  get  the  signs  of  what  may 
be  called  his  resistance  to  it.  "So  we  learn  that  the  un- 

i*Ibid.,  p.  315. 


2l6  THE   MONIST. 

conscious  idea  is  excluded  from  consciousness  by  living 
forces,  which  oppose  themselves  to  its  reception ;  while  they 
do  not  object  to  other  ideas, — the  foreconscious  ones."  At 
the  present  state  of  our  knowledge,  therefore,  he  suggests 
the  following  as  the  most  probable  theory  that  can  be 
formulated:  "The  unconscious  is  a  regular  and  inevitable 
phase  in  the  processes  constituting  our  psychical  activity; 
every  psychical  act  begins  as  an  unconscious  one,  and  it 
may  either  remain  so,  or  go  on  developing  into  conscious- 
ness, according  as  it  meets  with  resistance  or  not."  Freud 
illustrates  this  view  by  referring  to  ordinary  photography. 
The  first  stage  of  the  photograph  is  the  "negative" ;  every 
picture  has  to  pass  through  the  negative  process ;  and  those 
negatives  which  on  examination  prove  to  be  satisfactory 
are  admitted  to  the  positive  process,  ending  in  the  picture ; 
those  which  do  not  are  rejected.  Such  is  the  distinction 
between  the  foreconscious  and  unconscious  ideas  or  activ- 
ities. In  reply  to  his  critics  that  an  unconscious  idea  is 
inconceivable,  he  declares  that  "the  existence  of  an  un- 
conscious consciousness  is  still  more  objectionable." 

Sidis  gives  three  definitions  of  the  subconscious  which 
may  be  called  the  medico-popular,  the  metaphysical  and 
the  scientific,  respectively.  In  one  place  he  defines  the  sub- 
conscious "as  mental  processes  of  which  the  individual  is 
not  directly  aware."  In  another  place  he  refers  to  it  "as  a 
diffused  consciousness  below  the  margin  of  personal  con- 
sciousness"; and  on  a  third  occasion  he  defines  it  "as  con- 
sciousness below  the  threshold  of  attentive  personal  con- 


sciousness." 


The  subconscious  like  the  conscious  may  be,  according  to 
Sidis,  of  three  types :  desultory,  synthetic,  or  recognitive.20 
Sidis  would  almost  banish  the  term  subconscious  from 
literature,  and  what  is  commonly  called  subconscious  he 

19  The  Theory  of  the  Unconscious,  p.  319. 

20  Psychology  of  Suggestion,  p.  201. 


THE  PRESENT  STATUS  OF  THE  UNCONSCIOUS.          217 

would  call  conscious,  while  that  which  is  commonly  known 
as  the  conscious  he  would  call  the  self-conscious.  The  self- 
conscious  is  that  form  of  mentation  which  is  aware  of  itself ; 
it  is  "the  knowledge  of  consciousness  within  the  same  mo- 
ment of  consciousness."  and  in  that  sense  it  is  identical 
with  personality,21  On  the  other  hand,  the  secondary  or 
subconscious  self  must  not  be  regarded  as  an  individual; 
"it  is  only  a  form  of  mental  life";  it  is  a  coordination  of 
many  series  of  moments-consciousness," — i.  e.,  pulses  of 
consciousness.  And  it  is  these  moments-consciousness  that 
are  at  the  heart  of  the  subconscious.  Therefore,  subcon- 
scious experience  is  not  wn-conscious  experience.  The  proof 
is  this:  Normal  memory  is  a  reproduction  of  conscious 
states.  Now,  when  a  subject  is  hypnotized  he  can  be  made 
to  recall  an  experience  which  he  does  not  remember  in  his 
waking  state;  and  in  this  he  displays  memory  like  normal 
memory.  Therefore,  we  have  proof  that  his  experience 
was  accompanied  by  consciousness  at  the  time  it  occurred. 
Or,  to  use  Sidis's  own  words,  "that  in  subconscious  states 
there  is  really  present  a  subconscious  consciousness."22 

It  is  to  be  noticed  that  this  is  not  the  same  thing  as 
saying  that  the  ego  or  the  personality  was  aware  of  that 
experience,  but  on  the  contrary,  there  was  an  awareness 
of  which  the  attending  self  had  no  consciousness. 

Having  eliminated  the  subconscious  from  literature, 
therefore,  there  are  only  two  forms  of  awareness  to  be 
considered,  according  to  Sidis:  consciousness  as  such  and 
self-consciousness.  The  difference  between  these  two  states 
may  be  made  clear  in  the  words  of  Hoffding.  "Many  feel- 
ings and  impulses  stir  within  us,  without  our  clearly  ap- 
prehending their  nature  and  direction.  A  man  who  has 
this  feeling  does  not  know  what  is  astir  in  him;  perhaps 
others  see  it,  or  he  himself  gradually  discovers  it;  but  he 

« ibid.,  P.  198. 

22  The  Theory  of  the  Unconscious,  p.  331. 


2l8  THE  MONIST. 

has  the  feeling  that  his  conscious  life  is  determined  in  a 
particular  way."23  What  Hoffding  means  is  that  there  are 
"mental  states  of  which  we  have  consciousness,  but  which 
do  not  reach  the  personal  consciousness."  This  is  the  dis- 
tinction that  Sidis  makes  between  the  subconscious  and  the 
self-conscious. 

It  naturally  follows  from  the  above,  and  there  are  many 
facts  in  support  of  the  conclusion,  that  "the  stream  of  sub- 
waking  consciousness  is  broader  than  that  of  the  waking 
consciousness,  so  that  the  submerged  subwaking  self  knows 
the  life  of  the  upper,  primary  self,  but  the  latter  does  not 
know  the  former."  He  admits,  however,  that  there  are  cases 
on  record  showing  that  the  two  streams  may  flow  in  sep- 
arate channels ;  that  the  two  selves  may  be  ignorant  of  each 
other.24 

On  the  basis  of  the  foregoing  view,  the  phenomenon  of 
double  personality  is  not  difficult  to  explain,  thinks  the 
author.  When  a  sufficient  number  of  the  submerged  mo- 
ments of  consciousness  have  accumulated  they  tend  to  be- 
come synthesized,  to  group  themselves  in  constellations  and 
break  forth  into  attentive  consciousness,  as  do  hallucina- 
tions, for  example.  In  this  manner  the  secondary  con- 
sciousness attains  self-consciousness,  and  appears  as  a  new 
and  independent  personality.  Now  and  then  it  "rises  to 
the  surface  and  assumes  control  over  the  current  of  life." 
This  secondary  self  is  aware  of  and  passes  judgment  on  the 
primary  self,  while  the  latter,  when  it  returns,  has  not  the 
least  knowledge  of  the  intruding  ego. 

It  is  apparent  that  the  views  of  Freud  and  Sidis  are 
essentially  the  same.  The  argument,  therefore,  that  exists 
between  these  two  writers  is  purely  verbal  and  meaningless. 
There  is  no  fundamental  difference  between  an  unconscious 
idea  and  an  unconscious  moment-consciousness,  or  even  an 

23  Quoted  by  Sidis  in  The  Theory  of  the  Unconscious,  p.  339. 

24  Psychology  of  Suggestion,  p.  198. 


THE  PRESENT  STATUS  OF  THE  UNCONSCIOUS.          219 

unconscious  consciousness.  There  may  be  a  difference  in 
quantity  but  not  in  quality.  Yet  we  find  Freud  declaring 
that  if  philosophers  find  it  difficult  to  accept  the  existence 
of  unconscious  ideas,  the  existence  of  unconscious  con- 
sciousness is  still  more  objectionable.  To  which  Sidis  re- 
torts: "An  idea  is  essentially  of  a  conscious  nature.  To 
speak,  therefore,  of  unconscious  ideas  is  self-contradictory, 
— it  is  equivalent  to  the  assumption  of  an  unconscious  con- 
sciousness."2 I  do  not  see  why  Sidis  should  find  fault  with 
this  conclusion,  since  it  is  the  very  assumption  with  which 
he  opens  his  own  discussion  on  the  theory  of  the  subcon- 
scious. There  he  defines  the  subconscious  as  mental  proc- 
esses of  which  the  individual  is  not  aware.  But  what  are 
mental  processes  if  not  ideas,  images  and  perceptions? 
His  definition,  therefore,  turns  out  to  be  precisely  the  same 
as  Freud's. 

Though  the  views  of  neither  of  these  men  lend  them- 
selves to  acceptance  in  the  light  of  the  fundamental  postu- 
late of  psychology, — namely,  that  every  psychosis  has  its 
neurosis  (but  not  the  reverse),  still  Freud's  doctrine  of  the 
subconscious  is  somewhat  more  palatable  than  that  of  Sidis. 
At  least  it  is  capable  of  interpretation  in  terms  of  our 
existing  knowledge  of  neurology;  it  does  not  assume  too 
much  and  does  not  pretend  to  offer  a  solution  of  all  mental 
phenomena.  The  view  of  Sidis,  on  the  other  hand,  is  en- 
tirely out  of  harmony  with  the  fundamental  postulate  of 
psychology,  and  it  is  so  all-embracing  and  metaphysical 
in  nature  as  almost  to  remind  one  of  the  teachings  of  von 
Hartmann. 

This  is  demonstrated  by  the  vigorous  but  wholly  un- 
justifiable attack  that  Sidis  launches  against  the  theory  of 
unconscious  cerebration.  This  doctrine,  it  will  be  recalled, 
states  that  physiological  processes  may  go  on  in  the  sen- 
sorium  which  enable  the  organism  to  adapt  itself  to  its 

28  The  Theory  of  the  Unconscious,  p.  337. 


22O  THE  MONIST. 

environment  without  any  consciousness  on  its  part.  If  this 
is  so,  says  Sidis,  then  there  is  no  reason  why  similar  adap- 
tations which  are  accompanied  by  consciousness  should  not 
also  be  purely  mechanical  and  automatic.  If  the  writing 
of  letters  during  somnambulism  is  automatic,  then  the  cor- 
respondence of  waking  life  must  be  carried  on  in  the  same 
manner.  But,  he  asks,  "Can  unconscious  physiological 
processes  write  rational  discourse?  It  is  simply  wonderful, 
incomprehensible."  Assuming  that  every  sense  impression 
leaves  behind  it  a  trace,  or  a  slight  modification  of  nerve 
tissue,  he  says,  still  this  does  not  explain  why  it  is  that  a 
series  of  sensations,  ideas,  and  images  experienced  at  dif- 
ferent times  "should  become  combined,  brought  into  a  unity, 
felt ....  like  copies  of  one  original  experience."2  Conse- 
quently the  subconscious  must  be  considered  not  as  "an 
unconscious  physiological  automatism,"  but  as  "a  secondary 
consciousness,"  as  a  secondary  self.27 

It  is  doubtful  whether  the  theory  of  unconscious  cere- 
bration can  account  for  the  whole  of  unconscious  phenom- 
ena, but  there  is  no  doubt  that  Sidis's  notion  does  not 
account  for  even  a  fraction  of  it.  The  weakness  of  his 
logic  is  seen  in  such  passages  as  the  following :  "Reactions 
to  environment  accompanied  by  intelligence  in  us  are  rightly 
judged  to  have  the  same  accompaniment  in  others."  From 
which,  of  course,  he  would  have  us  draw  the  conclusion  that 
since  we  guide  our  footsteps  on  the  crowded  street,  or 
build  a  fire,  with  some  degree  of  waking  consciousness  or 
intelligence,  therefore  the  stroller  who  is  absorbed  in  his 
newspaper  or  the  somnambulist  who  builds  a  fire  is  also 
guided  by  awareness.  This  conclusion  would  be  correct, 
provided  the  proposition  on  which  it  is  based  were  not  re- 
versible. But  it  is  reversible.  It  is  precisely  because  we 
perform  many  so-called  intelligent  actions  (as  judged  by 

26  Psychology  of  Suggestion,  p.  125. 

27  Ibid.,  p.  128. 


THE  PRESENT  STATUS  OF  THE  UNCONSCIOUS.          221 

their  end  product)  without  any  consciousness  in  our  normal 
life,  that  we  rightly  claim  such  actions  to  be  devoid  of  in- 
telligence or  active  consciousness  in  other  beings  when 
performed  under  the  same  conditions,  or  when  those  beings 
are  abnormal.  The  above  proposition,  therefore,  stands 
incomplete  without  its  complement,  which  says  with  equal 
right:  Reactions  to  environment  not  accompanied  by  in- 
telligence and  attentive  consciousness  in  us  are  rightly 
judged  to  be  devoid  of  these  accompaniments  in  others, 
especially  when  those  others  can  give  no  direct  testimony 
as  to  the  presence  of  consciousness. 

Let  us  take  an  instance  of  so-called  intelligent  action 
which  is  accompanied  without  consciousness  so  far  as  mem- 
ory can  testify,  and  see  whether  it  must  be  explained  only 
on  the  basis  of  unconscious-consciousness,  or  whether  a 
better  explanation  cannot  be  found.  The  case  of  the  per- 
son who,  though  absorbed  in  his  magazine,  still  picks  his 
way  through  the  crowded  thoroughfare  will  do  quite  well. 
Now  two  wholly  unrelated  streams  of  thought  cannot  oc- 
cupy the  same  mind  at  the  same  time.  To  be  sure,  we  may 
dream  and  know  that  we  are  dreaming,  or  dream  and  ex- 
perience a  desire  to  wake  up ;  or  experience  both  the  music 
and  the  color  effect  of  an  opera  at  the  same  time ;  but  these 
are  somewhat  related  mental  complexes:  at  least  they  are 
logically  related.  We  certainly  can  not  solve  mathematical 
problems  and  at  the  same  time  think  of  our  social  engage- 
ments. Suppose,  then,  we  assume  that  our  hypothetical 
person  is  strongly  conscious  of  his  reading  material  only, 
and  is  oblivious  to  the  people  on  the  sidewalk.  How  shall 
we  explain  his  ability  to  pick  his  way  through  the  crowd? 

The  process  may  be  described  thus :  Two  sorts  of  stimuli, 
diverse  in  nature,  impinge  on  a  single  sensory  organ,  the 
eye.  The  one  stimulus  is  the  words  on  the  printed  page, 
which  falls  in  the  center  of  visual  regard ;  the  other  stim- 
ulus is  the  people  on  the  sidewalk,  perceived  in  the  periph- 


222  THE  MONIST. 

ery  of  vision.  Tracing  these  diverse  impressions  it  seems 
reasonable  to  assume  that  the  impression  of  the  printed  page 
is  conducted  to  the  occipital  lobes,  from  there  to  the  associa- 
tion centers,  and  from  these  the  nerve  energy  is  distributed 
to  the  other  centers,  including  the  motor  center,  so  that 
when  the  individual  reaches  the  bottom  of  the  page  he 
makes  a  conscious  and  coordinated  movement  with  the  hand 
to  turn  over  a  new  page.  The  other  vague  impressions 
which  fall  on  the  periphery  of  vision  are  also  conducted  to 
the  occipital  lobes,  but  the  path  to  the  association  centers 
is  already  blocked.  Naturally  the  nerve  energy  seeks  an 
outlet  in  some  other  direction.  Now  in  the  course  of  the 
individual's  life,  strong  association  bonds  had  been  formed 
between  visual  perceptions  of  the  kind  that  now  impinge 
on  the  periphery  of  his  vision  and  specific  organic  reactions, 
i.  e.,  seeing  a  body  coming  toward  him  and  moving  out  of 
its  way.  Psychophysically  speaking,  these  strong  associa- 
tion bonds  are  smoothly  working  conduction-paths  between 
the  visual  and  motor  centers.  Consequently  when  now  a 
visual  impression  of  the  same  kind  reaches  the  visual  cen- 
ter, it  immediately  discharges  itself  through  the  path  of 
least  resistance,  and  upon  reaching  the  motor  center  re- 
leases the  customary  response  which,  of  course,  is  an  adap- 
tation to  the  external  situation.  Since  all  this  takes  place 
without  reaching  the  association  centers,  we  have  uncon- 
scious "intelligent"  action. 

But  it  will  be  asked:  How  does  this  view  account  for 
the  fact  that  if  the  individual  is  hypnotized  he  can  be  made 
to  give  an  account  of  persons  and  places  he  had  passed 
though  wholly  oblivious  of  them  at  the  time  ?  The  answer 
to  this  question  involves  the  physiological  theory  of  the 
unconscious,  and  it  is  to  this  that  we  turn  next. 

Generally  stated,  this  theory  means  that  the  subcon- 
scious is  not  psychical  at  all,  but  purely  physiological ;  that 
the  presence  of  awareness  cannot  be  measured  by  adaptive- 


THE  PRESENT  STATUS  OF  THE  UNCONSCIOUS.          223 

ness  of  action,  for  there  are  many  glands  and  thousands  of 
cells  in  the  human  body  performing  very  complex  adaptive 
acts,  or  acts  designed  for  the  preservation  of  the  organism ; 
yet  we  do  not  say  that  these  are  mental.  Why  should  we 
expect  less  from  the  tissue  of  the  central  nervous  system 
than  we  do  of  all  other  tissue  ?  Or  in  the  words  of  Miinster- 
berg,  "Why  cannot  they,  too,  produce  physiological  proc- 
esses that  yield  to  well-adjusted  results?,"  i.  e.,  to  pur- 
posive sensorial  excitements  and  motor  impulses.28 

The  same  view  is  advanced  by  Ribot,  who  declares  that 
the  psychological  solution  of  the  unconscious  rests  on  the 
assumption  that  consciousness  is  a  quantity  which  may 
decrease  indefinitely  without  ever  reaching  zero.  But  there 
is  no  justification  for  this  postulate ;  he  says :  "The  results 
of  psychophysists  with  regard  to  the  threshold  of  con- 
sciousness seem  to  justify  the  opposite  view,  namely,  the 
perceptible  minimum  appears  and  disappears  instantane- 
ously, and  this  fact  is  unfavorable  to  the  hypothesis  of  an 
increasing  and  decreasing  continuity  of  consciousness." 
The  physiologic  solution,  moreover,  is  simple,  inasmuch  as 
it  maintains  that  subconscious  activity  is  purely  cerebral.29 

The  same  theory  is  shared  by  Jastrow.  He  deems  it  a 
fundamental  requisite  of  any  adequate  conception  of  the 
subconscious  that  it  make  a  vital  connection  with  normal 
mental  activity;  it  must  find  a  natural  place  in  an  evolu- 
tionary interpretation  of  psychic  functions,  and  like  normal 
activity  it  must  be  interpreted  in  terms  of  neural  disposi- 
tions. He  proposes  a  criterion,  therefore,  for  the  measure 
of  awareness.  "The  measure  of  awareness  that  shall  ac- 
crue to  any  given  nervous  structure  to  an  environmental 
situation,  in  order  to  render  the  response  advantageous.  . . 
will  be  determined  by  the  status  of  the  need  thus  satisfied 
in  the  organic  life  of  the  individual.  The  simplest,  recur- 

2SJourn.  Abnorm.  Psychol.,  II,  1907,  p.  30. 
29  Op.  cit.,  p.  35. 


224  THE  MONIST. 

rent  and  constant  needs  will  be  sufficiently  met  by  neural 
dispositions  without  conscious  states,  or  with  the  lowest 
type  thereof."30 

Irving  King  advances  the  same  view  and  almost  in  the 
same  words.  "Neural  processes,"  he  says,  "are  accom- 
panied by  psychical  processes  only  when  there  is  some  need 
for  them."3  According  to  him,  consciousness  is  definitely 
related  to  the  facilitation  of  reactions  and  adjustments  re- 
quired by  the  life  process,  but  which  the  automatic  arrange- 
ments of  the  organism  cannot  meet.  Consciousness  either 
is  or  is  not.  It  may  be  more  intense  at  one  moment  than 
at  another,  but  it  does  not  consist  of  different  degrees  of 
intensity,  as  James's  theory  of  the  "fringe"  would  imply. 
On  the  neural  side,  however,  we  do  have  a  system  which 
may  be  spatially  represented.  In  terms  of  this  system  con- 
sciousness is  not  "the  sum  of  the  organization  of  psychic 
elements,  but  rather  the  unique  and  single  accompaniment 
of  a  peculiar  organization  of  neural  processes."  From  this 
definition  it  follows  that  each  neural  element  will  determine 
the  complexion  of  consciousness.  If  it  is  in  the  center 
of  the  system,  it  has  dynamic  conscious  value ;  if  outside  of 
that  system,  it  has  potential  value  only.  The  subconscious, 
therefore,  is  not  to  be  conceived  as  dim  consciousness,  but 
rather  as  a  "physical  mass  of  neural  dispositions,  tensions 
and  actual  processes  which  are  in  some  degree,  perhaps 
organized,  the  remnants  of  habits  and  experiences,  both 
those  which  have  lapsed  from  consciousness  and  those  which 
have  never  penetrated  the  central  plexus." 

On  the  basis  of  these  definitions  it  becomes  fairly  easy 
to  understand  most  of  the  phenomena  that  come  under  the 
heads  of  the  conscious  and  the  unconscious.  "When  con- 
sciousness is  present,"  say  King,  "the  neural  processes  in- 
volved are  much  more  intense  than  otherwise."  The  dream 
consciousness  is  a  condition  in  which  the  central  activity 

80  The  Unconscious,  p.  411.  31  Op.  cit.,  p.  42. 


THE  PRESENT  STATUS  OF  THE  UNCONSCIOUS.          22$ 

is  so  subdued  that  more  or  less  fragmentary  neural  dispo- 
sitions are  aroused.  In  hypnosis,  again,  the  center  of  activ- 
ity is  shifted  in  more  or  less  degree,  resulting  in  the  tem- 
porary lapse  from  consciousness  of  some  processes  and  the 
incorporation  of  others  which  were  previously  mere  neural 
dispositions.  While  in  multiple  personality  there  are  one 
or  more  strongly  organized  potential  systems  of  neural 
elements  which,  under  appropriate  conditions,  can  sep- 
arately become  sufficiently  active  to  be  conscious.32 

It  is  to  be  noted  that  the  chief  characteristic  of  the  ex- 
ponents of  the  physiologic  theory  is  that  they  do  not  endow 
the  subconscious  with  any  mysterious  powers,  they  do  not 
regard  it  as  the  reservoir  of  consciousness,  but  on  the  con- 
trary, they  consider  subconscious  events  as  very  much  like 
the  ordinary  facts  of  waking  consciousness;  and  their 
method  of  explanation  is  to  proceed  in  a  true  scientific  man- 
ner from  the  known  to  the  unknown,  from  the  facts  of  the 
conscious  to  those  of  the  unconscious.  And  although  Mor- 
ton Prince  does  not  hold  this  view  in  its  entirety,  it  is 
nevertheless  in  this  fashion  that  he  commences  the  presen- 
tation of  what  is  without  doubt  the  most  able  and  most 
cogent  theory  of  the  unconscious  that  has  appeared  in  re- 
cent years. 

The  problem  of  the  subconscious,  according  to  him,  is 
the  problem  of  memory.  Whoever  solves  the  latter  will 
also  have  solved  the  former.  Memory  should  be  considered 
from  two  points  of  view :  as  a  process  and  as  an  end  result. 
As  a  process  it  is  composed  of  three  factors, — registration, 
conservation  and  reproduction.  The  last  is  the  end  result, 
but  to  understand  this  we  must  know  something  of,  or  at 
least  have  a  plausible  theory  concerning,  impression  and 
conservation. 

Instances  of  the  conservation  of  forgotten  experiences 
abound  both  in  normal  and  pathological  life.  They  are 

32  Ibid.,  pp.  4Sff. 


226  THE  MONIST. 

such  as  lapses  of  memory,  forgotten  acts,  failure  to  recog- 
nize, or  in  abnormal  cases  they  become  manifest  in  auto- 
matic writing  and  speech,  in  post-hypnotic  suggestions,  and 
so  forth.  After  examining  the  facts  in  great  detail,  Prince 
comes  to  the  conclusion  that  it  does  not  matter  at  what 
period  of  life  or  in  what  state  experiences  have  occurred, 
"or  how  long  a  time  has  intervened  since  their  occurrence, 
they  may  still  be  conserved.  They  become  dormant,  but 
under  favorable  conditions,  they  may  be  awakened  and 
may  enter  conscious  life."3  Naturally  these  experiences 
must  be  conserved  in  some  form ;  and  whatever  the  nature 
of  this  form  may  be  it  is  obvious  that  the  experiences  them- 
selves must  have  "a  very  specific  and  independent  existence, 
somewhere  and  somehow,  outside  of  the  awareness  of  con- 
sciousness." 

Now  in  order  to  account  for  normal  memory  we  must 
posit  that  ideas  which  have  passed  through  the  mind  have 
been  conserved  through  some  residuum  left  by  the  original 
experience.  This  residuum  must  be  either  psychological 
or  physiological.  Suppose  we  consider  the  former  alterna- 
tive first.  We  shall  have  to  assume  that  sensations,  per- 
ceptions, emotions  and  even  complex  systems  of  ideas  are 
capable  of  pursuing  "autonomous  and  contemporaneous 
activity  outside  of  the  various  systems  of  ideas  that  make 
up  the  personal  consciousness."  This  is  an  untenable  view, 
for  it  would  necessitate  the  storing  up  of  millions  of  ideas 
and  infinite  forms  of  associations.  Let  us,  therefore,  con- 
sider the  other  alternative,  namely,  conservation  as  phys- 
ical residua.  This  view  is  based  on  the  assumption  that 
whenever  we  have  a  mental  experience  of  any  sort  some 
change  or  trace  is  left  in  the  neurones  of  the  brain.  This 
does  not  necessarily  mean  that  the  neural  modification  is 
the  cause  of  the  conscious  process.  On  the  contrary,  it 
assumes  the  postulate  of  psychophysical  parallelism  and 

83  The  Unconscious,  pp.  82ff. 


THE  PRESENT  STATUS  OF  THE  UNCONSCIOUS.          227 

declares  that  with  every  passing  state  of  conscious  ex- 
perience, with  every  idea,  emotion  and  perception,  the  brain 
process  that  is  functioning  leaves  some  trace,  some  residua 
of  itself  within  the  neurones  and  in  the  functional  arrange- 
ments among  them.  This  physiological  conception  is  at  the 
basis  of  the  association  theory,  wherein  it  is  assumed  "that 
whenever  a  number  of  neurones  involved  in  a  coordinated 
sensory-motor  act  are  stimulated  into  functional  activity, 
they  become  so  associated  and  the  paths  between  them  be- 
come so  opened  or  sensitized,  that  a  disposition  becomes 
established  for  the  whole  group  to  function  together  and  to 
reproduce  the  original  reaction  when  either  one  or  the 
other  is  afterward  stimulated  into  activity.  This  'dispo- 
sition' is  spoken  of  in  physiological  language  as  a  lowering 
of  the  threshold  of  excitability.  This  change  we  may  speak 
of  as  a  residuum,"34  says  Prince. 

We  are  now  in  a  position  to  answer  the  question  raised 
a  while  ago  concerning  the  ability  of  a  hypnotized  person 
to  recall  a  forgotten  experience  or  one  that  he  was  not 
aware  of  at  the  time  of  its  occurrence. 

The  neurones  in  retaining  the  residua  of  the  original 
process  have  become  organized  into  a  functioning  system 
corresponding  to  the  system  of  mental  states — whether 
ideas,  perceptions  or  emotions — which  accompanied  that 
original  experience  and  are  now  capable  of  reproducing  it. 
Hence  when  we  reproduce  the  original  ideas  in  the  form  of 
memory  it  is  because  there  is  a  refunctioning  of  the  physio- 
logical neural  process.  On  hypnotizing  a  person,  therefore, 
and  asking  him  to  recall  a  forgotten  event,  we  simply  start 
that  process  by  introducing  what  may  be  called  a  catalytic 
agent,  i.  e.,  we  stir  one  neurone  or  one  brain  cell,  or  one  part 
of  the  system,  and  that  sets  the  entire  system  working  pre- 
cisely as  it  did  on  the  original  occasion.  This  physiological 
functioning  now  reaches  consciousness  or  motor  expression, 

**Ibid.t  pp.  119-120. 


228  THE  MONIST. 

because  all  other  mental  processes  are  arrested  for  the  time 
being,  thus  facilitating  a  greater  discharge  of  nerve  energy 
in  this  one  direction. 

The  same  is  true  of  crystal  gazing  and  automatic  writ- 
ing. In  the  former  occurrence  there  is  an  intense  concen- 
tration of  primary  attention.  That  is,  the  subject  does  not 
attend  to  any  idea  or  to  a  situation  from  which  he  tries  to 
derive  meaning,  but  merely  to  a  visual  stimulus.  In  this 
manner  all  distracting  influences  and  mental  processes 
which  do  not  harmonize  with  the  original  experience,  of 
which  it  must  be  said  the  individual  has  some  intimation 
to  begin  with,  are  arrested.  Thus  the  resumption  of  the 
original  neural  process  is  facilitated  and  with  it,  of  course, 
the  psychical  accompaniment.  Anything  that  will  hold  the 
attention  will  do  as  well  as  a  crystal.  A  soft  light  will 
work  just  as  well. 

Equally  well  can  automatic  writing  be  explained  on  the 
basis  of  this  theory.  The  writing  habit  is  very  highly  and 
delicately  "developed  in  us  writing  mortals,"  to  use  a 
phrase  of  Pierce,  and  it  is  no  wonder  that  it  may  operate 
mechanically,  when  for  some  reason  its  neural  system  has 
become  detached  from  that  other  system  which  constitutes 
self-consciousness.  Nor  do  the  specimens  of  automatic 
writing  show  this  phenomenon  to  be  essentially  different 
from  the  uncontrolled  movements  of  the  hands  and  bodily 
twitching  that  most  of  us  have  at  times ;  and  by  no  means 
is  it  different  from  such  nervous  troubles  as  chorea  and 
locomotor  ataxia.  The  hand  has  been  observed  to  write 
backwards,  to  write  mirror  script,  to  follow  indefinitely  a 
direction  given  to  it  by  the  experimenter  such  as  moving 
in  a  circle,  it  misplaces  and  omits  letters.  "Surely,"  says 
Pierce,  "such  occurrences  point  clearly  to  a  disordered 
neural  mechanism,  rather  than  to  a  perverse  or  humorously 
inclined  secondary  consciousness."3 

35  Carman,  Studies  in  Phil,  and  Psychol,  1906,  pp.  327-328. 


THE  PRESENT  STATUS  OF  THE  UNCONSCIOUS. 

We  see,  then,  that  most  if  not  all  subconscious  phenom- 
ena can  best  be  explained  in  terms  of  cerebration.  Now  it 
is  necessary  to  have  some  term  to  designate  the  separate 
neurological  modifications,  and  Prince  calls  these  "neuro- 
grams."  A  neurogram,  therefore,  is  a  brain  record;  and, 
just  as  a  phonogram  characterizes  the  form  in  which  the 
physical  aspect  of  spoken  thought  is  recorded,  so  a  neuro- 
gram characterizes  the  form  in  which  thoughts  and  other 
mental  experiences  are  recorded  in  the  brain  tissue.  Of 
course  this  is  merely  a  theoretical  concept,  like  atoms  and 
moments  of  force. 

Though  memory  is  regarded  in  psychology  as  a  con- 
scious process,  it  is  evident  that  on  the  basis  of  the  fore- 
going view  any  process  that  consists  of  the  three  factors, 
registration,  conservation  and  reproduction  of  experiences, 
must  be  considered  as  memory,  "whether  the  final  result 
be  the  production  of  a  conscious  experience  or  of  one  to 
which  no  consciousness  was  ever  attached."3 

That  memory  is  ultimately  a  physiological  phenomenon 
was  demonstrated  by  the  experiments  of  Rothmann  who 
showed  that  decorticated  animals  can  be  educated,  i.  e., 
new  dispositions  and  new  associations  may  be  established 
in  the  lower  centers  "without  the  intervention  of  the  in- 
tegrating influence  of  the  cortex  or  conscious  intelligence."37 
The  bearing  of  this  fact  is  that  unconscious  processes  are 
capable  of  being  conserved  in  the  form  of  physiological 
memory. 

If  we  accept  the  psychophysiological  theory  of  memory, 
then,  we  may  define  the  unconscious  as  the  brain  residua, 
the  physiological  dispositions  or  neurograms  in  which  the 
experiences  of  life  are  conserved.  The  co-conscious,  on  the 
other  hand,  means  "a  coexisting  consciousness  of  which 
the  personal  consciousness  is  not  aware.  And  since  these 

36  Prince,  The  Unconscious,  p.  135. 
87  The  Unconscious,  p.  238. 


23O  THE  MONIST. 

two  function  together  we  need  an  inclusive  term,  one  that 
will  embrace  them  both,  and  that  is  the  subconscious."3 

Here  the  truly  scientific  discussion  ends.  The  rest  that 
Prince  has  to  say  about  the  subconscious  is  metaphysical, 
and  not  unlike  the  views  of  Sidis,  von  Hartmann  and  My- 
ers. He  declares,  for  instance,  that  the  subconscious,  rather 
than  the  conscious,  is  the  important  factor  in  personality 
and  intelligence;  that  the  subconscious  furnishes  the  ma- 
terial out  of  which  our  judgments  and  beliefs,  our  ideals 
and  characters,  are  shaped.  Yet  I  can  hardly  see  how  he 
squares  this  statement  with  the  next  in  which  he  says  that 
the  unconscious  complexes  are  kept  in  check  by  the  normal 
inhibitions  and  the  counterbalancing  influences  of  the  nor- 
mal mental  mechanism.39  Evidently,  then,  it  is  the  normal 
mental  mechanism,  by  which  I  suppose  he  means  attentive 
consciousness  or  intelligence,  which  exercises  a  determining 
control  over  the  unconscious  complexes.  Hence  it  is  the 
conscious  and  not  the  unconscious  which  is  at  the  basis 
of  our  beliefs,  our  ideals  and  character. 

Be  this  as  it  may,  Prince's  metaphysical  interpretation 
does  not  change  the  facts  nor  the  value  of  his  scientific 
concepts  which  so  excellently  explain  those  facts.  For  by 
resolving  the  subconscious  into  unconscious  physiological 
dispositions  on  the  one  hand,  and  coactive  conscious  states 
on  the  other,  we  are  able  to  understand  more  clearly  the 
nature  of  lapsed  memory,  absent-mindedness,  post-hypnotic 
suggestion,  artificial  hallucinations,  hysteria,  psychoneu- 
rosis  and  multiple  personality. 

With  respect  to  bridging  the  gap  between  the  conscious 
and  the  subconscious,  Prince  declares  that  no  gap  exists. 
What  belongs  to  one  at  times  passes  into  the  other,  and 
vice  versa.  Consciousness  may  be  conceived  of  as  a  round 
disk  with  attention  or  the  focus  of  awareness  at  the  center. 

38  Ibid.,  p.  253. 

39  Ibid.,  p.  262. 


THE  PRESENT  STATUS  OF  THE  UNCONSCIOUS.          23! 

Surrounding  this  is  a  zone  which  constitutes  the  fringe  of 
awareness.  Embracing  that  is  the  co-conscious,  i.  e.,  un- 
conscious mentation,  while  the  outermost  zone  comprises 
the  unconscious  processes.  There  is  a  gradual  shading 
from  the  center  to  the  edge  of  this  figurative  disk  or  sphere 
of  consciousness.  But  here  again  Prince  treads  on  the 
metaphysical,  and  we  have  not  the  time  to  follow  him. 

The  space  at  our  disposal  only  permits  us  to  suggest 
that  more  original  work  ought  to  be  done  in  this  field.  Too 
many  writers  weave  their  theories  around  the  same  cases 
of  somnambulism  and  double  personality.  The  cases  ex- 
amined by  Morton  Prince,  Janet  and  Bernheim  constitute 
a  sort  of  stock-in-trade  making  their  rounds  in  the  litera- 
ture on  the  subconscious.  But  a  theory  does  not  gain 
credence  by  hopping  about  on  the  same  crutches;  it  must 
gather  new  facts  if  it  would  increase  in  strength.  In  this 
respect  Professor  Lillien  Martin  has  shown  a  good  way 
in  her  experimental  investigation  of  the  subconscious.  She 
does  not  add  anything  new,  but  her  method  of  investigation 
which  consisted  in  having  normal  subjects  permit  images 
to  arise  of  themselves  and  then  introspect  on  them,  is  more 
reliable  than  the  questionnaire  method  used  by  some  au- 
thors, or  the  observation  of  pathological  cases  upon  which 
still  others  have  built  their  concepts.  Experimental  research 
under  strictly  controlled  conditions,  should  be  the  slogan 
of  psychologists  in  the  field  of  the  subconscious  as  it  is  in 
that  of  the  conscious. 

Thus  we  stand  at  the  present  moment  with  three  theories 
of  the  unconscious  before  us.  The  psychometaphysical, 
the  psychophysiological  with  metaphysical  leanings,  and 
the  psychoneurological  with  scientific  leanings.  The  first 
declares  that  the  whole  universe  is  permeated  with  con- 
sciousness, that  there  is  intelligence  in  all  animals,  in  plants 
and  even  in  inorganic  matter.  This  notion  is  held  by  writ- 
ers like  von  Hartmann,  Myers,  Delboeuf  and  persons  in- 


232  THE  MONIST. 

terested  in  psychical  research.    It  is  obvious,  however,  that 
this  view  will  not  bring  us  anywhere. 

The  psychophysiological  theory  with  metaphysical  lean- 
ings also  maintains  that  there  is  consciousness  in  all  organ- 
isms, only  it  is  not  conscious  of  itself.  That  in  living  organ- 
isms this  consciousness  is  accompanied  by  physiological 
changes,  but  these  changes  are  not  necessarily  the  cause 
of  conscious  phenomena.  Neurological  modifications  are 
only  conceptions  assumed  for  the  purpose  of  explaining 
unconscious  activity.  But  the  psyche  is  the  fundamental 
principle.  The  unconscious  is  the  source  of  all  intelligence. 
This  view  is  held  explicitly  or  implicitly  by  Freud,  Sidis, 
Prince,  Lloyd  Morgan  and  Janet. 

Finally,  the  psychophysiological  theory  with  scientific 
leanings  asserts  that  neurological  modifications  are  the  es- 
sential factors  of  conscious  and  unconscious  phenomena. 
That  consciousness  appears  only  when  the  neurones  attain 
a  certain  tension,  or  function  in  a  certain  relation;  that 
consciousness  may  or  may  not  accompany  so-called  intelli- 
gent actions  performed  under  pathological  conditions ;  that 
it  is  certainly  not  present  in  instinctive  functioning  which 
characterizes  the  life  of  lower  animals ;  that  the  unconscious 
is  not  the  storehouse  of  the  conscious,  that  there  is  nothing 
mysterious  or  wonderful  about  it,  and  that  with  further 
investigation  its  precise  nature  and  place  in  the  scale  of 
psychogenesis  will  be  at  the  command  of  psychologists. 
This  view  is  held  by  writers  like  Ribot,  Pierce,  King  and 
Jastrow. 

Such  are  the  three  views  that  present  themselves  for 
our  consideration.  There  is  no  doubt  about  the  one  that 
scientists  will  adopt  as  leading  to  a  greater  extension  of 
human  knowledge. 

GUSTAVE  A.  FEINGOLD. 

HARTFORD,  CONN. 


NIRVANA 

THE  BUDDHIST'S  FINAL  GOAL. 

NIRVANA,  state  of  rest  unbroken,  where 
Benign  extinction  of  all  'passion  rules — 
A  rest  so  deep  that  in  eternity 
It  shall  not  be  disturbed — I  long  for  thee! 
After  life's  storm  and  stress  thou  grantest  peace. 
Weary  of  this  world's  wild  anxieties, 
Its  pains  and  empty  pleasures,  I  will  seek 
The  everlasting  in  blank  vacancy, 
Thus  to  attain  the  boon  of  dreamless  sleep 
From  which  nor  rancor  of  a  villainous 
Intrigue,  planned  by  malevolence  or  hate, 
Nor  the  misfortune  of  a  sorry  slip 
Of  my  misguided  feet,  will  waken  me, 
But  unconcerned  and  calm  I  shall  remain 
In  perfect  quietude.    For  I'll  be  safe 
From  all  the  worry  and  from  all  the  trouble 
That  restlessly  stirs  life  and  keeps  it  moving. 
The  bustle  of  the  world,  its  vulgar  noise 
With  its  deplorable  afflictions,  trials 
And  eke  malicious  slander,  will  be  hushed. 

There  is  a  refuge,  vainly  sought  for  here, 
And  in  its  sanctuary  I'll  find  shelter 
From  life's  great  woes  and  small  annoyances. 
There  paltry  problems  will  no  longer  vex 
Nor  will  demand  immediate  solution. 


234  THE  MONIST. 

I  shall  no  longer  be  disquieted 

By  urgent  needs  to  be  responded  to 

In  energetic  action.    E'en  my  ego 

With  its  ambitions,  wants  and  vanities; 

Its  recollection  of  the  past  with  all 

Its  sweet  and  bitter  memories — all  that, 

My  very  consciousness,  will  be  extinct. 

I  shall  be  left  in  tranquil  emptiness 

And  in  a  soothing  void  of  non-existence, 

A  clean,  pure  state  of  rest  most  absolute, 

Without  the  slightest  ripple  of  disturbance, 

A  panacea  for  all  earthly  ills, 

An  anodyne  for  any  pang  or  pain. 

In  former  ages  mankind  felt  assured 

Of  a  survival  of  the  soul.    The  savage 

Met  his  dead  parents  and  his  friends  in  dream. 

He  saw  them,  he  conversed  with  them,  and  dreams 

Were  real  to  him  just  as  actual  life. 

When  man  grew  wiser,  he  began  to  doubt 

And  he  grew  anxious  for  convincing  proof. 

Though  proofs  were  negative,  yet  he  still  clung 

To  hope  expressing  his  desire  to  live 

And  to  prolong  his  life  beyond  the  grave. 

Oh  foolish  man,  why  dost  thou  shrink  from  death 
And  yearnest  greedily  for  prolongation 
Of  thy  ill-favored  self?    Thy  selfishness 
Thou  wishest  to  preserve,  thy  abject  foibles, 
Instead  of  gladly  hiding  them  into 
The  darkness  of  a  taciturn  forgetting, 
As  in  wise  justice  Nature  has  intended, 
Thou  wouldst  perpetuate  with  petulance 
And  peevish  childishness  that  of  thyself, 
Exactly  that,  whose  riddance  should  be  welcomed 


NIRVANA.  235 

As  a  great  boon,  a  seemly  liberation 

From  slavery,  its  drudgery  and  curse. 

Why  should  we  cling  to  chains  that  burden  us 

When  we  might  cast  them  off  and  free  ourselves  ? 

Why  should  we  serve  new  terms  as  sentenced  convicts 

When  duly  our  acquittal  is  pronounced 

And  a  reprieve  has  graciously  been  granted? 

Mara,  the  Evil  One  will  envy  me 
In  my  benign  repose;   he  will  continue 
His  vicious  persecution.    Shall  I  help  him 
And  do  the  wrong  myself  unto  myself 
By  pressing  from  a  place  of  safety  into 
My  prison  with  its  ugly  bars  ?    Oh  no ! 
No,  I  shall  not!    For  I  prefer  my  freedom! 
There  I  shall  be  where  most  malignant  foes 
Shall  not  be  able  to  do  any  harm. 
And  if  they  should  go  on  abusing  me 
I  shall  no  longer  heed  their  defamation; 
I'll  leave  them  to  their  fate  which  in  full  justice 
Will  come  to  them  without  my  interference. 
Their  lies  no  longer  touch  me.    In  Nirvana 
I  shall  be  free;  the  vicious  will  remain 
In  a  gehenna  builded  by  themselves. 

The  wild  desires  of  my  hot  pulsing  heart 
Will  then  be  calmed,  all  hunger  will  be  stilled, 
All  thirst  be  quenched  in  deepest  satisfaction. 
And  mine  shall  be  the  glory  of  Nirvana ; 
Having  achieved  the  conquest  of  all  pain, 
Having  attained  final  emancipation, 
It  will  be  mine,  Nirvana  will  be  mine. 
I  shall  be  free  when  I  have  closed  mine  eyes, 
To  enter  death,  life's  solemn  grand  finale, 
Its  fruitage,  benison  and  consummation. 
Nirvana's  holy  peace  shall  then  be  mine. 


236  THE  MONIST. 

Indeed  it  is  mine  here ;  I  live  it  now 
If  I  but  understand  the  art  of  living 
The  truth :  It  is  and  will  be  mine,  when  I 
Surrender  transient  things  to  transiency 
And  live  in  that  alone  which  will  endure. 

Oh,  the  inanities  of  self!  how  puny, 
How  paltry  are  they;  and  how  kind  is  death 
To  brush  them  off  with  gently  sweeping  stroke 
Like  spider  webs  out  of  a  gloomy  corner, 
Together  with  the  spider  who  has  built  them. 

O  let  them  go  without  regret  and  sorrow, 

The  ego  with  its  portion  is  not  worthy 

Of  preservation.    It  is  but  the  burden 

Of  our  existence,  the  receptacle 

In  which  the  weaknesses  and  faults  of  life 

Are  bred,  in  which  its  plagues  are  caught  and  stored. 

So  let  them  go  and  bless  their  disappearance. 

They  are  like  painful  sores  that  should  be  healed, 

And  when  our  ego  passes  they  are  cured. 

The  right  ideas  only  which  we've  thought, 

The  good  deeds  too  which  we  have  done  and  things 

Of  beauty  we  have  shaped,  they  shall  survive. 

They  are  our  better  selves ;  they  will  be  helpful, 

Helpful  to  others,  to  the  generations 

That  are  to  come,  helpful  like  gifts  of  God, 

Like  rain  or  sunbeams,  showered  down  on  earth, 

Profuse,  unstinted,  and  with  utter  lack 

Of  egotism.    But  do  not  cling  to  self, 

Nor  yearn  for  any  undue  preservation 

Of  personality.    Our  ego's  life 

And  all  that's  of  an  accidental  nature 

Be  handed  over  to  its  destination 

Which  is  a  dissolution  into  naught. 


NIRVANA.  237 

Our  conscious  ego  has  originated, 

It  has  been  growing,  and  'twill  pass  away. 

Such  is  its  destiny  and  so  'tis  best. 

But  I  will  glory  in  my  future  lot — 

Nirvana's  boon,  the  state  of  perfect  peace. 

Yea,  I  can  enter  even  now  into 

Nirvana's  hallowed  temple  where  my  soul 

Is  liberated  from  all  transiency 

And  will  be  ready  for  a  final  exit 

Out  of  existence  with  its  narrowness 

Into  the  better  and  superior  realm, 

The  realm  of  bliss,  Nirvana's  noble  bliss. 

Praised  be  Nirvana,  glorious  radiant  state 
Of  biding  peace,  hope  of  all  living  creatures 
And  comfort  of  the  dead.    Holy  asylum 
Which  grander  is  than  highest  joy  in  heaven 
And  more  divine  than  the  divinity 
Of  Brahma  and  his  gods  in  all  their  splendor. 
Praised  be  Nirvana,  goal  of  all  the  Buddhas! 
And  blest  is  he  who  enters  there,  who  lives 
There  in  Nirvana;  lives  there  in  the  truth 
Which  therein  is  revealed;    he  who  is  free 
From  vain  attachment,  who's  above  temptation. 
'Tis  he  in  whom  all  passion  is  extinct ; 
Who  has  attained  life's  final  aim  Nirvana, 
Goal  of  the  wise,  and  of  the  blessed  Buddhas. 
He  who  has  reached  it  is  the  Conqueror, 
The  conqueror  of  Evil,  the  great  Jina, 
He's  the  Enlightened  One :  he  is  the  Buddha ! 
And  he  is  blest ;  the  Buddha,  yea !  is  blest. 
Pathfinder  to  Nirvana !    Praised  be  he ! 
Namo  tassa  Bhagavato  Buddhassa. 

PAUL  CARUS. 


THE  MANUSCRIPTS  OF  LEIBNIZ  ON  HIS  DIS- 
COVERY OF  THE  DIFFERENTIAL  CALCULUS.* 

PART  II. 
§m. 

The  following  notes,  on  certain  MSS.  which  Gerhardt  does  not 
give  in  full,  are  taken  from  G.  1848,  p.  20  et  seq.  (see  also  G.  1855, 
p.  55  et  seq.) 

In  a  manuscript  of  August,  1673,  bearing  the  title  Methodus 
nova  investigandi  Tangentes  linearum  curvarum  ex  datis  applicatis, 
vel  contra  Applicatis  ex  datis  productis,  reductis,  tangentibus,  per- 
pendicularibus,  secantibus,  Leibniz  begins  at  once  with  an  attempt 
to  find  a  method  that  is  applicable  to  any  curve  for  the  determination 
of  its  tangent.  "But  if,"  says  Leibniz  with  regard  to  the  classifica- 
tion of  curves  which  Descartes  laid  down  as  fundamental  for  his 
method  of  tangents,  "the  figure  is  not  geometrical  —  such  as  the 
cycloid — it  does  not  matter;  for  it  will  be  treated  as  an  example 
of  a  geometrical  curve,  by  supposing  that  there  is  a  relation  between 
the  straight  lines  and  curves  by  which  they  are  made  known  to  us ; 
in  this  way,  tangents  can  be  drawn  just  as  well  to  either  geometrical 
or  ageometrical  curves,  as  far  as  the  nature  of  the  figure  allows." 
He  considers  the  curve  as  a  polygon  with  an  infinite  number  of 
sides,  and  here  already  he  constructs  what  he  calls  the  "Character- 
istic Triangle,"  whose  sides  are  an  infinitely  small  arc  of  the  curve, 
and  the  differences  between  the  ordinates  and  between  the  abscissae ; 
this  is  similar  to  the  triangle  whose  sides  are  the  tangent,  the  sub- 
tangent  and  the  ordinate  for  the  point  of  contact.  In  just  the  same 
manner  as  used  by  Descartes,  Leibniz  seeks  the  tangent  by  means 

*  Part  I  appeared  in  The  Monist  of  October,  1916. 


THE  MANUSCRIPTS  OF  LEIBNIZ.  239 

of  the  subtangent ;  he  denotes  the  infinitely  small  differences  of  the 
abscissae  by  b,  and  verifies  for  the  parabola,  that  his  method  works 
out  correctly,  when  the  terms  of  the  equation  that  contain  the  in- 
finitely small  quantities  are  neglected.  The  omission  of  these  terms, 
however,  does  not  appear  to  Leibniz  to  be  a  method  to  be  relied 
upon.  In  fact,  he  says:  "It  is  not  safe  to  reject  multiples  of  the 
infinitely  small  part  b,  and  other  things;  for  it  may  happen  that 
through  the  compensation  of  these  with  others,1  the  equation  may 
come  to  a  totally  different  condition."  So  he  seeks  to  obtain  the 
determination  of  the  subtangent  in  some  other  way.  "The  whole 
question  is,  how  the  applied  lines  can  be  found  from  the  differences 
of  two  applied  lines,"  are  his  own  words.  He  then  finds  that  the 
solution  of  this  problem  reduces  to  the  summation  of  a  series,  of 
which  the  terms  are  the  differences  of  consecutive  abscissae. 

At  the  end  of  the  manuscript  Leibniz  proceeds  to  speak  of  the 
inverse  problem:  "It  is  an  important  subject  for  investigation, 
whether  it  is  possible,  by  retracing  our  steps,  to  proceed  from  tan- 
gents and  other  functions  to  ordinates.  The  matter  will  be  most 
accurately  investigated  by  tables2  of  equations ;  in  this  way  we  may 
find  out  in  how  many  ways  some  one  equation  may  be  produced 
from  others,  and  from  that  which  of  them  should  be  chosen  in  any 
case.  This  is,  as  it  were,  an  analysis  of  the  analysis  itself,  but  if 
that  is  done  it  forms  the  fundamental  of  human  science,  as  far  as 
this  kind  of  things  is  concerned."  Ultimately  Leibniz  obtains  the 
following  result:  "The  two  questions,  the  first  that  of  finding  the 
description  of  the  curve  from  its  elements,  the  second  that  of  find- 
ing the  figure  from  the  given  differences,  both  reduce  to  the  same 
thing.  From  this  fact  it  can  be  taken  that  almost  the  whole  of  the 
theory  of  the  inverse  method  of  tangents  is  reducible  to  quadra- 
tures." 

According  to  this,  Leibniz  has  in  the  middle  of  the  year  1673 
already  attained  to  the  knowledge  that  the  direct  and  the  so-called 
inverse  tangent-problem  have  an  undoubted  connection  with  one 
another ;  he  has  an  idea  that  the  latter  may  be  ..capable  of  reduction 
tc  a  quadrature  (i.  e.,  to  a  summation). 

Again,  in  a  manuscript  dated  October  1674,  i.  e.,  fourteen 
months  later,  which  bears  the  title  Schediasma  de  Methodo  Tan- 

1  It  is  impossible  to  see,  without  a  fuller  knowledge  of  the  context.whether 
this  refers  to  "compensation  of  errors,"  or  whether  Leibniz  is  alluding  to  the 
possibility  of  all  the  finite  terms  cancelling  one  another. 

2  Leibniz  comes  back  to  this  point  later ;  see  §  IV. 


240  THE  MONIST. 

gentium  inversa  ad  circulum  applicata,  he  is  able  to  say  for  certain 
that  "the  quadratures  of  all  figures  follow  from  the  inverse  method 
of  tangents,  and  thus  the  whole  science  of  sums  and  quadratures 
can  be  reduced  to  analysis,  a  thing  that  nobody  even  had  any  hopes 
of  before." 

After  Leibniz  thus  recognized  the  identity  between  the  inverse 
tangent-problem,  of  which  the  general  solution  had  not  been  found 
by  Descartes,  and  the  quadrature  of  curves,  he  applied  himself  to 
the  investigation  of  series  by  the  summation  of  which  quadratures 
were  then  obtained.  In  a  very  extensive  discussion,  bearing  the 
date  of  October,  1674,  and  the  title  Schediasma  de  serierum  summis, 
et  seriebus  quadraticibus,  Leibniz  starts  from  the  series 


and  obtains  the  following  general  rule:  "By  calling  the  variable 
ordinates  x,  and  the  variable  abscissae  y,  and  b  the  abscissa  of  the 
greatest  ordinate  e,  and  d  the  abscissa  of  the  least  ordinate  h,"  are 
Leibniz's  own  words,  "we  have  the  following  rules  : 


hzw      dzh  x 

—-  +  —-  =  xy--,e_k  =  w, 

*      w2 

f7/l  —  -    _    - 

2        2' 

yw  =  x  in  decreasing  values,  for  in  ascending  or  increasing  values 
yw  =  eb  -  x"s 

Leibniz  then  goes  on  to  remark  :  "These  rules  are  to  be  altered 
slightly  according  as  the  series  increase  or  decrease;  also  mention 
of  the  least  ordinate  may  be  omitted,  if  it  is  always  understood  to 
be  the  last  ordinate;  on  the  other  hand,  w  can  always  be  inserted 
wherever  mention  is  made  of  w.  All  series  hitherto  found  are  con- 
tained in  the  one  by  means  of  these  rules,  except  the  series  of 
powers,  which  is  to  be  obtained  by  taking  differences." 

3  This,  without  either  proof  or  figure,  is  a  hopeless  muddle  ;  and  yet  it  is 
repeated  word  for  word,  without  any  addition  or  remark,  in  Gerhardt's  1855 
publication.  Goodness  knows  what  the  use  of  it  was  supposed  to  be  in  this 
form!  Unless  Leibniz  has  omitted  some  length,  which  he  has  supposed  to  be 
unity,  the  dimensions  are  all  wrong. 


THE  MANUSCRIPTS  OF  LEIBNIZ.  24! 

In  the  same  essay,  Leibniz  makes  use  of  a  theorem,  which  he 
has  probably  found  to  be  general  at  an  earlier  date,  namely: 

"Since  BC  is  to  BD  as  WL  to  SW,  there- 
fore BC^SW,4  that  is,  the  sum  of  every  BC 
[applied  to  AC],  is  equal  to  BD^WL,  that  is, 
the  sum  of  every  BD  applied  to  the  base ;  more- 
over, the  sum  of  every  BD  applied  to  the  base  is 
equal  to  half  the  square  on  the  greatest  BD. 
Further,  it  is  evident  that  the  sum  of  every  WL 
is  equal  to  the  greatest  BD." 

Accordingly,  Leibniz  comes  to  the  further 
conclusion  that  the  method  of  Descartes,  which 
uses  a  subsidiary  equation  with  two  equal  roots,  to  solve  the  general 
inverse-tangent  problem,  is  unsatisfactory.  In  a  manuscript  of 
January,  1675,  Leibniz  says :  "Thus  at  last  I  am  free  from  the  un- 
profitable hope  of  finding  sums  of  series  and  quadratures  of  figures 
by  means  of  a  pair  of  equal  roots,  and  I  have  discovered  the  reason 
why  this  argument  cannot  be  used;  this  has  worried  me  for  quite 
long  enough."5 

§  IV. 

The  manuscript  that  comes  next  in  da'te  is  one  that  is 
given  in  G.  1855.  It  really  consists  of  three  short  notes, 
(i)  a  theorem  on  moments,  (2)  a  continuation  of  the  idea 
started  at  the  end  of  the  manuscript  of  August,  1673 
(§  III),  namely  the  formation  of  tables  of  equations  that 
are  derivable  from  certain  standard  equations,  with  the 
appropriate  substitutions  for  each  case,  (3)  a  return  to 
the  consideration  of  moments. 

This  is  the  first  appearance  of  the  word  "moment,"  but 
from  the  context  it  is  evident  that  Leibniz  has  done  some 
considerable  amount  of  work  upon  the  idea  before.  If  the 
theorem  that  is  first  given  is  written  in  modern  notation, 

4  The  sign  °  signifies  multiplication. 

6  Observe  that  as  yet  nothing  has  been  said  about  the  area  of  surfaces  of 
revolution  or  moments  about  the  axis,  although  we  should  expect  them  to  be 
mentioned  in  connection  with  the  figure  that  is  given ;  for  the  next  manuscript 
shows  that  in  October  1675,  Leibniz  has  already  done  a  considerable  amount 
of  work  on  moments. 


242 


THE  MONIST. 


it  takes  the  form  of  an  "integration  by  parts"  and  serves 
to  change  the  independent  variable.    Thus  we  have 


and  it  is  readily  seen  that  if  x  can  be  expressed  as  a  square 
root  of  a  simple  function  of  y,  as  for  the  circle  and  the 
conic  sections,  then  the  integral  on  the  right-hand  side 
has  no  irrationality.  This,  I  take  it,  is  the  connection 
between  this  theorem  and  those  which  follow. 

The  proof  is  not  so  clear  as  it  might  be  on  account  of 
two  errors,  both  I  think  errors  of  transcription  or  mis- 
prints. The  first  a  should  be  an  x,  and  the  second  a  should 
be  the  preposition  a  (=  from)  ;  also,  for  modern  readers 
the  figure  might  be  improved  by  showing  the  variable  lines 
AB  (=x),  BC  (=30  as  in  the  accompanying  diagram. 
The  argument  then  is  as  follows: 

Moment  of  BC(=30  about  AD  is  xy,  when  it  is  applied 
to  AB  for  the  summation ;  for  this  brings  in  the  infinitesi- 
mal breadth  of  the  line. 


B 


Moment  of  DE  (=  x)  about  AD  is  xz/2,  when  applied 
to  AD,  so  as  to  include  the  infinitesimal  breadth  of  the 
line,  and  assuming  that  the  line  may  be  considered  to  be 
condensed  at  its  center  of  gravity.  The  theorem  follows 
at  once. 

Note  the  use  of  the  sign  n  as  a  symbol  of  equality, 
which  I  have  allowed  to  stand  in  the  opening  paragraph. 
Leibniz  adopts  the  ordinary  sign  two  months  later,  or  Ger- 


THE  MANUSCRIPTS  OF  LEIBNIZ.  243 

hardt  makes  the  change,8  so  I  have  not  thought  it  necessary 
to  adhere  to  it,  but  only  to  show  it  in  the  opening  para- 
graph. 

The  only  remark  that  seems  to  be  necessary  with  regard 
to  the  second  part  of  this  manuscript  is  that  Weissenborn7 
argues  from  the  continued  allusion  by  Leibniz  to  the  de- 
sirability of  forming  tables  of  curves  whose  quadratures 
may  be  derived  from  those  of  others,  especially  the  conic 
sections,  (starting  with  the  manuscript  of  November,  1675, 
where  Weissenborn  states  that  it  is  first  hinted),  that 
Leibniz  had  probably  either  seen  or  heard  of  the  Cata- 
logus  curvarum  ad  conicas  sectiones  relatarum  of  Newton. 
The  point  is  that  Weissenborn  seems  to  have  missed  the 
clear  reference  to  the  reduction  of  curves  to  those  of  the 
second  degree,  in  this  manuscript  of  October,  1675.  It 
may  of  course  be  just  possible  that  G.  1855,  m  which  this 
MS.  appears,  was  not  at  Weissenborn's  hand  at  the  time 
that  he  wrote,  for  Weissenborn's  book  was  published  in 
1856. 

With  regard  to  the  third  part,  it  will  be  found  in  the 
original  Latin-  that  Leibniz,  after  apparently  starting  with 
perfect  clearness,  gets  rather  into  a  muddle  toward  the  end. 
This  is  however  only  apparent,  being  partly  due  to  an  in- 
accurate figure,  and  partly  to  what  I  am  convinced  is  an 
error  of  transcription.  This  incorrect  sentence  makes  Leib- 
niz write  apparently  absolute  nonsense ;  but  if  a  correction 
is  made  according  to  the  suggestion  in  the  footnote,  and 
reference  is  made  to  the  corrected  diagram  that  I  have 
added  on  the  right  of  the  figure  of  Leibniz,  as  given  by 
Gerhardt,  then  the  proof  given  by  Leibniz  reads  perfectly 
smoothly  and  sensibly. 

6  Gerhardt  has  a  footnote  to  the  effect  that,  as  nearly  as  possible  he  has 
retained  the  exact  form  of  this  and  the  manuscripts  that  immediately  follow ; 
except  in  the  matter  of  this  one  sign  I  have  adhered  to  the  form  given  by 
Leibniz. 

7  Weissenborn,  Principien  der  hoheren  Analysis,  Halle,  1856. 


244 


THE  MONIST. 


25  October,  1675. 

Analysis  Tetragonistica  Ex  Centrobarycis. 
Analytical  quadrature  by  means  of  centers  of  gravity. 

Let  any  curve  AEC  be  referred  to  a  right  angle  BAD  ;  let  AB  n 
DCna,8  and  let  the  last  xnb;  also  let  BCnADn^,  and  the  last 
ync.  Then  it  is  plain  that 


ornn.  yx  to  x  =  —  —  omn.  ~^~  to  y. 


(1) 


For,  the  moment  of  the  space  ABCEA  about  AD  is  made  up 
of  rectangles  contained  by  BC  (=  y)  and  AB  (=  x}  ;  also  the  moment 


B 


about  AD  of  the  space  ADCEA,  the  complement  of  the  former 

/     x2\ 
is  made  up  of  the  sum  of  the  squares  on  DC  halved  (  =  — ) ;  and  if 

this  moment  is  taken  away  from  the  whole  moment  of  the  rectangle 

i2 

ABCD  about  AD,  i.  e.,  from  c  into  omn.  x?  or  from  — ,  there  will 

remain  the  moment  of  the  space  ABCEA.    Hence  the  equation  that 
I  gave  is  obtained ;  and,  by  rearranging  it,  it  follows  that 


omn.  yx  to  x  +  omn.  —  to  y  = 


(2) 


In  this  way  we  obtain  the  quadrature  of  the  two  joined  in  one 
in  every  case ;  and  this  is  the  fundamental  theorem  in  the  center  of 
gravity  method. 

Let  the  equation  expressing  the  nature  of  the  curve  be 

ay*+6x*+cxy+dx+ey+/=0, (3) 


and  suppose  that  xy=z,  -  •  •  •  (4),  then  y  =  — . 


x 


(5) 


Substituting  this  value  in  equation  (3),  we  have 


8  This  a  should  be  x. 

9  Here,  in  the  Latin,  "ac  in  omn.*-"  should  be  "a  c  in  omn.*." 


THE  MANUSCRIPTS  OF  LEIBNIZ.  245 

*+/-0.  (6) 

X"  X 

and,  on  removing  the  fractions, 

az2  +  bx*  +  cx2z  +  dxs  +  exz  +  fx2  =  0 (7) 

Again,  let  x2  =  2w   (8)  ;  then,  substituting  this  value  in 

equation  (3),  we  have 

and  therefore 

x_-a*-2bw -ey-f      (10) 

ey+d 
=  V2w;    (11) 

and,  squaring  each  side,  we  have10 

azy2  +  4aby2w  +  2aeya  +  2afy2  +  4b2w2  +  4bewy  +  4bfw 

+  ezy2  +  2fey  +  f2-  2c2y2w  -  4cdyw  -  2d2w  =  0.    . .  ( 12) 

Now,  if  a  curve  is  described  according  to  equation  (7),  and 
also  another  according  to  equation  (12),  I  say  that  the  quadrature 
of  the  figure  of  the  one  will  depend  on  the  quadrature  of  the  figure 
of  the  other,  and  vice  versa. 

If,  however,  in  place  of  equation  (3),  we  took  another  of 
higher  degree,  the  third  say,  we  should  again  have  two  equations 
in  place  of  (7)  and  (12)  ;  and  continuing  in  this  manner,  there  is 
no  doubt  that  a  certain  definite  progression  of  equations  (7)  and 
(12)  would  be  obtained,  so  that  without  calculation  it  could  be 
continued  to  infinity  without  much  trouble.  Moreover,  from  one 
given  equation  to  any  curve,  all  others  can  be  expressed  by  a  general 
form,  and  from  these  the  most  convenient  can  be  selected. 

If  we  are  given  the  moment  of  any  figure  about  any  two 
straight  lines,  and  also  the  area  of  the  figure,  then  we  have  its 
center  of  gravity.  Also,  given  the  center  of  gravity  of  any  figure 
(or  line)  and  its  magnitude,  then  we  have  its  moment  about  any 
line  whatever.  So  also,  given  the  magnitude  of  a  figure,  and  its 
moments  about  any  two  given  straight  lines,  we  have  its  moment 
about  any  straight  line.  Hence  also  we  can  get  many  quadratures 
from  a  few  given  ones.  Moreover,  the  moment  of  any  figure  about 
any  straight  line  can  be  expressed  by  a  general  calculation. 

The  moment  divided  by  the  magnitude  gives  the  distance  of  the 
center  of  gravity  from  the  axis  of  libration. 

10  In  view  of  this  accurate  bit  of  algebra,  the  faulty  work  in  subsequent 
manuscripts  seems  very  unaccountable. 


246 


THE  MONIST. 


Suppose  then  that  there  are  two  straight  lines  in  a  plane,  given 
in  position,  and  let  them  either  be  parallel  or  meet,  when  produced 
in  F.  Suppose  that  the  moment  about  BC  is  found  to  be  equal  to 
ba?,  and  the  moment  about  DE  is  found  to  be  ca2.  Call  the  area 
of  the  figure  v\  then  the  distance  of  the  center  of  gravity  from  the 

ba2 
straight  line  BC,  namely  CG,  is  equal  to  — ,  and  its  distance  from 

v 

2 

the  straight  line  DE,  namely  EH,  is  equal  to  -  — ;  therefore  CG  is  to 

v 
EH  as  b  is  to  c,  or  they  are  in  a  given  ratio.11 


GERHARDT'S  DIAGRAM. 


SUGGESTED  CORRECTION. 


Now  suppose  that  the  straight  line  EH,  remaining  in  the  plane, 
traverses  the  straight  line  DE,  always  being  perpendicular  to  it,  and 
that  the  straight  line  CG  traverses  the  straight  line  BC,  always  per- 
pendicular to  it,  and  that  the  end  G  leaves  as  it  were  its  trace,  the 
straight  line  G(N),  and  the  end  H  the  straight  HN.  Then,  if  BC 
and  DE  meet  anywhere,  G(N)  and  HN  must  also  meet  somewhere, 
either  within  or  without  the  angle  at  F.  Let  them  meet  at  L ;  then 
the  angle  HLG  is  equal  to  the  angle  EFC,  and  PLQ  (supposing 
that  PL  =  EH  and  LQ  =  CG)  will  be  the  supplement  of  the  angle 
EFC  between  the  two  straight  lines,  and  will  thus  be  a  given  angle. 
If  then  PQ  is  joined,  the  triangle  PQL  is  obtained,  having  a  given 
vertical  angle,  and  the  ratio  of  the  sides  forming  the  vertex,  QL :  LP, 
also  given. 

When  then  BL  is  taken,  or  (B)(L),  of  any  length  whatever, 
since  the  angle  BLP  always  remains  the  same,  and  in  addition  we 
have  BL  to  LP  as  (B)  (L)  to  (L)  (P),  therefore  also  BLto  (B)  (L) 
as  LP  to  (L)  (P)  ;  and  this  plainly  happens  when  FL  is  also  propor- 

11  This  proves  the  fundamental  theorem  given  lower  down,  with  regard 
to  a  pair  of  parallel  straight  lines;  and  he  now  goes  on  to  discuss  the  case 
of  non-parallel  straight  lines. 


THE  MANUSCRIPTS  OF  LEIBNIZ.  247 

tional  to  these,  that  is,  when  a  straight  line  passes  through  F,  L, 
(L), 

Hence,  since  we  are  not  here  given  several  regions,  it  follows 
that  the  locus  is  a  straight  line.  Therefore,  given  the  two  moments 

of  a  figure  about  two  straight  lines  that  are  not  parallel, , 

the  area  of  the  figure  will  be  given,  and  also  its  center  of  gravity.12 

Behold  then  the  fundamental  theorem  on  centers  of  gravity.  If 
two  moments  of  the  same  figure  about  two  parallel  straight  lines 
are  given,  then  the  area  of  the  figure  is  given,  but  not  its  center  of 
gravity. 

Since  it  is  the  aim  of  the  center  of  gravity  method  to  find 
dimensions  from  given  moments,  we  have  hence  two  general  the- 
orems : 

If  we  are  given  two  moments  of  the  same  figure  about  two 
straight  lines,  or  axes  of  libration,  that  are  parallel  to  one  another, 
then  its  magnitude  is  given;  also  when  the  moments  about  three 
non-parallel  straight  lines  are  given.  From  this  it  is  seen  that  a 
method  for  finding  elliptic  and  hyperbolic  curves  from  given  quad- 
ratures of  the  circle  and  the  hyperbola  is  evident.13  But  of  this  in 
a  special  note. 

§  V. 

The  next  manuscript  to  be  considered  is  a  continuation 
of  the  preceding,  and  is  dated  the  next  day.  Its  character 
is  of  the  nature  of  disjointed  notes,  set  down  for  further 
consideration. 


12  The  passage  in  Gerhardt  reads : 

Datis  ergo  duobus  momentis  figurae  ex  duabus  rectis  non  parallelis,  dabi- 
tur  figurae  momentis  tribus  axibus  librationis,  qui  non  sint  omnes  parallel! 
inter  se,  dabitur  figurae  area,  et  centrum  gravitatis. 

For  this  I  suggest : 

Datis  ergo  tribus  momentis  figurae  ex  tribus  rectis  non  parallelis,  aliter 
figurae  momentis  tribus  axibus  librationis,  qui  non  sunt  omnes  paralleli  inter 
se 

The  passage  would  then  read: 

Given  three  moments  of  a  figure  about  three  straight  lines  that  are  not 
parallel,  in  other  words,  the  moments  of  the  figure  about  three  axes  of  libra- 
tion, which  are  not  all  parallel  to  one  another,  then  the  area  of  the  figure  will 
be  given  and  also  the  center  of  gravity. 

If  the  alternative  words  are  written  down,  one  under  the  other,  and  not 
too  carefully,  I  think  the  suggested  corrections  will  appear  to  be  reasonable. 

18  Apparently,  here  Leibniz  is  referring  back  to  the  theorem  at  the  beginning 
of  the  article. 


248  THE  MONIST. 

26  October,  1675. 

Another  tetragonistic  analysis  can  be  obtained  by  the  aid  of 
curves.  Thus,  let  the  same  curve  be  resolved  into  different  elements, 
according  as  the  ordinates  are  referred  to  different  straight  lines. 
Hence  also  arise  diverse  plane  figures,  consisting  of  elements  similar 
to  the  given  curve ;  and  since  all  of  these  are  to  be  found  from  the 
given  dimension  of  the  curve,  it  follows  that  from  the  dimension 
of  any  one  of  the  curves  of  this  kind  the  rest  are  obtained. 

In  other  ways  it  is  possible  to  obtain  curves  that  depend  on 
others,  if  to  the  given  curve  are  added  the  ordinates  of  figures  of 
which  the  quadrature  is  either  known  or  can  be  obtained  from  the 
quadrature  of  the  given  one. 

Just  as  areas  are  more  easily  dealt  with  than  curves,  because 
they  can  be  cut  up  and  resolved  in  more  ways,  so  solids  are  more 
manageable  than  planes  and  surfaces  in  general.  Therefore,  when- 
ever we  divert  the  method  for  investigating  surfaces  to  the  con- 
sideration of  solids,  we  discover  many  new  properties;  and  often 
we  may  give  demonstrations  for  surfaces  by  means  of  solids  when 
they  are  with  difficulty  obtained  from  the  surfaces  themselves. 
Tschirnhaus  observed  in  a  delightful  manner  that  most  of  the  proofs 
given  by  Archimedes,  such  as  the  quadrature  of  the  parabola,  and 
dependent  theorems  on  the  sphere,  cone,  and  cylinder,  can  be  re- 
duced to  sections  of  rectilinear  solids  only,  and  to  a  composition  that 
is  easily  seen  and  readily  handled. 

Various  -ways  of  describing  new  solids. 

If  from  a  point  above  a  plane  a  rigid  descending  straight  line 
is  moved  round  an  area,  of  any  shape  whatever,  diverse  kinds  of 
conical  bodies  are  produced.  Thus  if  the  plane  area  is  bounded 
by  the  circumference  of  a  circle,  a  right  or  scalene  cone  is  produced. 
Also  if  the  figure  used  for  the  base,  or  the  plane  area,  has  a  center — 
an  ellipse  for  example — then  we  get  an  elliptic  cone,  which  is  a  right 
cone  if  the  given  point  is  directly  above  the  center,  and  if  not  it  is 
scalene.  Another  conic  gives  another  elliptic  cone. 

If  the  rigid  line  drawn  down  from  the  point  is  circular  or  some 
other  curve,  at  one  time  it  is  so  fixed  to  the  point  or  pole  that  it  has 
freedom  to  move  in  one  way  only,  say  round  an  axis,  in  which  case 
it  is  necessary  that  the  base  should  be  a  circle  and  that  the  fixed 
point  or  pole  should  be  directly  over  the  center.  At  another  time 
it  is  necessary  that  the  rigid  line  should  have  freedom  for  other 
motions,  such  as  an  up  and  down  motion,  or  some  other  motion, 


THE  MANUSCRIPTS  OF  LEIBNIZ.  249 

controlled  by  some  straight  line;  and  then  it  will  always  ascend 
or  descend  when  necessary,  so  that  it  ever  touches  the  given  plane 
area  by  its  rotation  round  the  axis ;  and  this  is  the  second  class  of 
cones.  A  third  class  consists  of  those  in  which,  besides  the  double 
motion  of  a  rotation  round  an  axis  and  an  up  and  down  motion, 
the  curve  alone,  or  the  axis  alone,  or  even  both  the  curve  and  the 
axis,  also  perform  other  motions  meanwhile,  or  even  the  point  itself 
moves. 

Here  is  another  consideration. 

The  moments  of  the  differences  about  a  straight  line  perpen- 
dicular to  the  axis  are  equal  to  the  complement  of  the  sum  of  the 
terms ;  and  the  moments  of  the  terms  are  equal  to  the  complement 
of  the  sum  of  the  sums,  i.  e., 


n  ult.;r,  omn.w,,  —  omn.  omn.w 


OTun.    ur 


az 

Let  xw  n  az,  then  w  "">  — ,  and  we  have 

x 

az  az 

omn.as  "  ult..r.  omn. omn.omn. —  : 

x  x   ' 

az  az  az 

hence  omn.  —  ""•  ult.^r    omn.  — =  -  omn.omn.  —5  ; 

X  X2  X2    ' 

inserting  this  value  in  the  preceding  equation,  we  have 


10  az       i,  az 

omn.a^r  *~>  ult.-r2  omn.  -=•  -  ult.^r.  omn.omn.  -=• , 
x2  x2 

14  I  have  given  this  equation,  and  those  that  immediately  follow  it,  in 
facsimile,  in  order  to  bring  out  the  necessity  that  drove  Leibniz  to  simplify 
the  notation. 

We  have  here  a  very  important  bit  of  work.  Arguing  in  the  first  instance 
from  a  single  figure,  Leibniz  gives  two  general  theorems  in  the  form  of  moment 
theorems.  The  first  is  obvious  on  completing  the  rectangle  in  his  diagram, 
and  this  is  the  one  to  which  the  given  equation  applies.  In  the  other  the  whole, 
of  which  the  two  parts  are  the  complements,  is  the  moment  of  the  completed 
rectangle ;  its  equivalent  is  the  equation 

omn^ey  =  ult.jr  omn.y  —  omn.  omn.y. 

Now,  although  Leibniz  does  not  give  this  equation,  it  is  evident  that  he  rec- 
ognized the  analogy  between  this  and  the  one  that  is  given ;  for  he  immediately 
accepts  the  relation  as  a  general  analytical  theorem  that  he  can  use  without 
any  reference  to  any  -figure  whatever,  and  proceeds  to  develop  it  further. 
This  would  therefore  seem  to  be  the  point  of  departure  that  led  to  the  Leib- 
nizian  calculus. 


250  THE  MONIST. 


ii:  az 

—  omn.  ult.jr.  own.—:  —  omn.omn.— 7  ; 
x2  x2 

and  this  can  proceed  in  this  manner  indefinitely. 

a  a  a 

Again,        omn.  —  •"»  x  omn.  -=-  —  omn.omn.  — z- 
x  x*  xf 

and  omn.a  n  ult  JT  omn.  —  omn.  omn.  —  ; 

x  x 

the  last  theorem  expresses  the  sum  of  logarithms  in  terms  of  the 
known  quadrature  of  the  hyperbola.15 

The  numbers  that  represent  the  abscissae  I  usually  call  ordinals, 
because  they  express  the  order  of  the  terms  or  ordinates.  If  to  the 
square  of  any  ordinate  of  a  figure  whose  quadrature  can  be  found, 
you  add  the  square  of  a  constant,  the  roots  of  the  sum  of  the  two 
squares  will  represent  the  curve  of  the  quadratrix.  Now  if  these 
roots  of  the  sum  of  the  two  squares  can  also  give  an  area  that  has 
a  known  quadrature,  then  also  the  curve  can  be  rectified.16 

15  Having  freed  the  matter  from  any  reference  to  figures,  he  is  able  to 
take  any  value  he  pleases  for  the  letters.  He  supposes  that  s=l,  and  thus 
obtains  the  last  pair  of  equations.  He  then  considers  x  and  w  as  the  abscissa 
and  ordinate  of  the  rectangular  hyperbola  JTW  =  a  (constant)  ;  hence  omn.a/T 
or  omn.  w  is  the  area  under  the  hyperbola  between  two  given  ordinates,  and 
therefore  a  logarithm;  and  thus  omn. omn.o/jr  is  the  sum  of  logarithms,  as 
he  states. 

*•  There  only  seem  to  be  two  possible  sources  for  this  paragraph,  (1) 
original  work  on  the  part  of  Leibniz,  and  (2)  from  Barrow.  For  we  know 
that  Neil's  methods  was  that  of  Walk's,  and  the  method  of  Van  Huraet  used 
an  ordinate  that  was  proportional  to  the  quotient  of  the  normal  by  the  ordinate 
in  the  original  curve. 

Now  Barrow,  in  Lect  XII,  §20,  has  the  following:  "Take  as  you  may 
any  right-angled  trapezial  area  (of  which  you  have  sufficient  knowledge), 
bounded  by  two  parallel  straight  lines  AK,  DL,  a  straight  line  AD,  and  any 
Hue  KL  whatever;  to  mis  let  another  such  area  be  so  related  that  when  any 
straight  line  FH  is  drawn  parallel  to  DL,  cutting  the  lines  AD,  CE,  KL  in  the 
points  F,  G,  H,  and  some  determinate  line  Z  is  taken,  the  square  on  FH  is 
equal  to  the  squares  on  FG  and  Z.  Moreover,  let  the  curve  AIB  be  such  that, 


if  the  straight  line  GFI  is  produced  to  meet  it,  the  rectangle  contained  by  Z 
and  FI  is  equal  to  the  space  AFGC;  then  the  rectangle  contained  by  Z  and 
the  curve  AB  is  equal  to  the  space  ADLK.  The  method  is  just  the  same, 
even  if  the  straight  line  AK  is  supposed  to  be  infinite. 

This  striking  resemblance,  backed  by  the  fact  that  there  seems  to  be  no 
connection  between  this  theorem  and  the  rest  of  the  paper,  that  Leibniz  gives 


THE  MANUSCRIPTS  OF  LEIBNIZ.  25! 

To  describe  a  curve  to  represent  a  given  progression. 

From  the  square  of  a  term  of  the  progression,  take  away  the 
square  of  a  constant  quantity;  if  the  figure  that  is  the  quadratrix 
of  the  roots  formed  from  the  two  squares  is  described,  it  will  give 
the  curve  required ;  it  does  not  follow  that  a  rectifiable  curve  can 
be  described. 

The  elements  of  the  curve  described  can  be  expressed  in  many 
different  ways.  Different  methods  of  expressing  the  elements  of 
a  curve  may  be  compared  with  different  methods  of  expressing 
a  figure  having  similar  parts  with  it,  according  as  it  is  referred  in 
different  ways.  Lastly,  a  solid  having  similar  parts  with  a  curve 
can  thus  far  be  expressed  in  many  ways,  and  so  also  for  a  surface 
or  figure  having  similar  parts  with  the  curve. 

§  VL 

Three  days  later,  Leibniz  considers  the  possibility  of 
being  able  to  find  the  quadratrix  in  all  cases,  or  when  that 
is  impossible,  some  curve  which  will  serve  for  the  quadra- 
trix very  approximately.  He  makes  an  examination  of  the 
difficulties  that  are  likely  to  be  met  with  and  the  means  to 
overcome  them,  and  he  seems  to  be  satisfied  that  the  method 
can  be  made  to  do  in  all  cases.  But  in  the  absence  of  an 
example  of  the  method  he  proposes  to  adopt,  he  seems  only 
to  have  been  wasting  his  time.  But  this  may  be  dismissed, 
for  it  is  not  here  that  the  importance  of  this  essay  lies;  it 
is  altogether  in  what  follows. 

The  rest  of  the  essay  is  in  the  form  of  disjointed  notes: 
it  is  just  the  kind  of  thing  that  any  one  would  write  as 
notes  while  reading  the  works  of  others.  This  is  what  I 
take  it  to  be ;  and  the  works  he  is  considering  are  those  of 

no  attempt  at  a  proof,  (indeed  I  very  much  doubt  whether  I  could  have  made 
out  his  meaning  from  the  original  unless  I  had  recognized  Barrow's  theorem) 
and  that  Leibniz  gives  1675  as  the  date  of  his  reading  Barrow,  almost  forces 
one  to  conclude  that  this  is  a  note  on  a  theorem  (together  with  an  original 
deduction  therefrom  by  himself)  which  Leibniz  has  come  across  in  a  book 
that  is  lying  before  him,  and  that  that  book  is  Barrow's.  Against  it,  we  have 
the  facts  of  the  use  of  the  word  "quadratrix,"  not  in  the  sense  that  Barrow 
uses  it,  namely  as  a  special  curve  connected  with  the  circle;  that  the  quad- 
ratrix is  one  of  the  special  curves  that  Barrow  considers  in  the  five  examples 
he  gives  of  the  Differential  Triangle  method;  and  that  another  example  of 
this  method  is  the  differentiation  of  a  trigonometrical  function  which  seems 
to  be  unknown  to  Leibniz. 


252  THE  MONIST. 

Descartes,  Sluse,  Gregory  St.  Vincent,  James  Gregory  and 
Barrow.  Descartes  he  has  already  dismissed  as  imprac- 
ticable in  the  manuscript  of  January,  1675;  but  there  are 
indications  that  the  former's  method  has  still  some  influ- 
ence. An  incidental  remark  leads  to  the  consideration  of 
the  ductus  of  Gregory  St.  Vincent ;  but  these  too  are  soon 
cast  aside,  truly  because  Leibniz  does  not  quite  grasp  the 
exact  meaning  of  Gregory.  He  then  either  remembers 
what  he  has  seen  in  Barrow  or  refers  to  it  again,  for  the 
next  thing  he  gives  is  some  work  in  connection  with  which 
he  draws  the  characteristic  triangle,  which  is  here  for  the 
first  time,  as  far  as  these  manuscripts  go,  the  Barrow  form 
and  not  the  Pascal  form.  He  immediately  obtains  some- 
thing important,  namely, 


omn.  I2 


=  omn.  omn.  /— . 
a 


Noting  that,  in  modern  notation,  /  is  dy,  and  a  is  dx, 
and  also,  since  a  is  also  supposed  to  be  unity,  that  the 
final  summation  on  the  right-hand  side  is  performed  by 
"applying  the  successive  values  to  the  axis  of  x,  while  the 
summation  denoted  by  omn./  is  a  straightforward  summa- 
tion, it  follows  that  the  equivalent  of  the  result  obtained 

dy 
by  Leibniz  is  %y*  =  fy  -r  dx. 

However,  in  attempting  to  put  this  theorem  into  words 
as  a  general  theorem  he  makes  an  error ;  he  quotes  omn./2  as 
the  "sum  of  the  squares"  instead  of  the  "square  of  the 
final  y."  This  I  think  is  simply  a  slip  on  the  part  of  Leib- 
niz, and  not,  as  suggested  by  Gerhardt  and  Weissenborn, 

-  an  indication  that  Leibniz  confused  omn./2  with  omn./2,  and 
considered  them  as  equivalent.  Neither  of  these  authori- 

,  ties  appears  to  have  noticed  the  fact  that  when  Leibniz 
has  invented  the  sign  /  (which  he  immediately  proceeds 
to  do)  he  carefully  makes  the  distinction  between  the 


THE  MANUSCRIPTS  OF  LEIBNIZ.  253 

equivalents  to  the  square  of  a  sum  and  the  sum  of  the 
squares.    Thus  we  find  that  his  equation  is  written  as 

J  4  =  J  Jl - ,         (note  the  vinculum) 

while  later  in  the  essay  we  have  j*/3  to  stand  for  the  sum 
of  the  cubes.  Further,  apart  from  this.  I  do  not  think  that 
any  one  can  impute  such  confusion  of  ideas  to  Leibniz,  if 
it  is  noted  that  so  far  this  is  not  the  differential  calculus, 
but  the  calculus  of  differences,  i.  e.,  /  is  still  a  very  small 
but  finite  line  and  not  an  infinitesimal ;  for  in  §  IV,  Leibniz 
had  squared  a  trinomial  successfully,  and  must  have  known 
that  the  sum  of  the  squares  could  not  be  equal  to  the  square 
of  the  sum.  Both  these  above-named  authorities  seem  to 
find  some  difficulty  over  the  introduction  of  the  letter  a, 
apparently  haphazard.  This  difficulty  becomes  non-exis- 
tent, if  it  is  remembered  that  a  is  taken  to  be  unity,  and 
the  remarks  made  about  dimensions  by  Leibniz  are  care- 
fully considered;  it  will  then  be  found  that  the  a  is  in- 
troduced to  keep  the  equations  homogeneous!  Weissen- 
born  also  remarks  that  Leibniz  jots  down  the  integral  of 
x2  without  giving  a  proof,  and  appears  to  be  in  doubt  how 
he  reached  it.  If  this  is  so,  it  confirms  the  opinion  that  I 
have  already  formed,  namely,  that  neither  Gerhardt  nor 
Weissenborn  tried  to  get  to  the  bottom  of  these  manu- 
scripts, being  content  with  simply  "skimming  the  cream." 
I  suggest  that  Barrow,  Gregory  St.  Vincent,  and  even 
Sluse,  now  join  Descartes  on  the  shelf  or  the  floor,  and  that 
the  rest  of  the  essay  is  all  Leibniz.  He  writes  the  two 
equations  he  has  found,  the  equivalents  to  two  theorems 
obtained  geometrically,  notes  the  fact  that  these  are  true 
for  infinitely  small  differences  (without,  however,  men- 
tioning that  they  are  only  true  in  such  a  case),  discards 
diagrams,  and  proceeds  analytically;  that  is,  the  y's  are 
successive  values  of  some  function  of  x,  where  the  values 


254  THE  MONIST. 

of  x  are  in  arithmetical  progression;  hence,  substituting 
x  for  /  in  the  equation 

omn.xl  =  omn./  —  omn.  omn./, 

and  remembering  that  omn.jir  ==  x2/?,  as  he  has  proved, 
we  have 

_2T  _2T  2T 

omn.  x*  =  x  — -  —  omn.  —  ,  or  omn.  x*  =  — -  . 
22  3 

/.j-3        jj/4 
—  =  —  correctly  (although 

O  T" 

there  is  an  obvious  slip  or,  as  I  think,  a  misprint  of  /  for  x)  ; 
this  could  have  been  obtained  in  the  same  way. 

x^  x^  x^ 

omn.  x*  =  x  —  —  omn.  —  ,  or  omn.  x*  =  —  . 

Similarly,  Leibniz  could  have  gone  on  indefinitely,  and 
thus  obtained  the  integrals  of  all  the  powers  of  x.  But 
his  brain  is  too  active ;  as  Weissenborn  says,  his  soul  is  in 
the  throes  of  creation.  He  merely  alludes  in  passing  to  the 
inverse  operation  to  /  as  being  represented  by  d,  which 
he  for  some  reason  writes  in  the  denominator  (probably 
erroneously  because  he  has  noted  that  /  increases  the  di- 
mensions) ;  and  then  he  harks  back  to  the  opening  idea  of 
the  essay,  the  obtaining  of  the  quadratrix  by  means  of 
transformation  of  equations,  an  idea  truly  as  hopeless  as 
the  method  of  Descartes  which  he  has  discarded.  Never- 
theless, even  then  he  obtains  something  remarkable,  noth- 
ing more  or  less  than  the  inverse  of  the  differentiation  of  a 
product.  This  fundamental  theorem  is  obtained  geomet- 
rically; the  proof  of  the  little  theorem  on  which  the  final 
result  is  founded  is  not  given,  neither  is  there  a  diagram. 
It  cannot  therefore  be  supposed  but  that  Leibniz  is  work- 
ing from  a  diagram  already  drawn,  and  I  suggest  he  was 
referring  to  one  of  those  theorems,  with  which  he  had 
filled  "hundreds  of  pages"  between  1673  and  1675.  The 


THE  MANUSCRIPTS  OF  LEIBNIZ.  255 

proof  follows  quite  easily  by  the  use  of  the  characteristic 
triangle,  and  is  given  in  a  footnote.  This  theorem  is  not 
in  Barrow,  nor  can  I  remember  seeing  it  in  Cavalieri; 
I  have  not  yet  been  able  to  procure  a  Gregory  St.  Vincent  ; 
it  may  be  in  James  Gregory. 

The  benefits  of  this  discovery  are  lost  as  before,  for 
Leibniz  once  more  alludes  to  the  transformation  of  equa- 
tions for  the  purpose  of  obtaining  the  quadratrix. 

Summing  the  whole  essay,  we  can  say  that  in  it  is  the 
beginning  of  the  Leibnizian  analytical  calculus. 

29  October,  1675. 

Analyseos  Tetragonisticae  pars  secunda. 
(Second  part  of  analytical  quadrature.) 

I  think  that  now  at  last  we  can  give  a  method,  by  which  the 
analytical  quadratrix  may  be  found  for  any  analytical  figure,  when- 
ever that  is  possible  ;  and,  when  it  can  not  be  done,  it  will  yet  always 
be  possible  that  an  analytical  figure  may  be  described,  which  will 
act  as  the  quadratrix  as  nearly  as  is  required.  This  is  how  I  look 
at  it: 

Suppose  the  equation  of  the  curve,  of  which  the  quadratrix 
is  required,  is  given,  and  that  the  unknowns  in  it  are  x  and  v.  Let 
the  equation  to  the  curve  required  be17 

v  =  b  +  cx  +  dy  +  ex2  +  fy2  +  gyx  +  hys  +  lx*  +  mxyy  +  yxx  +  etc.  ;    ...    (i) 
let  it  be  set  in  order  for  tangents,  as  follows  : 
-dy-  2fy2  -  gyx  -  3hy3  -  2mxy2  -  mxzy  -  etc. 

(ii) 


17  This  is  either  a  misprint,  v  instead  of  O,  or  else  Leibniz  is  in  error. 
For  Slusius's  method  there  must  be  only  two  variables  in  the  equation.  In  the 
Phil.  Trans,  for  1672  (No.  90),  Sluse  gives  his  method  thus: 

If  y5  -f-  by*  =  2qqv3  —  yyv3,  then  the  equation  must  be  written  y5  -f-  by4  -f- 
yy3  =  2qqv3  —  yyv3  ;  then  multiply  each  term  on  the  left-hand  side  by  the 
number  of  y's  in  the  term,  and  substitute  t  in  place  of  one  y  in  each  ;  similarly 
multiply  each  term  on  the  left-hand  side  by  the  exponent  of  v;  the  equation 
obtained  will  give  the  value  of  t. 

The  use  of  the  letters  v  and  y  is  to  be  noted  in  connection  with  Leibniz's 
use  of  the  same  letters  ;  it  does  not  seem  at  all  necessary  that  Leibniz  should 
have  seen  Newton's  work,  with  this  ready  to  the  former's  hand,  as  a  member 
of  the  Royal  Society.  I  suggest  that  Sluse  obtained  his  rule  by  the  use  of  a 
and  e,  as  given  in  Barrow.  Can  Barrow's  words  usitatum  a  nobis  (in  the 
midst  of  a  passage  written  in  the  first  person  singular)  have  meant  that  the 
method  was  common  property  to  himself  and  several  other  mathematicians 
that  were  contemporary  with  him?  This  would  explain  a  great  deal. 


256  THE  MONIST. 

Now,  t/y  =  a/v;  hence,  if  from  the  equation  t/y  =  a/v,  we  elim- 
inate t  and  y  by  the  help  of  equations  (i)  and  (ii),  that  equation 
should  be  produced  which  represents  the  figure  that  has  to  be 
quadratured ;  and  by  comparing  the  terms  of  the  equation  thus  ob- 
tained with  the  given  equation,  unless  indeed  there  is  no  possibility 
of  comparing  them,  we  shall  obtain  the  quadrature. 

But  if  an  impossibility  arises,  it  is  then  known  that  the  given 
analytical  figure  has  no  analytical  quadratrix.  But  it  is  quite  clear 
that  if  we  add  to  it  such  as  will  change  it  almost  imperceptibly,  then 
a  quadrible  figure  may  be  obtained,  since  this  plainly  produces  an- 
other equation.  However,  as  an  impossible  case  may  arise,  we  must 
consider  the  difficulties. 

Say  that  the  equation  that  is  obtained  is  of  infinite  prolixity, 
while  the  given  one  is  finite.  My  answer  is,  that  in  comparing  the 
one  with  the  other  it  will  be  seen  how  far  at  most  the  powers  of 
the  unknowns  in  the  indefinite  equation  can  go.  The  retort  may 
be  made,  that  it  may  happen  that  the  indefinite  equation  obtained 
may  have  more  terms  than  the  finite  equation  that  is  given  and  yet 
may  be  reduced  to  it,  for  it  may  be  divided  by  something  else  that 
is  either  finite  or  indefinite.  This  difficulty  hindered  me  for  a  long 
time  a  year  ago,  but  now  I  see  that  we  should  not  be  stopped  by  it. 
For  it  may  happen  that  from  a  certain  determinate  figure  (whose 
equation  is  not  divisible  by  a  rational)  by  the  method  of  tangents 
there  may  arise  an  ambiguous  figure ;  for  it  is  impossible  to  say 
that,  for  any  figure,  there  shall  be  only  one  tangent  at  any  one  point. 
Hence  the  produced  equation  can  neither  be  divided  by  a  finite  nor 
by  an  indefinite  quantity ;  for  in  truth  indefinite  figures,  or  those 
whose  ordinates  are  represented  by  an  infinite  equation,  have  some- 
times these  very  ordinates  finite,  and  these  ought  to  satisfy  the 
equation.  Notwithstanding  that,  I  foresee  another  difficulty ;  for 
indeed  it  seems  that  sometimes  it  may  happen  that  all  the  roots  of 
the  equation  will  not  serve  for  the  solution  of  the  problem.  Yet, 
to  tell  the  truth,  I  believe  they  will  do  so. 

Now  here  is  a  difficulty  that  really  is  great.  It  may  happen 
that  a  finite  equation  may  also  be  expressed  as  an  indefinite  one, 
so  that  the  equation  obtained  may  really  be  the  same  as  the  given 
equation  although  it  does  not  appear  to  be.  For  example, 

<y2  =  x/(  1  +  x*)  =  x  -  xz  +  x3  -  x*  +  x*  -  x6  +  etc. ; 

and  in  the  same  way  others  can  be  formed  by  various  compositions 
and  divisions.    This  I  confess  is  truly  a  difficult  point,  but  it  can  be 


THE  MANUSCRIPTS  OF  LEIBNIZ. 


257 


answered  thus :  If  a  figure  has  an  analytical  quadratrix  of  any  sort, 
in  all  cases  it  may  be  assumed  to  be  an  indefinite  one ;  and  then  it 
will  not  in  all  cases  give  an  indefinite,  but  sometimes  a  finite,  equa- 
tion that  is  equivalent  to  the  given  equation.  In  the  same  way, 
it  is  certain  that  the  quadratrix  of  a  given  curve  as  it  is  usually 
investigated,  whenever  there  is  one,  will  also  be  determined ;  and 
that  too  given  uniquely  and  not  ambiguously,  so  that  any  that  differs 
from  it,  differs  only  in  name.  There  is  still  one  difficulty  left;  it 
seems  impossible  to  determine  which  is  the  end  or  first  term  of  the 
indefinite  equation  that  is  obtained ;  for  it  may  happen  that  the  terms 
of  lower  degree  are  cut  out,  and  then  it  is  divisible  by  y  or  x 
or  yx  or  powers  of  these;  nor  do  I  see  that  there  is  anything  to 
prevent  this.  There  is  the  same  difficulty  whether  you  start  from 
the  lowest  or  the  highest  degree  in  the  equation  assumed  to  begin 
with  as  indefinite.  Suppose  then  that  in  the  equation  obtained  this 


division  is  possible,  then  it  is  necessary  that  the  constant  term 
should  be  absent,  and  also  all  those  terms  in  which  x  alone  or,  if 
you  like,  all  the  terms  in  which  y  alone  is  absent ;  and  if  we  examine 
this  continuously  we  may  light  upon  an  impossibility. 

In  this  general  calculus  then,  we  may  take  it  as  certain  that 
this  difficulty  is  solved,  and  that  such  a  division  after  the  calcula- 
tion can  never  happen;  or  if  it  is  possible  for  it  to  happen,  then 
the  terms  will  go  out,  one  after  the  other,  so  that  the  equation  can 
be  depressed  and  the  comparison  be  made ;  and  then  it  is  to  be  seen 
whether  this  difficulty  cannot  be  overcome  in  general,  and  the  com- 
parison proceed  as  we  proceed  with  the  elimination.  Perhaps  if 
the  figure  to  be  quadratured  is  reduced  beforehand  to  its  simplest 
equation  possible,  impossibilities  will  the  more  readily  be  detected. 
For  then  presumably  the  quadratrix  must  become  more  simplified. 
In  addition  we  have  another  source  of  assistance ;  for  various  cal- 


258  THE  MONIST. 

culations  leading  to  the  same  thing,  though  obviously  differing  from 
one  another,  can  be  contrived,  from  which  equations  are  comparable. 

Let  BL  =      WL  =  /,  EP  =   ,  TB  =  t,  GW  =  a,  then     =  omn.l. 


Incidentally  I  may  remark  that  there  are  composite  numbers 
that  cannot  be  added  or  subtracted  from  one  another  by  parts, 
namely  those  denominated  by  powers,  or  by  sub-powers  or  surds. 
There  are  also  other  denominate  numbers  which  cannot  be  multi- 
plied together  by  parts,  such  as  numbers  representing  sums  ;  for 
instance,  omn.l  cannot  be  multiplied  by  omn.p,  nor  can  we  have 
y  =  2omn.  omn.pl.  However,  as  such  a  multiplication  may  be  im- 
agined to  occur  under  certain  conditions,  we  must  consider  it  as 
follows  : 

We  require  the  space  that  represents  the  product  of  all  the 
p's  into  all  the  I's;  we  cannot  make  use  of  the  ductions  of  Gregory 
St.  Vincent,  where  figures  are  multiplied  by  figures,  for  by  this 
method  one  ordinate  is  not  multiplied  by  all  the  others,  but  one  into 
one.  You  may  say  that  if  one  ordinate  is  multiplied  by  all  the  rest 
it  will  produce  a  sursolid  space,  namely,  the  sum  of  an  infinite  num- 
ber of  solids.  For  this  difficulty  I  have  found  a  remedy  that  is 
really  admirable.  Let  every  /  be  represented  by  an  infinitely  short 
straight  line  WL,  that  is,  we  want  the  quadratrix  line  representing 
omn.  /  ;  well,  the  line  BL  =  omn.  /  ;  and  if  this  is  multiplied  by  every 
p,  each  represented  by  a  plane  figure,  then  a  solid  is  produced. 
If  all  the  I's  are  straight  lines  and  all  the  p's  are  curves,  a  curved 
surface  is  produced  by  a  duction  of  the  same  sort.  But  these  things 
are  all  old;  now,  here  is  something  new. 

If  upon  WL,  MG,  or  every  single  /,  is  superimposed  the  same 
curve  representing  all  the  p's,  where  the  curve  p  is  originally  all  in 
the  same  plane  and  is  carried  along  the  curve  AGL  while  its  plane 
always  moves  parallel  to  itself,  then  what  we  require  will  be  ob- 
tained. In  place  of  a  curve  a  plane  may  be  carried  along  the  curve 
in  the  same  manner,  and  a  solid  will  be  obtained,  whereas  by  the 
former  method  it  was  a  curvilinear  surface;  and  both  for  the  sur- 
face and  for  the  solid  the  section  always  remains  the  same.  It 
remains  to  be  seen  whether  a  number  of  analytical  surfaces  cannot 
be  ascertained,  as  in  the  case  of  analytical  lines  ;  but  this  is  men- 
tioned only  incidentally. 

N.  B.  The  curvilinear  surface  formed  by  the  motion  of  a 
curve  parallel  to  itself  along  the  curve  will  be  equal  to  the  cylinder 


THE  MANUSCRIPTS  OF  LEIBNIZ.  259 

of  the  curve  under  BL,  the  sum  of  all  the  I's  but  this  is  also  men- 
tioned incidentally. 

To  resume,  —  =  — — —  =  y,    therefore  p=~^—  /.     Hence, 
a        omn.  /  a 

omn.  y  -  does  not  mean  the  same  thing  as  omn.y  into  omn./,  nor  yet 
a 


y  into  omn./ ;  for,  since  p  =  —  /  or  -  ™—  /,  it  means  the  same  thing 

a  a 

as  omn./  multiplied  by  that  one  /  that  corresponds  with  a  certain 

p;    hence,   omn./>  =  omn. — /.     Now   I   have  otherwise   proved 

a 


omn.p=  ¥-,  i.  e.,  = — ir~'>  therefore  we  have  a  theorem  that  to  me 

Lt  £ 

seems  admirable,  and  one  that  will  be  of  great  service  to  this  new 
calculus,  namely, 


omn.  I2  -  ,  /  ,  , 

-  —  =  omn.  omn./—,  whatever  /  may  be; 
2  a 

that  is,  if  all  the  I's  are  multiplied  by  their  last,  and  so  on  as  often 
as  it  can  be  done,  the  sum  of  all  these  products  will  be  equal  to  half 
the  sum  of  the  squares,  of  which  the  sides  are  the  sum  of  the  /'s 
or  all  the  I's.  This  is  a  very  fine  theorem,  and  one  that  is  not  at  all 
obvious. 

Another  theorem  of  the  same  kind  is: 

omn.^r/  =  x  omn./  -  omn.omn./  , 

where  /  is  taken  to  be  a  term  of  a  progression,  and  x  is  the  number 
which  expresses  the  position  or  order  of  the  /  corresponding  to  it  ; 
or  x  is  the  ordinal  number  and  /  is  the  ordered  thing. 

N.  B.  In  these  calculations  a  law  governing  things  of  the  same 
kind  can  be  noted  ;  for,  if  omn.  is  prefixed  to  a  number  or  ratio,  or 
to  something  indefinitely  small,  then  a  line  is  produced,  also  if  to 
a  line,  then  a  surface,  or  if  to  a  surface,  then  a  solid  ;  and  so  on 
to  infinity  for  higher  dimensions. 

It  will  be  useful  to  write  J  for  omn.,  so  that 

j*/  =  omn./,  or  the  sum  of  the  I's. 


Thus, 

From  this  it  will  appear  that  a  law  of  things  of  the  same  kind 


26O  THE  MONIST. 

should  always  be  noted,  as  it  is  useful  in  obviating  errors  of  cal- 
culation. 

N.  B.  If  (I  is  given  analytically,  then  /  is  also  given  ;  therefore 
if  j*  J7  is  given,  so  also  is  /;  but  if  /  is  given,  J*/  is  not  given  as  well. 
In  all  cases  (x  =  x'*/2. 

N.  B.  All  these  theorems  are  true  for  series  in  which  the 
differences  of  the  terms  bear  to  the  terms  themselves  a  ratio  that  is 
less  than  any  assignable  quantity. 

/&  _  •*_ 
"  3 

Now  note  that  if  the  terms  are  affected,  the  sum  is  also 
affected  in  the  same  way,  such  being  a  general  rule  ;  for  example, 

I  —  I  =  ^-  x   I  /  ,  that  is  to  say,  if  ^  is  a  constant  term,  it  is  to  be 
^   b        b       J  b 

multiplied  by  the  maximum  ordinal  ;  but  if  it  is  not  a  constant  term, 
then  it  is  impossible  to  deal  with  it,  unless  it  can  be  reduced  to  terms 
in  /,  or  whenever  it  can  be  reduced  to  a  common  quantity,  such  as 
an  ordinal. 

N.  B.  As  often  as  in  the  tetragonistic  equation,  only  one  letter, 
say  I,  varies,  it  can  be  considered  to  be  a  constant  term,  and  J/  will 
equal  x.  Also  on  this  fundamental  there  depends  the  theorem: 


/     -/? 


//,  that 

Hence,  in  the  same  way  we  can  immediately  solve  innumerable 
things  like  this  ;  thus,  we  require  to  know  what  e  is,  where 


c 

1/ 
J 


/3  _f_     I   /3   _   ^3. 
S    a~  J 

we  have 

«j  CJC^  j     9  ^  o 

ar<?  =   —  +  barx  +   T~  +  **• 

«5  TC  • 

For  indeed   Cl3  =  x,  because  /  is  supposed  to  be  equal19  to  a  for  the 

purpose  of  the  calculation ;       —  =  x. 

J  a 

18  There  is  evidently  a  slip  here ;  /  should  be  x. 

19  This  is  an  instance  of  the  care  which  Leibniz  takes ;  in  the  work  above 
/  has  been  the  difference  for  y,  and  a  the  difference  for  x ;  he  is  now  integrating 
an  algebraical  expression,  and  not  considering  a  figure  at  all.;  hence  /==o,  and 
o  is  equal  to  unity,  and  therefore  /  I3  =  fix  —  azx  =  x  \     Thus  what  is  gen- 
erally considered  to  be  a  muddle  turns  out  to  be  quite  correct.    The  muddle 
is  not  with  Leibniz,  it  is  with  the  transcriber.    It  is  certain  that  these  manu- 
scripts want  careful  republishing  from  the  originals ;  won't  some  millionaire 
pay  to  have  them  reproduced  photographically  in  an  edition  de  luxe? 


THE  MANUSCRIPTS  OF  LEIBNIZ.  26l 


da. 


Also  fcJp  =  C-?~,  that  is  =  CJJ-    f  ba2  =    f  / 

3a?  '  ^ 

Also  it  is  understood  that  a  is  unity.  These  are  sufficiently  new  and 
notable,  since  they  will  lead  to  a  new  calculus. 

I  propose  to  return  to  former  considerations. 

Given  /,  and  its  relation  to  x,  to  find  j*/. 

This  is  to  be  obtained  from  the  contrary  calculus,  that  is  to  say, 
suppose  that  fl  =  ya.  Let  l  =  ya/d  ;  then  just  as  J  will  increase,  so  d 
will  diminish  the  dimensions.  But  J*  means  a  sum,  and  d  a  differ- 
ence. From  the  given  y,  we  can  always  find  y/d  or  /,  that  is,  the 
difference  of  the  y's.  Hence  one  equation  may  be  transformed  into 

the  other;  just  as  from  the  equation   I  c  J  I2  =  c          ,  we  can  ob- 

J  3a3 

tain  the  equation  c  (~ft—        ^ 
"  3a*d 

N-B  .    f*?  +  f^=:   (*?+£?«.     Andinthe 
J  b        J    e        J  b  e 

,         x*  .  x^a 

X  Xrd          -  H  -- 

same  manner,  —  4-  -;  —  —  b         e    . 
do      de 


d 

But  to  return  to  what  has  been  done  above.  We  can  investi- 
gate J/  in  two  ways;  one,  by  summing  y  and  seeking  ya/d  =  l; 
the  other,  by  summing  z2/2a  =  y,  or  by  summing  ^/2ay  =  z,  and 
then  zz/t  =  p  =  l=ya/d.  Hence,  if  in  an  indefinite  equation,  we 
eliminate  y  by  substituting  in  its  place  zz/2a,  and  investigate  the 
t  of  this  new  equation  which  is  indefinite  like  the  first,  and 
then  by  the  help  of  the  value  zz/t  =  l,  and  after  that  by  the  help  of 
the  new  value  of  t,  eliminate  z  from  the  indefinite  equation  con- 
taining z  and  t,  there  will  remain  out  of  the  (three)  letters  x,z,t,l, 
the  letter  /  alone ;  and  again  we  ought  to  get  an  equation  which 
should  be  the  same  not  only  as  the  given  one,  but  also  the  same  as 
the  one  that  was  obtained  a  little  while  ago.  Hence,  since  we  have 
two  indefinite  equations,  containing  not  only  the  principle  quanti- 
ties, but  also  arbitrary  ones,  yet  not  altogether  unlike  the  former; 
and  these  ought  to  be  identical ;  it  will  appear  to  show  whether 
certain  terms  cannot  be  eliminated,  whether  it  is  not  possible  that  a 
comparison  should  be  made,  and  other  things  of  the  sort ;  and,  what 
is  really  the  most  important  thing,  which  terms  are  really  the 
greatest  and  the  least,  or  the  number  of  terms  of  the  equation. 

Moreover,  since  in  the  similar  triangles  TBL,  GWL,  LBP,  no 


262 


THE  MONIST. 


mention  has  yet  been  made  of  the  abscissa  x  or  of  the  fixed  point  A, 
let  us  then  suppose  that  through  the  fixed  point  A  there  is  drawn 
an  unlimited  straight  line  AIQ,  parallel  to  LB,  meeting  the  tangent 
LT  in  I;  and  let  AQ  =  BL;  bisect  AI  in  N;  then  I  say  that  the 
sum.  of  every  QN  will  always  be  equal  to  the  triangle  ABL,  as  can 
easily  be  shown  by  what  I  have  said  in  another  place.20 


N  \l 


B 

(B) 


These  considerations  give  once  more  a  fresh  fundamental  theo- 
rem for  the  calculus.  For  xv/2  =  y,  where  we  suppose  that  ~BL,  =  v 
and  QN  =  /,  and  y=  J7; 

t—x 


,         AI       t-x 
but    —  =  — 
v  t 


therefore  AI  = 


xv 


and   QI  =  v  —  AI  =  v  —  v  +  — ,  i.  e.  QI  =  — , 

*  2*  t 


(21) 


2  /      '    2  2/  2t 

Now,  by  the  help  of  the  equation  (xv+tv)/2t  =  l,  and  of  the  former 
equation  y=xv/2,  and  taking  once  more  the  first  indefinite  or  gen- 
eral equation  as  a  third,  and  eliminating  first  of  all  y,  then  t  by 
means  of  the  value  found  for  the  ratio  of  t  to  x  from  the  indefinite 
equation  containing  x  and  v,  and  lastly  v  by  the  help  of  the  equation 
(xv+tv)/2t  =  l,  in  which  the  principal  quantities  x  and  /  alone  re- 
main, as  before;  and  this  again  should  be  identical  with  the  given 
equation. 

Thus  we  have  found  three  equations  obtained  in  different  ways, 
which  should  all  be  identical  with  one  another  and  with  the  given 
equation ;  and  these  three  are  not  only  identical  but  should  also 

20  Since  the  triangles  QLI,  WL(L)   are  similar,  QI.B(B)  =  QL.Q(Q), 
hence  omn.QI   (applied  to  AB)=omn.  QL  (to  AQ)  =  figure  AQLA,  hence 
omn.(QI  +  QA)  =  rect.  ABLQ  =  2AABL. 

21  Since  /  is  the  difference  for  y,  therefore  21  is  the  difference  for  xv ; 
this  is  shown  to  be  {xv  +  tv)/t  or  x(v/t)  +v;  and  this  is  the  equivalent  to 
(since  v/t  =  dv/dx  —  dv) 

d(xv)  =  xdv  +  v  =  xdv  +  vdx. 


THE  MANUSCRIPTS  OF  LEIBNIZ. 


263 


consist  of  the  same  letters  and  signs;  and  whether  this  is  possible, 
will  immediately  appear  on  being  worked  out  analytically. 

§  VII. 

The  next  manuscript  is  a  further  continuation  of  the 
preceding,  written  two  days  later.  In  this  Leibniz  returns 
to  the  idea  that  he  has  found  so  prolific,  namely,  the  mo- 
ments of  a  figure.  It  is  to  be  observed  that  he  speaks  of 
the  method  of  breaking  up  an  area  into  segments  as  some- 
thing that  he  has  already  worked  out ;  this  will  be  remarked 
upon  in  a  note  on  a  later  manuscript,  where  it  will  help  to 
clear  up  a  small  difficulty.  The  accuracy  of  the  rather  in- 
volved algebraical  work  is  also  a  point  to  be  noticed. 
1  November,  1675. 

Analyseos  Tetragonisticae  pars  tertia. 
(Third  part  of  Analytical  Quadrature.) 

It  was  some  time  ago  that  I  observed  that,  being  given  the 
moment  of  a  curve  ABC,  or  of  a  curvilinear  figure  DABCE,  about 
two  straight  lines  parallel  to  one  another,  such  as  GF,  LH  (or  MN, 


M 

0 

n 

L 

G, 

r 

XB 

e 

\ 

\ 

L 

H 

Q 

N 

PQ),  then  the  area  of  the  figure  could  be  obtained;  because  the 
two  moments  differed  from  one  another  by  the  cylinder  of  the 
figure,  where  the  altitude  was  the  distance  between  the  parallels. 

Now,  this  is  true  of  every  progression,  whether  of  numbers 
or  of  lines ;  that  is,  even  if  we  do  not  use  curvilinear  figures  but 
ordinated  polygons;  in  other  words,  where  the  differences  between 
the  terms  are  not  infinitely  small.  Suppose  we  have  any  such 
ordinated  quantity  z,  and  let  the  ordinal  number  be  x,  then 

b  omn.z  •""  ±  omn.zx  q=  omn.zx+ b 
and  this  is  evident  by  the  calculus  alone. 

By  the  help  of  this  rule,  the  sums  of  terms  of  an  arithmetical 


264 


THE  MONIST. 


progression  refolded  reciprocally  ;22  and  this  multiplication  takes 
place  when  it  is  required  to  find  the  moment  of  the  ordinates  about 
a  straight  line  perpendicular  to  the  axis.  But  if  the  moment  about 
any  other  straight  line  is  required,  there  is  the  following  general 
rule: 

From  the  center  of  gravity  of  each  of  the  quantities  of  which 
the  moment  is  required,  a  perpendicular  is  drawn  to  the  axis  of 
libration ;  then  the  sum  of  the  rectangles  contained  by  the  distances 
or  perpendiculars  and  the  quantities  will  be  equal  to  the  moment 
about  the  given  straight  line. 

Hence,  if  the  given  straight  line  is  the  axis  of  equilibrium, 
it  immediately  follows  that  the  moment  of  the  figure  about  the  axis 
is  equal  to  the  sum  of  the  half-squares.  Also  when  it  is  parallel 
to  that,  it  will  differ  from  the  foregoing  by  a  known  quantity. 

Now,  let  us  take  another  straight  line :  for  the  circle  for  instance, 
let  ABCD  be  a  quadrant,  vertex  B,  and  center  D ;  let  another  straight 
line  be  given,  that  is  to  say,  let  the  prependicular  DF  be  given  and 


also  EF  where  it  meets  the  diameter,  and  thus  also  DE;  let  HB 
be  the  general  ordinate  to  the  circle,  and  L  its  middle  point;  let 
LM  be  drawn  perpendicular  to  EF. 

Then  it  is  clear  that  the  triangles  EFD,  EMN   (where  N  is 
the  intersection  of  ML  and  AD),  and  LHN  are  similar. 

y 


Let 


=  *,  then  HL  =  -  = 


NH 


But,  on  account  of 


the  similar  triangles,  ^^r  =  ^^7  —  T\  > 
ri  L>       r  r/  (^  =/  ) 

therefore 


22  The  meaning  of  this  is  probably  a  series  such  as  that  considered  by 
Wallis.  If  a,  a  -f  d,  a  -\-  2d,  etc.  is  the  arithmetical  progression,  and  /,  /  —  d, 
I  —  2d,  etc.  is  the  series  reversed,  then  the  series  refolded  reciprocally  is  al, 
(o  +  rf)(/  —  d),  (a-\-2d)(l —  2d),  etc.  It  may  however  mean  the  sum  of  the 
squares  of  the  arithmetical  progression.  B.ut  the  point  is  not  very  important. 


THE  MANUSCRIPTS  OF  LEIBNIZ.  265 

Hence,  EN  =  DE(=0 -HD(=*) -NH  (  =  -^  =<-*-^. 


Now    NL= 

MN       NH                      NH.EN 
and  =77-  =  TTT  »  or  MN  =  — rrj ;  thus  we  have 


and 

'-D* 

hence,  since  ^=  -<ffi—d2,    we  have  23 


and  this  calculation  is  general  for  any  curve,  so  long  as  x  is  always 
taken  as  the  abscissa  and  y  as  the  ordinate. 

Therefore  the  rectangle  contained  by  ML  and  HB  (=y),  or  the 
moment  of  each  ordinate  taken  with  regard  to  the  straight  line  EF, 
or  wa,  will  be  equal  to 


Hence,  omn.w  will  be  obtained  from  the  known  values  of 
omn.jr,  omn.xy,  and  omn.y2 ;  also,  if  any  three  of  these  four  are 
given,  the  fourth  is  also  known. 

Now,  omn..ry  will  be  equal  to  the  moment  of  the  figure  about 
the  vertex,  omn.  y2  will  be  equal  to  the  moment  of  the  figure  about 
the  axis ;  hence,  given  three  moments  of  the  figure,  that  is  to  say, 
the  moments  about  two  straight  lines  at  right  angles  and  any  third, 
the  area  is  given. 

This  theorem,  however,  is  less  general  than  the  one  that  was 
given  before,  in  the  first  part  of  this  essay,  where  it  does  not  matter 

28  The  accuracy  of  the  algebra  is  noteworthy  in  comparison  with  the  in- 
accuracies that  occur  later.  There  is  however  a  slip :  ez  =  fj  -\-  d2  and  not 
f2  —  d2 ;  this  must  be  a  slip  and  not  a  misprint,  because  it  persists  throughout. 
It  should  be  noted  that  the  figure  given  by  Gerhardt  is  careless  in  that  LM  is 
made  to  pass  through  A. 


266  THE  MONIST. 

what  the  angle  between  the  straight  lines  may  be,  if  only  we  are 
given  three  moments;  but  it  is  always  understood  that  they  are  in 
the  same  plane.  ( Meanwhile,  however,  this  theorem  will  suffice  for 
the  curve  of  the  primary  hyperbola ;  for,  if  /  is  infinite,  or  if  FE 
and  ED  are  parallel,  dy  +  y2/2  =  wa,  as  has  already  been  proved.) 

It  is  to  be  observed  that  by  other  calculation  the  area  of  a 
quantity,  whose  center  of  gravity  lies  in  a  given  plane  (even  though 
the  whole  quantity  does  not),  can  be  found  from  three  given 
moments  about  three  straight  lines  in  that  plane.  From  this  it  is 
to  be  seen  whether  the  results  obtained,  when  compared  with  one 
another,  will  not  produce  something  new. 

If  instead  of  the  moment  of  a  figure  we  require  the  moment 
of  all  the  arcs  BP,  PC,  etc.,  the  perpendiculars  are  to  be  drawn 
from  the  points  B,  P,  C,  etc.  only,  to  the  straight  line;  for  it  will 
make  no  difference  whether  they  are  drawn  from  the  end  or  from 
the  middle  of  BP,  for  instance,  for  the  difference  between  two  such 
perpendiculars  is  infinitely  small.  Hence,  calling  the  element  of 
the  curve  z,  the  moment  of  the  curve  about  the  straight  line  EF  is 


d  V/2  -  dzz  -  dxz  +  fyz 
~ 


Most  of  the  theorems  of  the  geometry  of  indivisibles  which 
are  to  be  found  in  the  works  of  Cavalieri,  Vincent,  Wallis,  Gregory 
and  Barrow,  are  immediately  evident  from  the  calculus;  as,  for 
instance,  that  the  perpendiculars  to  the  axis  are  equal  to  the  surface 
or  moment  of  the  curve  about  the  axis,  for  you  find  that  a  perpen- 
dicular is  equal  to  the  rectangle  contained  by  an  element  of  the 
curve  and  the  ordinate.  Therefore  I  do  not  set  any  value  on  such 
theorems,  or  on  those  about  applications  of  intercepts  on  the  axis 
(intercepted  between  the  tangents  and  the  ordinates)  to  the  base. 
Such  theorems  bring  forth  nothing  new,  except  maybe  they  afford 
formulas  for  the  calculus. 

But  my  theorem  about  the  dimensions  of  the  segments  does 
bring  out  a  new  thing,  because  the  space  whose  dimension  is  sought 
is  broken  up  in  a  different  way,  that  is  to  say,  not  only  into  ordi- 
nates but  into  triangles.  Also  perhaps  the  Centrobaric  method 
yields  something  new.  Maybe  an  easy  method  can  be  obtained,  by 
which  without  diagrams  those  things  which  depend  on  a  figure  can 
be  derived  by  calculus.  Gregory's  theorem,  on  ductions  of  two 


THE  MANUSCRIPTS  OF  LEIBNIZ.  267 

parabolas,24  one  under  the  other,  equal  to  a  cylinder,  is  immediately 
evident  by  calculus;  for  the  ordinate  of  a  circle  y=^/a2-x2,  that  is, 
the  product  of  -\/a  +  x  and  ^a-x;  and  in  the  same  way,  -\/2av-v2 

=  y,  which  gives  y=V^  mto  ^2a-v;  and  these  come  to  the  same 
thing. 

If  the  same  ordinate  y  is  multiplied  by  some  quantity  z,  and 
afterward  by  the  same  z  ±  some  known  or  constant  number  b,  the 
difference  between  the  sums  produced  will  be  equal  to  the  cylinder 
of  the  figure;  so  that 

zy,,-zy  +  by  """  by. 

Although  this  is  evident  in  general  by  itself,  yet  applications  of  it 
are  not  always  evident.  For  instance,  let 

x2  x2 

y  — 


ax-b*  , 

x2 


then,  multiplying  by  ifax  +  bt  we  have 


—  o 


and,    multiplying  by  ^ox  —  b,  we  have 1= — ~ —  5 

•\ax  +  b 

ax2  bzx 

but,    since  instead  of  — — ,   we  can  have   x  +  TT — , 

ax  —  e>2  ax  —  b£ 

which  depends  on  the  quadrature  of  the  hyperbola ;  and  thus  if  one 
of  the  two  things,  (A)  or  (B),  is  given,  then  the  other  is  also 
known,  supposing  that  the  quadrature  of  the  hyperbola  is  known. 

Suppose  that  at  the  points  C,  D,  E  of  a  curve  situated  in  any 
plane  there  are  imposed,  perpendicular  to  the  plane,  the  ordinates 
of  another  curve  FGH  (not  necessarily  of  the  same  constitution), 
in  such  a  manner  that  the  middle  point  of  each  of  these  ordinates 
lies  in  the  plane;  then  it  is  evident  that  LG,  MD,  NE,  multiplied 
by  FL,  GM,  HN,  (that  is,  the  lines  imposed  at  C,  D,  E  of  the  curve 
BCDE)  or  the  rectangles  FLG,  GMD,  HNE,  or  the  duction  of 
these  two  planes  into  one  another,  will  be  equal  to  the  moment  of 
every  LC,  MD,  NE,  etc.  Hence,  if  PR  is  another  axis,  and  the 
interval  between  it  and  QL  is  the  straight  line  PQ,  the  moment 


24  Such  theorems  are  also  considered  in  Wallis,  where  it  is  shown  that 
the  products  for  two  equal  parabolas  are  the  squares  on  the  ordinates  of  a 
semicircle ;  the  axes  of  the  parabolas  being  coincident,  but  set  in  opposite  sense. 


268 


THE   MONIST. 


about  PR  differs  from  that  about  QL  by  the  cylinder  whose  base 
is  LC,  MD,  etc.,  and  whose  height  is  PQ.25 

But,  if  the  moment  about  the  straight  line  PQ,  and  also  that 
about  some  other  straight  line  in  another  position,  as  TS,  of  all  the 


ordinates  LF  of  the  same  figure,  imposed  at  the  points  C,  then  we 
shall  have  the  cylinder  corresponding  to  all  the  LF's,  as  I  will  now 
prove. 

/        g 
If  we  call  QL,  x,  and  CL,  y,  then  TC=  —  x+  -y  +  h\  and  this 

multiplied  by  z,  where  FL  or  MG  =  z,  will  give 

/          g 

—  xz  +  —  yz  +  hz . 
a  a 

Now  xz  is  given,  being  the  supposed  moment  about  PQ,  which  is 
the  same  whether  the  s's  are  placed  where  they  were  in  the  lines 
LF,  MG,  etc.,  or  at  the  points  C,  D,  E.  Also  yz  is  given,  either 
as  the  rectangle  FLC  or  as  the  duction,  by  hypothesis.  Hence,  if  in 
addition  there  is  given  one  moment  of  the  ordinates  imposed  upon 

25  This  is  obviously  wrong ;  the  base  of  the  cylinder  is  the  area  made  up 
of  FL,  GM,  HN,  etc.  The  whole  of  this  last  passage  proved  to  be  difficult  to 
make  out ;  Leibniz  has  not  completed  his  figure,  by  showing  the  surface  formed 
by  placing  the  ordinates  FL,  GM,  HN  with  their  middle  points  at  C,  D,  E, 
and  the  ordinates  themselves  perpendicular  to  the  plane  of  the  curve  BCDE, 
which  figure  I  have  added  on  the  right-hand  side  of  Leibniz's  figure.  Even 
when  this  is  given,  there  is  another  difficulty  added  because  as  given  by  Ger- 
hardt,  CS  is  the  tangent  at  D  instead  of  the  proper  line,  namely,  the  perpen- 
dicular from  C  to  TS ;  in  addition  through  a  misprint,  this  line  is  afterward 
referred  to  as  TC.  Lastly,  "the  rectangle  FLG"  is  a  misprint  for  FLC,  which 
with  Leibniz  stands  for  FL.LC;  this  notation  for  a  rectangle  is,  as  far  as  I 
can  remember,  used  by  Wallis  and  Cavalieri. 

When  all  these  errors  are  revised,  what  at  first  sight  seemed  to  be  rather 
a  muddle  turns  out  to  be  an  exceedingly  neat  idea  in  connection  with  the 
moments  of  a  figure,  and  their  use  to  find  an  area,  although  mostly  imprac- 
ticable. 

Note.  The  values  f,  g,  a,  h,  are  the  lengths  of  TQ,  QP,  PT,  and  the  per- 
pendicular from  Q  on  PT. 


THE  MANUSCRIPTS  OF  LEIBNIZ.  269 

the  curve  at  the  points  C,  D,  E,  and  this  is  taken  to  be  equal  to 
/         K 
—  xz  +  -ys  +  hz,  then  we  have  hz  or  the  cylinder  required. 

Hence,  the  curve  BCDE  is  to  be  chosen  such  that  the  ordi- 
nates  of  the  given  curve  can  be  multiplied  by  different  ordinates  of 
the  former,  drawn  either  to  the  axis  QL  or  to  the  axis  TS,  with 
some  advantage  of  simplicity;  and  the  curves  that  are  suitable  for 
this  are  those  that  have  several  suitable  axes,  such  as  the  circular 
or  primary  hyperbola,  which  has  a  pair  of  asymptotes,  or  an  axis 
and  a  conjugate  axis. 


§  VIII. 

Much  comment  has  been  made  on  the  fact  that  the  date 
of  the  next  manuscript  was  originally  "n  November 
1675";  that  the  5  had  been  altered  to  a  3,  the  ink  being 
of  a  darker  shade;  and  that  it  is  almost  certain  that  this 
alteration  in  date  was  made  for  some  ulterior  motive  by 
Leibniz  himself.  Hence,  if  he  was  capable  of  falsifying  a 
date  in  one  particular  case,  then  he  is  not  to  be  trusted  in 
others, .  . .  . ,  and  so  on.  Instead  of  trying  to  explain  away 
this  alteration,  let  us  try  to  find  an  explanation  as  to  the 
reason  of  its  having  been  made  by  Leibniz;  I  offer  the 
following  as  at  least  feasible. 

The  essay  starts  with  the  words,  '7am  superiore  anno 
mihi  proposueram  qucstionem, .  . .  . "  I  suppose  that  by  this 
Leibniz  intended:  "A  year  or  two  ago,  I  set  myself  the 

question, "  This  conforms  with  what  follows ;  the 

theorem  that  he  sets  down  is  one  such  as  those  that  were 
suggested  to  him  by  Huygens,  and  further  theorems  that 
came  to  him  as  deductions  during  his  first  intercourse  with 
Huygens.  Years  later,  I  therefore  suggest,  Leibniz  refers 
to  this  manuscript,  reads  his  own  Latin,  superiore  anno,  as 
"in  the  above  year,"  gets  no  further,  recognizes  the  theo- 
rem by  its  figure  as  one  of  the  Huygens-time  batch,  and 
says  to  himself  "1675  ?  No,  that's  wrong,  should  be  1673," 


2/O  THE  MONIST. 

and  proceeds  to  alter  it  to  what  he  remembers  was  the 
date  for  the  first  consideration  of  the  theorem. 

N.  B.  Gerhardt  himself  has  remarked  on  the  darker 
tint  of  the  ink  used  in  the  alteration ;  hence  my  argument, 
made  at  a  later  date. 

The  date  1675  is  incontestable;  for  this  composition  is 
quite  glaringly  a  development  of  the  work  that  has  been 
so  efficiently  started  in  that  of  November  i,  1675.  Progress 
is  still  delayed  by  the  idea  that  has  obsessed  Leibniz  up  till 
now,  that  of  the  transformation  of  equations,  so  as  to  be 
able  to  eliminate  more  unknowns  than  the  original  number 
of  his  equations  warrant.  He  sets  himself  the  problem: 
"To  determine  the  curve  in  which  the  distance  between  the 
vertex  and  the  foot  of  the  normal  is  reciprocally  propor- 
tional to  the  ordinate,"  i.  e.,  the  solution  of  the  equation 
x  +  y  dy/dx  =  a?/y,  in  modern  notation.  This  is  a  very 
unlucky  choice  for  him:  for  I  have  it  on  the  authority  of 
Prof.  A.  R.  Forsyth  that  this  is  incapable  of  solution  in 
ordinary  functions  or  even  by  a  series  in  which  the  law 
of  the  series  is  easily  and  simply  expressible — at  least  he 
confesses  that  he  is  unable  to  obtain  such  a  solution,  which 
I  take  it  comes  to  the  same  thing. 

Leibniz  professes  to  have  found  the  solution  and  gives 
(y2  -(-  xz)  (a*  —  yx)  =  2y2  logy,  and  unfortunately  this 
false  success  but  enhances  the  value  in  his  eyes  of  the 
method  mentioned  above.  But  from  the  equation  given  as 
the  solution  we  may  draw  an  incontestable  conclusion ;  for 
in  a  previous  problem  Leibniz  verifies  his  solution  by  the 
method  of  tangents,  i.  e.,  by  differentiation,  although  the 
method  does  not  as  yet  convey  that  idea  to  him ;  but  he  does 
not  verify  the  solution  in  this  case,  because  he  is  unable  at 
this  date  to  differentiate  the  product  y2  logy. 

The  introduction  of  dx  instead  of  x/d  marks  a  further 
advance,  more  important  perhaps  than  the  use  of  fy  dy ; 


THE  MANUSCRIPTS  OF  LEIBNIZ, 

for  he  still  writes  $x,  considering  dx  to  be  constant  and 
equal  to  unity.  He  is  beginning  to  grasp  the  infinitesimal 
nature  of  his  calculus,  and  that  infinitesimals  are  not  to  be 
neglected  because  of  their  intrinsic  smallness,  but  because 
of  their  smallness  with  respect  to  other  quantities  which 
come  into  the  same  equations  and  are  finite;  but  he  is  far 
from  being  certain  about  it  as  yet,  as  is  evidenced  by  the 
discussion  as  to  whether  d(v/ty}  =  dv/dty  or  not.  How- 
ever, the  whole  manuscript  marks  a  distinct  advance  on 
anything  that  has  gone  before.  From  now  on  he  probably 
discards  geometry,  and  only  refers  to  Descartes,  Gregory 
and  Barrow  for  examples  to  show  how  much  superior  is 
his  method  to  theirs.  I  put  his  final  reading  of  Barrow 
down  to  the  interval  between  the  date  of  this  manuscript, 
ii  November,  1675,  and  November,  1676;  it  is  at  this 
time  that  he  inserts  his  sign  of  integration  in  the  margins 
of  the  theorems.  The  next  person  that  examines  the  orig- 
inals of  these  manuscripts  (I  am  convinced  that  this  is 
very  necessary),  should  carefully  see  whether  the  ink  used 
for  the  note  "novi  dudum"  (which  I  have  mentioned)  is 
the  same  as  that  used  for  the  sign  of  integration ;  also  the 
other  books  that  were  used  by  Leibniz  in  his  self-education 
should  be  searchingly  scrutinized  for  clues. 

The  last  remark  I  have  to  make  is  one  of  astonishment 
at  the  errors  in  the  algebraical  work  which  brings  this 
essay  to  a  close,  and  to  a  less  degree  throughout  the  essay ; 
for  we  have  seen  the  accuracy  to  which  Leibniz  has  at- 
tained in  a  previous  manuscript ;  of  course,  a  great  deal  of 
erroneous  work  can  be  explained  by  supposing  none  too 
careful  transcription ;  but  a  re-examination  of  the  whole  of 
the  Leibnizian  remains  should  include  a  careful  scrutiny 
on  the  point  as  to  whether  some  of  the  extracts  given  by 
Gerhardt  are  not  the  work  of  pupils  of  Leibniz,  whose 
writing  would  naturally  be  somewhat  similar.  Perhaps 
too  some  of  those  early  geometrical  theorems  might  be  un- 


2/2 


THE  MONIST. 


earthed ;  and  this  would  well  reward  the  most  painstaking 
search.  Nobody  can  assert  that  anything  like  an  adequate 
tale  of  the  progress  of  the  Leibnizian  genius  has  so  far 
been  told. 


11  November,  1673.28 

Methodi  tangentium  inversae  exempla. 
(Examples  of  the  inverse  method  of  tangents.) 

A  year  or  two  ago  I  asked  myself  the  question,  what  can  be 
considered  one  of  the  most  difficult  things  in  the  whole  of  geometry, 
or,  in  other  words,  what  was  there  for  which  the  ordinary  methods 
had  contributed  nothing  profitable.  To-day  I  found  the  answer 
to  it,  and  I  now  give  the  analysis  of  it. 

Find  the  curve  C(C),  in  which  BP,  the  interval  between  the 
ordinate  BC  and  PC  the  normal  to  the  curve,  taken  along  the  axis 
AB(B),  is  reciprocally  proportional  to  the  ordinate  BC. 

Let  AD(D)  be  another  straight  line  perpendicular  to  the  axis 
AB(B),  and  let  ordinates  CD  be  drawn  to  it,  so  that  the  abscissae 


AD  along  the  axis  AD(D)  are  equal  to  the  ordinates  BC  to  the 
axis  AB(B),  and  the  ordinates  CD  to  the  axis  AD(D)  are  equal 
to  the  abscissae  AB  along  the  axis  AB  ( B ) .  Let  us  call  AD  =  BC  =  y, 
and  AD  =  BC  =  *;  also  let  BP  =  w  and  B(B)=*.  Then  it  follows 
from  what  I  have  proved  in  another  place  that 

29  See  Cantor,  III,  p.  183 ;  but  neither  Cantor  nor  Gerhardt  appears  to 
offer  any  suggestion  as  to  why  this  date  should  have  been  altered. 


THE  MANUSCRIPTS  OF  LEIBNIZ.  273 

y  y2  27 

Jwz=  2'  °rWZ2d 

yZ 

But  from  the  quadrature  of  a  triangle  it  is  evident  that— 3=.?; 

and  therefore  wz  =  y. 

Now,  from  the  hypothesis,  w  =  b/y,  for  thus  w  and  3;  will  be 
reciprocally  proportional  to  one  another.  Hence  we  have 

bz  y2 

—  =y,  and  thus  z  =  ^  . 

/y2 
-r   ;  and  from  the  quadrature  of  the 

/y2  y3  y3 

-r  =  —r-  ',  hence,  x=  -=^—  ;   and  this   is   the   required 

equation  expressing  the  relation  between  the  ordinates  3;  and  the 
abscissae  x  of  the  curve  C(C),  which  was  to  be  found.  Therefore 
we  consider  that  the  curve  has  been  found  and  it  is  analytical;  in 
short,  it  is  the  cubical  parabola  whose  vertex  is  A. 

We  will  therefore  see  whether  the  truly  remarkable  theorem  is 
not  true,  namely,  in  the  cubical  parabola  C(C),  the  intervals  BP 
between  the  normals  to  the  curve,  PC,  and  the  ordinates  to  the 
axis,  BC,  taken  along  the  axis  ABP,  are  reciprocally  proportional 
to  the  ordinates,  BC. 

The  truth  of  this  is  easily  shown  by  the  calculus  of  tangents. 
For  the  equation  to  the  cubical  parabola  is  xc*  =  y3 ;  taking  c  to  be 
the  latus  rectum,  and  supposing  that  for  c2  we  put  3ba,  or  c=^3ba, 
we  have  3xba  =  y3. 

Now,  by  Slusius's  method  of  tangents,  we  have  t  =  y*/3ba, 
where  t  is  put  for  BT,  the  interval  along  the  axis  between  the 
tangent  and  the  ordinate. 

v2 

y9  —      ba 

But  BP=w=— ,  and  therefore  w=  y^  == —  ;  hence,  the  w's 

1  ba       y 

and  the  y's  are  reciprocally  proportional  as  was  to  be  proved. 

27  This  was  obtained  in  the  form  omn./>  =  y*/2,  previous  to  October,  1674, 
from  the  Pascal  form  of  the  characteristic  triangle ;  it  is  quoted  as  a  known 
theorem  in  the  essay  dated  29  October,  1675.  See  §§  III,  VI. 

It  is  probably  at  this  date  that  he  began  to  revise  his  ideas  as  to  d  dimin- 
ishing the  dimensions ;  being  forced  to  reconsider  them  by  the  occurrence  of 
such  equations  as  wz  =  y.  It  is  seen  in  the  next  paragraph  how  careful  he  is 
to  keep  his  dimensions  equal;  for  he  introduces  an  apparently  irrelevant 
a(=  1)  for  this  purpose.  It  gradually  dawns  on  him  that  neither  /  nor  d  alter 
the  dimensions,  but  that  a  "sum  of  lines"  is  really  a  sum  of  rectangles,  on 
account  of  the  fact  that  they  are  applied  in  a  certain  fixed  way  to  an  axis; 
he  is  not  quite  certain  of  this  however  until  well  on  in  the  next  year,  when 
we  find  him  using  fdx  y. 


2/4  THE  MONIST. 

The  artifice  of  this  analysis28  consisted  in  obtaining  the  abscissa 
from  the  ordinate;  and  this  idea  was  never  previously  thought  of. 
It  is  not  a  more  difficult  question  either,  if  the  curve  is  required 
in  which  BP,  the  interval  between  the  normals  and  the  ordinates, 
is  reciprocally  proportional  to  the  abscissae  AB.  Indeed,  iv=az/x; 
but  w  =  yz/2 ;  hence,  we  have 


—    12  C  w  or      /2  i  a2 

~V  J       \  J  *• 


Now  fw  cannot  be  found  except  by  the  help  of  the  logarithmic 
curve.29  Hence,  the  figure  that  is  required  is  that  in  which  the 
ordinates  are  in  the  subduplicate  ratio  of  the  logarithms  of  the 
abscissae;  and  this  curve  is  one  of  the  transcendental  curves. 

Now,  in  truth,  it  is  a  much  harder  question,30  if  the  curve,  in 
which  AP  is  reciprocally  proportional  to  the  ordinate  BC  is  re- 
quired. 

a2  y2  f 

For  then       x  +  w=  —  and  wz=—-.  also   \z  =  x, 
y  2<t         J 

,  =  %',  thus,  «^  =  |1,  and  «^w£; 

y2       x      a? 
hence,  *+_w_=_. 

If  we  suppose  that  the  x's  are  in  arithmetical  progression  then 
x/d-z  will  be  constant,  and  we  shall  have 


/     a2       r      Cc?    / 

±— —  —  or    I  #=  I — , 

2d      y       J        J  y       2 


y 

therefore 


*2      ^       (*       v 

-  +  —  =  I  -  or  d 

J 


2       2         y  y 

28  It  is  difficult  to  see  exactly  what  Leibniz  means  by  this  statement  ;  I 
can  only  guess  at  substitution  by  means  of  the  theorem  ws  =  y,  the  equivalent 
to  the  recognition  of  the  fact  that  y  dy/dx  .  dx  =  ydy.   The  wording  is  however 
impersonal,  and  may  mean  that  he  himself  had  never  thought  of  the  idea 
before. 

29  Required  y  =  /(*),  such  that  y  dy/dx  =  a2/x;  the  solution  is  y2  =  2a2 
logeA^r.     Weissenborn  remarks  on  the  omission  of  the  o  as  being  incorrect; 
from  Leibniz's  standpoint  I  cannot  agree  with  him.    Leibniz,  from  Mercator's 
work,  connects  az/x  with  the  ordinate  of  the  equilateral  hyperbola  xy  =  a2, 
and  its  integral  with  the  quadrature  of  this  curve.     The  omission  of  the  a2 
only  alters  the  base  of  the  logarithm,  and  Leibniz  merely  states  that  the  solu- 
tion is  of  a  logarithmic  nature  without  attempting  to  give  it  exactly. 

30  How  does  he  know  until  he  has  tried  it?    This  rather  combats  the  idea 
that  these  were  mere  exercises  ;  it  gives  this  essay  the  appearance  of  being  a 
fair  copy  intended  either  for  publication  or  for  one  of  his  correspondents.    If 
this  were  the  case,  the  errors  later  in  algebraical  work  are  all  the  more  un- 
intelligible.   The  idea  that  Leibniz  was  a  man  who  was  accustomed  to  writing 
down  his  thoughts  as  he  went  along  does  not  appeal  to  me  at  all  ;  this  is  the 
method  of  the  slow-working  mind,  rather  than  that  of  genius. 


THE  MANUSCRIPTS  OF  LEIBNIZ.  275 


but,  if  we  join  AC,  A(C),  then  these  are  equal  to  ^/*2  +  y2;  and  if 
with  center  A  and  radius  AC  we  describe  an  arc  CE  to  cut  the 
straight  line  AE(C)  in  E,  then  E(C)  will  be  the  difference  between 

AC  and  A(C)  ;  that  is,  E(C)  =e  =  dxTTf 

•'•  e  =  2a?/y. 

If  then  it  were  allowable  to  assume  that  the  y's  were  also  in 
arithmetical  progression,  we  should  have  what  was  required;  yet 
it  seems  that  it  does  not  make  any  difference  even  if  the  x's  have 
been  assumed  to  be  in  arithmetical  progression.  For  if  we  do 
assume  that  the  JF'S  are  in  arithmetical  progression,  it  follows  that 
the  AD's,  or  the  y's  are  the  reciprocals  of  the  E(C)'s  or  the  e's. 
Moreover,  if  they  are  so  at  any  one  time  they  are  so  at  all  times.  Also, 
the  sums  of  an  infinite  number  of  reciprocal  proportionals,  no  matter 
what  the  progression  may  be  of  which  they  are  taken  as  the  recip- 
rocal proportionals ;  for  in  this  case  there  is  not  any  consideration 
of  rectangles,  where  there  is  need  of  equal  altitudes,  but  a  sum  of 
lines  is  calculated,  that  of  all  the  E(C)'s.31  Hence  I  see  the  difficulty 
arise  from  the  fact  that  the  sum  of  every  e,  or  every  2az/y,  or  every 
E(C),  cannot  be  obtained,  unless  we  know  to  what  progression  the 
y's  belong.  In  this  case,  that  information  is  not  given ;  for  it  is 
necessary  that  the  .r's  should  be  in  arithmetical  progression,  and 
hence  that  the  y's  are  not  so. 

On  the  other  hand,  if  we  suppose  in  the  above  equation, 

y*       x       a? 
+  2d^d  =  J' 
that  the  y's  are  in  arithmetical  progression,  then  we  have 

y       a2  / 

x  +  —  =  —  or  xy  +  —  =a2  ; 
dx       y  dx 

and,  finally,  by  assigning  the  progression  to  neither  x  nor  y,  we  have 
in  general 


xy+y      = 


But  we  have  not  as  yet  really  obtained  anything.  Let  us 
therefore  consider  it  from  the  standpoint  of  "indivisibles"  ;  let  PCS 
produced  meet  AD  in  S  ;  then  the  sum  of  every  AP  applied  to  AB 

31  This  seems  to  be  the  root  of  the  error  into  which  he  falls  ;  he  has  not 
yet  perceived  that  the  e's  have  to  be  applied  to  some  axis,  before  he  can  sum 
them  ;  and  this  is  to  a  great  extent  due  to  the  omission  of  the  dx,  taken  as 
constant  and  equal  to  unity.  He  is  thus  bound  to  fall  back  on  the  algebraical 
summation  of  a  series. 


2/6  THE  MONIST. 

is  equal  to  the  sum  of  every  AS  applied  to  AD  ;82  or  calling  DS,  v, 
we  have 

dy  fy  +  dy  $v  =  dx  §x  +  dx  j*  w, 
or 


by  the  hypothesis  of  the  question. 

Now,  if  we  take  the  y's  to  be  in  arithmetical  progression,  we 
have 

r2      .x2 

~  +  y  =  </*Logy.     33 

But  just  above,  making  the  same  supposition  that  the  y's  were  in 
arithmetical  progression,  we  had 


xy+  —  =  a  or  dx=  -» 


» 
dx  a—xy 

and  now  we  have 

£,:*+£, 

dx-     -  --- 
*  Logy 

Hence  at  length  we  obtain  an  equation,  in  which  x  and  y  alone 
remain,  and  unshackled,  namely 

yz  +  xz,  a?-yx=2y2L,ogy  ; 

and  this  equation,  since  it  is  determinate,  will  give  the  required 
locus. 

This  then  is  an  exceedingly  remarkable  method,  for  the  reason 
that  when  it  is  not  in  our  power  to  have  as  many  equations  as  there 
are  unknowns,  yet  often  we  shall  be  able  to  obtain  some  more 
equations,  by  the  help  of  which  we  shall  be  able  to  eliminate  certain 
terms,  as  the  term  d.x  in  this  case,  which  alone  stood  in  our  way. 
Either  of  the  two  equations,  by  itself,  contained  the  whole  nature 
of  the  locus,  although  from  neither  of  them  could  the  solution  be 
derived,  because  so  far  easy  means  were  lacking;  yet  the  combina- 
tion of  the  two  equations  gave  the  solution  at  once. 

I  see  that  the  same  thing  could  be  otherwise  obtained  by 
moments  ;  and  here  there  comes  to  my  mind  a  new  consideration 
that  is  not  altogether  inelegant. 

32  From  the  characteristic  triangle,  AS  :  AP  =  dx  :  dy. 

33  This  is  of  course  nonsense.    The  error  seems  to  arise  from  the  dx  being 
placed  outside  the  integral  sign  ;  thus  he  assumes  that  dx  is  constant,  while,  for 
the  integration,  he  also  assumes  that  the  dy  is  constant. 

We  cannot  argue  from  this  equation  that  Leibniz  did  not  at  this  date 
appreciate  what  an  infinitesimal  was,  on  account  of  the  infinitesimal  being 
equated  to  a  finite  ratio  ;  for  since  he  is  assuming  that  dy  is  an  infinitely  small 
unit,  dx  really  stands  for  dx/dy. 


THE  MANUSCRIPTS  OF  LEIBNIZ. 


277 


In  the  attached  figure,  let  EC  =  y,  FC  =  dy;  let  S  be  the  middle 
point  of  FC;  then  it  is  evident  that  the  moment  of  FC  is  the 


urn  «s» 

rectangle  contained  by  FC  and  BS,  i.  e.,  the  rectangle  BFC;  this 
follows  from  the  fact  that  it  is  equal  to  BFC+SFC,  and  the  latter 
can  be  neglected  as  being  infinitely  small  compared  to  the  former.34 

Hence  fydy  =  y2/2,  or  the  moment  of  all  the  differences  FC 
will  be  equal  to  the  moment  of  the  last  term,  and  ydy  =  d(yz/2),  or 
y*dy  =  y  dy2/2. 

Now,  just  above,  in  equation  (A),  by  making  x  arithmetical, 
we  had 


y  d—  =  a*-xy  ,  or  d—  = 


a  —xy 


a  —xy 


but  this  is  the  same  thing  as  y  dy  ;  hence  y  dy  =  --  -  ,  and  therefore 


—  —       f~*     x* 
y  dy  —  I  ~  —  — 

J      V  i 


But  we  have  already  found  that 


C~=L    y* 
I  y  dy=^\ 

J  i 


therefore  y2  +  x*  =  2  I  —  ,  as  before;  i.  e.,  dx3  +  y*  = 


From  this  there  follows  something  to  be  noted  about  these  equa- 
tions, in  which  occur  j*  and  d,  where  one  quantity,  in  this  case  for 
instance  the  x,  is  taken  to  proceed  arithmetically,  namely,  that  we 
cannot  make  a  change,  nor  say  that  the  value  of  x  is  found,  thus, 
x=2(az/y)  -dy2;  for  dy2  cannot  be  understood  unless  the  nature 
of  the  progression  of  the  y's  is  determinate.  But  the  progression  of 
the  y's,  in  order  that  it  may  be  used  for  d  y2,  must  be  such  that  the 
xfs  are  in  arithmetical  progression  ;  hence  the  dy's  depend  on  the 
^r's,  and  therefore  the  x's  cannot  be  found  from  the  dy's.  For  the 
rest,  by  this  artifice  many  excellent  theorems  with  regard  to  curves 
that  are  otherwise  intractable  will  be  capable  of  being  investigated, 
namely,  by  combining  several  equations  of  the  same  kind. 

In  order  that  we  may  be  better  trained  for  really  very  difficult 

34  Note  the  advance  in  ideas  suggested  by  the  words  "infinitely  small 
compared  with  the  former."  Here,  of  course,  the  notation  BFC  is  the  usual 
notation  of  the  period  for  BF.FC,  the  rectangle  contained  by  BF  and  FC. 


278  THE  MONIST. 

considerations  of  this  kind,  it  will  be  a  good  thing  to  attempt  just 
one  more,  as  for  instance  when  the  AP's  are  reciprocally  propor- 
tional to  the  AB's. 


Here  x  +  a/=  —  ,  and2w  =  a—t  and  z=dx;  and  so  we  obtain 

X  £  • 


d~ 


w—  —  =  —  ,  hence  x  +  —  =  x  ' 
z         dx  dx 

The  solution  of  this  is  not  now  difficult  ;  for  if  we  suppose  that 
the  x's  are  arithmetical,35  we  have 

(36) 


J 


x+      = 


Hence,  ~\/x2  +  yz  =  AC=  \/2LogAD;  and  this  is  a  simple  enough 
expression  for  the  curve.  In  this  however  the  AP's  are  required 
to  be  in  arithmetical  progression  ;  but  on  the  other  hand,  if  the  y's 
are  taken  to  be  in  arithmetical  progression,  we  have  x  +  y/dx  =  a?/x  ; 
and  from  this  latter  the  nature  of  the  curve  is  not  easily  obtained. 

Let  us  see  whether  there  can  be  a  curve  in  which  AC  is  always 
equal  to  BP;  in  this  case  y/xz  +  yz  =  w}  and  w  =  dyz/2dx.  Let  the 


^s  be  in  arithmetical  progression  then  (f^/*2  +  y2=)  JAC  =  y2; 
this,  however,  is  not  sufficient  to  describe  the  curve  practically, 
that  is  to  say,  by  points  following  one  another  consecutively.  When 

*=1,  let  BC=(y);  then  V1+  (/)  =  (y2),  or  1  +  (y2)  =  (?*). 
Whence  (y)  may  be  obtained  ;  thus,  from  the  equation 

,yc         (37) 
,  or  (y)  =         . 


Further,  in  the  same  way, 


AC  A(C) 

and  thus  again  ((y))  can  be  found.     By  the  help  of  this  a  third 

35  Note  in  general  that  this  is  Leibniz's  equivalent  of  the  modern  phrase, 
"integrate  with  respect  to  x." 

36  This  I  think  is  more  likely  to  be  a  slip  on  the  part  of  Leibniz,  than  a 
misprint;  for  in  the  next  line  he  has  AD,  which  is  the  correct  equivalent  of  y. 
Further,  AP  varies  inversely  as  x,  hence  the  AP's  have  to  be  in  harmonical 
progression,  not  arithmetical,  otherwise  x  is  not  equal  to  x*/2.     If  on  the 
other  hand,  we  assume  three  errors  of  transcription,  and  replace  x  for  y,  AB 
for  AD,  AB  for  AP,  the  whole  thing  is  correct  with  an  arbitrary  base. 

37  It  is  hardly  necessary  to  point  out  the  error  in  the  arithmetical  solution 
of  the  quadratic ;  nor  is  it  important.    It  is  however  to  be  noted  that  if  AC  =  v, 
the  equation  reduces  to  vz=.x(x  -}-v),  and  the  solution  is  a  pair  of  straight 
lines. 


THE  MANUSCRIPTS  OF  LEIBNIZ.  279 

AC  can  be  found,  and  some  sort  of  polygon  can  be  found,  which 
is  more  and  more  like  the  curve  that  is  required,  in  proportion  as 
the  thing  taken  for  unity  is  less  and  less. 

That  the  x's  are  in  arithmetical  progression  signifies  that  the 
motion  (in  describing  it)  along  the  axis  AB  is  uniform.  But 
descriptions  that  suppose  any  motion  to  be  uniform  are  not  within 
our  power.38  For  we  cannot  produce  any  uniform  motion,  except  a 
continually  interrupted  one. 

Let  us  now  examine  whether  dxdy  is  the  same  thing  as  dxy, 

1C 

and  whether  dx/dy  is  the  same  thing  as  d—\  it  may  be  seen  that 
if  y  =  z2  +  bz,  and  x  =  cz  +  d;  then 

dy  =  z2  +  2pz  +  p2,  +  bz  +  bp,  -z2-bz, 
and  this  becomes  dy  =  2z  +  b/3. 

In  the  same  way  dx  =  +  cf$,  and  hence 
dx  dy  =  2z  +  b  eft2, 

But  you  get  the  same  thing  if  you  work  out  dxy  in  a  straight- 
forward manner.  For  in  each  of  the  several  factors  there  is  a 
separate  destruction,  the  one  not  influencing  the  other;  and  it  is 
the  same  thing  in  the  case  of  divisors. 

Now  let  us  see  if  there  is  any  distinction  when  we  seek  the 
sums  of  these  things.  We  have  ^dx-x,  fdy  =  y,  and  $dxy  =  xy. 
If  then  we  have  an  equation,  dxdy-x  say,  then  §dxdy  =  §x.  But 
J*JT  =  xz/2,  hence  xy  =  xz/2,  or  x/2  =  y ;  and  this  satisfies  the  equation 

dx  x*         (39) 

dxdy  =  x;  for  substituting  for  y  its  value,  ax—  =x,  or  a—  =x, 

£  £ 

which  is  known  to  be  true. 

In  sums  these  results  do  not  hold  good ;  for  §x  j*y  is  not  the 
same  thing  as  $xy,  the  reason  is  that  a  difference  is  a  single 
quantity,  while  a  sum  is  the  aggregation  of  many  quantities.  The 
sum  of  the  differences  is  the  latest  term  obtained.  However,  from 
the  sums  of  the  factors  we  can  find  the  sums  of  products,  not  indeed 
as  yet  analytically,  but  by  a  certain  method  of  reasoning;  such  as 
Wallis  has  done  in  this  class  of  thing,  not  by  proving  them,  but  by 
a  happy  method  of  induction.  Nevertheless  to  find  proofs  for  them 
would  be  a  matter  of  great  importance. 

38  This  is  strongly  reminiscent  of  Barrow,  Lect.  I   (near  the  beginning) 
and  Lect.  Ill  (near  the  end). 

39  Leibniz,  as  a  logician,  should  have  known  better  than  to  trust  a  single 
example  as  a  verification  of  an  affirmative  rule. 

With  regard  to  infinitesimals  note  the  equation  dx  dy  =  x ! 


28O  THE   MONIST. 

Suppose  J zy  to  be  the  sum  that  is  required.     Let  J  zy  =  wt 

dw  r        [  dw       0.  r        [  dw 

then  zy  =  dw,  and  y=  — ,  and  J  y= J  —  .     Similarly,  J  z  =  J  —  . 

&  *5  jr 

Suppose  that  Jy  is  known,  =  v,  and  that  Jz  is  known,  =<A;  then  y 

dw  ,.     dw          dv      2      „. 

=  dv=  — ,  and  z=dy=  — ,  and  -77  =— .     From  this  it  would  seem 
z  y  dy      y 

1)          2  1)  (  2  (2 

to  follow  that d-r  =  -,  and  therefore  that  7  =  I  -.    Therefore  I  -  = 
y     y  T     ~  y  •*  y 

-j- ,  which  is  obviously  incorrect.  (40)    Hence  it  follows  that   I  -r: 

v 
cannot  be  equal  to  r . 

What  then  can  it  be?  We  have  to  sum  the  difference  for  v 
divided  by  the  difference  for  y.  That  is,  not  every  one  of  the 
differences  for,  or  the  whole  of,  v  is  to  be  divided  by  each  single 
difference  for  the  y;  this  is  not  so,  I  say,  because  each  single  one 
of  the  first  set  is  only  divided  by  the  single  one  of  the  other  set 
that  corresponds  to  it,  and  not  by  all  of  them.  Therefore 

7  is  not  the  same  as  777,  or  -7.     Will  not  then  d—  be  something 
J  dy  fdy        v  y 

,  r  r  j 

different  from  -77  ?    If  it  is  the  same,  then  also  I  d-r  =  I  -jr ,  that  is 


-=  C—  = 
y  ~  J  dy  ~ 


dy 
which  is  absurd. 


Similarly,  if  we  can  suppose  that  dv$  =  dv  d$,  then  J  dvty,  or 
Jdvdty.       Now  -v^—jdv  Jdty;  hence,  jdvdfy  =  jd 


which  is  absurd. 

Hence  it  appears  that  it  is  incorrect  to  say  that  dvd\Jf  is  the 

same  thing  as  dvti,  or  that  -^=d^r  ;  although  just  above  I  stated 

ay        v 

that  this  was  the  case,  and  it  appeared  to  be  proved.     This  is  a 
difficult  point.    But  now  I  see  how  this  is  to  be  settled. 

If  we  have  v  and  \f/,  and  they  form  some  quantity,  say  <f>  =  v^/ 
or  v/\j/,  and  if  the  values  of  v  and  ^  are  expressed  as  rationals  in 
terms  of  some  one  thing,  for  instance,  in  terms  of  the  abscissa  x, 
then  the  calculus  will  always  show  that  the  same  difference  is  pro- 
duced, and  that  d<f>  is  the  same  as  dvd$  or  dv/dy.  But  now  I  see 

40  If  Leibniz  can  see  that  this  equality  is  "obviously  incorrect,"  what  is  the 
use  of  the  argument  that  has  preceded  this  sentence;  for  the  final  result  must 
also  be  obviously  incorrect. 


THE  MANUSCRIPTS  OF  LEIBNIZ.  28l 

the  former  can  never  happen,  nor  can  it  come  to  the  latter  by 
separation  of  parts ;  for  example, 

x  +  /?,  ^  x  +  ft,,  -,  x,  x,  becomes  2$x, 
which  is  quite  a  different  thing  from 

x  +  p,-x,,r^x  +  p,-x  which  gives  p2. 
Hence  it  must  be  concluded  that  dv$  is  not  the  same  as  dvdifr,  and 

.v  .  dv         (41) 

a  T  is  not  the  same  as  -77 . 

Take  an  equation  of  the  first  degree,  a + bx  +  cy  =  0.  Let  DV  =  6, 
AB  =  x}  BC  =  ;y,  and  TB  =  t.  Then,  by  making  use  of  the  method 
of  tangents,42  we  have  bt  =  -cy,  or  t=-cy/b.  In  the  same  way, 
0=-bx/c. 

T 


8 


Let  WC  =  w,  and  WS  =  /3,  then  it  is  evident  that  t/y  =  f$/w,  and 


therefore  w=—p-,  and  in  the  same  way,  fi=  —7—  . 

Second  degree.    a  +  bx  +  cy  +  dx2  +  ey2  +  fyx  =  0.    Making  use  of 
'the  method  of  tangents,  we  have 

bt  +  2dxt + fyt  =  -cy  -  2ey2  -  fyx ; 

41  Leibniz  here  justifiably  verifies  the  falsity  of  his  supposition  being  a 
general  rule  by  a  single  breach  of  it.    He  uses  v  =•  ^  =  x,  and  changes  x  into 
*  +  /3;  thus, 

d(xx)  =  (*  +  /8)(jr  +  /B)-.  xx       -  2£x 
dx  dx     =  (JF  +  /J  — *)(*  +  £  —  *)  =  02. 

Here  we  see  the  first  idea  of  the  method  that  is  the  same  as  that  used  by 
Fermat  and,  afterward  by  Newton  and  Barrow ;  this  consideration,  whatever 
the  source,  is  that  which  leads  him  later  to  the  substitution  x  -\-  dx,  y-\-dy  in 
those  cases  in  which  Barrow  uses  a  and  e. 

42  "ordinando  et  accommodando,"  literally  setting  in  order  and  adapting. 
It  is  to  be  remembered  that  Sluse  gave  only  a  rule,  and  not  a  demonstration 
of  the  rule.    Part  of  the  rule  was  that,  if  the  equation  in  two  variables  con- 
tained terms  containing  both  the  variables,  these  terms  had  to  be  set  down 
on  each  side  of  the  equation.    Thus,  for  the  equation  y3  =  bw  —  yw  would 
first  of  all  be  written 

y3-\-yw  =  bw —  yw ordinando  (?) 

then  each  term  on  the  left  is  multiplied  by  the  exponent  of  y,  and  each  term  on 
the  right  by  that  of  v,  thus, 

3ya  -}-  yw  =  2bw  —  2yw accommodando  (  ?) 

and  finally  one  y  on  the  left,  in  each  term,  is  changed  into  a  t,  where  t  is 
the  subtangent  measured  along  the  y  axis. 


282  THE  MONIST. 

hence  t=  —  7-  —  ^-r-  —  P—  •    From  this  it  is  quite  evident  that  t  can 
b  +  2dx+fy 

always  be  divided  by  y  (and  0  by  x),  and  since  w  =  fiy/t,  therefore 
we  have 


$b  +  2dx  +fy  -w  c+fx,^Pb  +  2  dx 

w  =  -  5  --  ,-  ,  and  y  =  --         f    0  -        —  , 
—  c  —  2ey  —jx  j+2e 

but  from  just  above  \=  -  ,  —  .  hence  we  have 

c  +  ey  +/x 


(43) 


-w, 


—  we  +/x,  —P6  +  2dx 


f+2e 

Hence  we  have  an  equation  in  which  there  is  no  longer  any 
y;44  and  all  figures  that  can  be  formed  from  this  equation  by  a 
variation  of  the  letters  that  stand  for  the  constants  can  be  squared  ; 
and  also  all  others  that  by  other  methods  can  be  shown  to  be  con- 
nected with  it. 

§  IX. 

In  the  manuscript  that  follows  we  must  refrain  from 
being  critical;  for,  as  suggested  by  the  opening  remark, 
it  contains  nothing  more  than  random  notes,  jotted  down 
as  they  came  into  Leibniz's  mind,  as  materials  for  further 
investigation.  In  the  ten  days  that  have  intervened  since 
the  date  of  the  last  MS.,  he  has  either  had  no  spare  time 
for  further  work  on  the  lines  of  this  last  manuscript,  or 
else  he  has  found  that  he  cannot  proceed  any  further  use- 

43  This  is  hopelessly  inaccurate  ;  all  except  one  error,  namely,  f  -\-  2e, 
which  should  be  Pf  +  2ew,  may  be  put  down  to  bad  transcription.  Even  if 
Leibniz's  writing  were  execrable,  the  correct  version  of  an  ambiguous  sign 
(through  bad  writing)  could  easily  have  been  settled,  by  working  through  the 
algebra.  Thus  the  first  of  the  last  pair  of  values,  in  Leibnizian  symbols 
should  be 


.._  —w,c  +  fx,—ft,  b  +  2dx,, 


—  •w,  c  +  fx,  —ft,  b  +  2dx,,  ^>—e, 
with  a  similar  correction  in  the  second  value. 

44  Even  if  Leibniz  had  worked  out  the  correct  result,  and  obtained  what 
he  was  trying  for,  namely,  w/P  in  terms  of  x,  he  would  have  got  a  very 
lengthy  quadratic,  and  the  roots  would  be  quite  beyond  his  power  to  use  at  any 
time.  But  he  convinces  himself  that  he  can  thus  find  the  quadrature  of  any 
conic,  or  figures  that  can  be  reduces  to  them. 


THE  MANUSCRIPTS  OF  LEIBNIZ.  283 

fully  until  he  has  perfected  the  method  he  had  in  hand. 
He  therefore  reverts  to  the  method  of  breaking  up  the 
figure  into  triangles  by  means  of  a  set  of  lines  meeting  in 
a  point,  coupled  with  the  ideas  of  the  moment  and  the 
center  of  gravity,  in  order  to  try  to  obtain  further  general 
theorems  for  analytical  use.  In  this  way,  he  again  comes 
across  the  differentiation  of  a  product  in  the  form  of  an 
"integration  by  parts" ;  but  he  does  not  recognize  in  it  the 
differentiation  of  a  product,  for  he  says  that  as  he  has 
obtained  this  before  he  can  get  nothing  new  from  it.  He 
is  still  wasting  his  energies  over  the  idea  of  obtaining 
dy/dx  as  an  explicit  function  of  x,  for  the  purposes  of 
integration  or  quadratures.  The  fact  that  he  can  use  the 
method  of  Slusius  as  an  unproved  rule  seems  to  have  hid- 
den from  him  the  necessity  of  pushing  on  his  investigations 
with  regard  to  the  laws  of  differentiation,  or  the  direct 
tangent  method. 

21  November  1675. 

Pro  methodo  tangentium  inversa  et  aliis  tetragonisticis  spe- 
cimina  et  inventa.  Trigonometria  indivisibilium.  Aequa- 
tiones  inadaequatae.  ordinatae  convergentes.  Usus  singu- 
laris  Centri  gravitatis. 

[Examples  and  discoveries  by  means  of  the  inverse  method  of 
tangents  and  other  quadratures.  Trigonometry  of  indivi- 
sibles. Inadequate  equations.  Converging  ordinates.  Spe- 
cial use  of  the  Center  of  Gravity.] 

Subject-matter  for  a  new  consideration  of  the  Center  of  Grav- 
ity method,  as  follows: 

A  segment  AECD  having  been  broken  up  into  infinite  tri- 
angles, AEC,  ACF,  etc.,  let  the  center  of  gravity  of  each  of  these 
triangles  be  found ;  this  is  a  simple  matter,  for  the  center  of  gravity 
is  always  distant  from  the  base  a  third  of  the  altitude.  Then,  since 
the  path  of  the  center  of  gravity  multiplied  by  the  area  of  the 
triangle  is  equal  to  the  solid  formed  by  its  rotation,  and  also  since 
the  products  of  the  AH's  and  the  infinitesimal  parts  of  the  axis  are 
twice  the  areas  of  the  triangle,  also  it  is  plain  that  the  AG's  multi- 


284 


THE  MONIST. 


plied  by  the  distances  of  the  centers  of  gravity  of  the  triangles  AEC 
from  the  axis  are  equal  to  the  moment  of  the  segment  about  the 
axis;  by  the  help  of  this  idea  a  number  of  things  can  be  at  once 
obtained  in  two  ways :  first,  by  taking  some  general  figure  and  mak- 
ing a  general  calculation,  and  then  so  expressing  it  that  the  center 
of  gravity  can  be  easily  found;  in  this  way  we  may  obtain  the 
moments  of  spaces  which  would  be  a  matter  of  difficulty  otherwise, 
if  they  were  investigated  by  the  ordinary  method  of  ordinates. 


Secondly,  on  the  other  hand,  if  figures  of  which  the  moments  are 
easily  obtained  in  the  ordinary  way  are  treated  by  this  method,  we 
shall  arrive  at  certain  very  difficult  curves,  the  dimensions  of  which  can 
always  be  deduced  from  some  that  are  easier.  Here  then  we  have 
a  remarkable  rule,  by  the  help  of  which  useful  properties  can  always 


be  obtained  from  any  method  however  complicated.  It  is  often 
useful  when  problems  arise  that  we  know  are  naturally  simple,  and 
from  other  reasons  are  soluble;  for  thus  many  notable  cases  are 
discovered.  See  what  Tschirnhaus  noted  about  the  Hastarian  line. 
In  irregular  problems,  such  as  cannot  be  treated  in  a  straight- 


THE  MANUSCRIPTS  OF  LEIBNIZ.  285 

forward  manner  or  reduced  to  an  equation  that  is  sufficiently  de- 
terminate, because,  say,  something  has  to  be  done  inversely,  it  is 
useful  to  compare  several  ways  with  one  another,  of  which  the 
results  should  be  identical.  This  seems  to  be  useful  for  the  inverse 
tangent  method.  Here  is  a  case  in  point. 

The  figure,  in  which  BP  and  AT  are  reciprocally  proportional, 
is  required. 

Let  TB  =  f,  then  AT  =  t-x,  andBP  =  a2/(*-*).  If  this  is  multi- 
plied by  t,  we  have 

hence.  ta2  =  ty2  -  xy2, 

or    t  =  xyz/(az-y2)  ;45  and  therefore  t/x=y2/(a2-y2),  or  all  the  f's 

together  equal  the  moment  about  the  vertex  of  every  y2/(a2-y2). 

But  from  other  reasons,  all  the  TP's  applied  to  the  axis  are 
equal  to  the  TC's  applied  to  the  curve. 

0y         _  Pa?  —y* 
Now  t/y={i/w,  and  therefore  w=          y*x  xy     . 

* ==     2 2 

But  fw  =  y,  therefore 

fW-£=y (A) 

xy 

Further,  wx=  —    — — ,  and  J  wx=-yx— J yft, 

/Par—~y* 
m§ 


n    2  _     2 

Also  w=dy,  dy—  -    •—*-,  and  therefore 


y 

Now  if  we  suppose  that  the  y's  are  in  arithmetical  progression, 
then  w  =  dy  is  constant  and  ft  is  variable; 


hence,  P=     -  3  -  2 

a*  -y' 

2          2 

But  from  equation  (B),  /?^-—  — 

y 
2 


hence,  ft—  =dyx. 

40  There  is  a  mistake  in  sign  ;  a2  —  yz  should  be  yz  —  a2  ;  hence  the  work 
that  follows  is  also  wrong. 


286  THE  MONIST. 

We  have  thus  obtained  two  equations  that  are  mutually  inde- 

d*  vie  (46' 

pendent,  the  first  f  =  -  ...............  (1) 

ay     a  +  y,  a—y 


and  the  second  dyx=  —    —  .......................  (2) 

V 
Let  us  seek  to  obtain  others  in  addition,  such  as 

J  t  dy  =  fy  dx. 
Now  this  furnishes  us  with  nothing  new;  but   Ctw+  Cxw  =  xy 

or  t  dy  +  x  dy  =  dxy.  and  t=  --  y;  hence  the  latter  =  —  *—  ^—^, 

dy  dy 

Therefore  dx  y  =  dxy  -  x  dy. 

Now  this  is  a  really  noteworthy  theorem  and  a  general  one 
for  all  curves.  But  nothing  new  can  be  deduced  from  it,  because 
we  had  already  obtained  it. 

However,  from  another  principle  we  shall  obtain  a  new  theo- 
rem; for  it  is  known  that  the  sum  of  every  BP  =  BC2/2;  that  is  to 

say,  BP=  —  ,  /=  &  =  —   y,  and  therefore 

t—x  w       dy   ' 


dxy-dyx       2 

We  therefore  have  two  equations,  in  which  dx  occurs,  namely, 
the  first  and  the  third  ;  by  the  help  of  these,  by  eliminating  dx,  we 
shall  have  an  equation  in  which  only  one  of  the  unknowns  remains 

shackled;  thus  from  equation  (1),  we  have  dx=  %  yx<>  ,  and  now 

a  —y 

from  equation  (3),  we  get  dxydy2-dydy2x=2a?dy.     Hence, 

2c?dy  +  dy  dy^x 

y  dy2 

We  have  therefore  an  equation  between  the  two  values  of  dx, 
in  which  only  the  y  remains  shackled.  From  this,  by  assuming 

46  Although  the  variables  are  separable,  Leibniz  does  not  recognize  the  fact 
that  he  can  make  use  of  this.  For  later  he  states  that  the  solution  of  a  prob- 
lem cannot  be  obtained  from  a  single  equation.  In  this  case  we  have 

dx        y    dy       dv    . 

-  =  -£  -  =5  =  —  ,  if  yz  —  a2  =  ±  v2. 
x         y2—  a?       v    ' 

Supposing  this  substitution  to  have  been  effected,  Leibniz  would  have  concluded 
that  x  =  v,  and  would  have  stated  that  he  had  solved  the  problem. 

But  here  again  he  has  made  an  unfortunate  choice,  for  the  origin  (A) 
cannot  fall  on  any  of  the  curves  Cx  =  v  or  Cxz  ±yz  =  ±  a2,  which  is  the  gen- 
eral solution  of  the  equation.  Hence  the  problem  is  impossible. 


THE  MANUSCRIPTS  OF  LEIBNIZ.  287 

the  y's  to  be  in  arithmetical  progression,  that  is  that  dy  =  fi  a  con- 
stant, and  dy2  =  z,  and  z  =  s*/2  -  y- ;  z  =  V 2  y  =  df."  Thus  we  have 
obtained  what  was  required. 

We  have  here  a  most  elegant  example  of  the  way  in  which 
problems  on  the  inverse  method  of  tangents  are  solved,  or  rather 
are  reduced  to  quadratures.  That  is  to  say  that  the  result  is  obtained 
by  combining,  if  possible,  several  different  equations,  so  as  to  leave 
one  only  of  the  unknowns  in  the  tetragonistic  shackle.  This  can 
be  done  by  summing  ordinates  in  various  ways,  or  on  the  other 
hand,  instead  of  ordinates,  converging  or  other  lines. 

Note.  If,  instead  of  x  or  y,  some  other  straight  line  can  be 
found,  either  one  that  is  oblique,  or  one  of  a  number  converging  to 
the  same  point,  by  the  employment  of  which  one  only  of  the  un- 
knowns is  left  in  bonds,  it  may  be  employed  with  safety.  Take 
for  instance  the  case  of  finding  the  relation  for  the  AP's ;  here  the 
sum  of  AP's  applied  to  the  axis  is  half  the  square  on  AC.  When- 
ever the  formula  for  the  one  unknown  that  is  left  in  shackles  is 
such  that  the  unknown  is  not  contained  in  an  irrational  form  or  as 
a  denominator,48  the  problems  can  always  be  solved  completely ; 
for  it  may  be  reduced  to  a  quadrature,  which  we  are  able  to  work 
out ;  the  same  thing  happens  in  the  case  of  simple  irrationals  or 
denominators.  But  in  complex  cases,  it  may  happen  that  we  obtain 
a  quadrature  that  we  are  unable  to  do.  Yet,  whatever  it  may  come 
to,  when  we  have  reduced  the  problem  to  a  quadrature,  it  is  always 
possible  to  describe  the  curve  by  a  geometrical  motion ;  and  this  is 
perfectly  within  our  power,  and  does  not  depend  on  the  curve  in 
question.  Further,  this  method  will  exhibit  the  mutual  dependence 
of  quadratures  upon  one  another,  and  will  smooth  the  way  to  the 
method  of  solving  quadratures.  Meanwhile  I  confess  that  it  may 
happen  that  there  may  be  need  for  a  very  great  number  of  inade- 
quate equations  (for  so  I  call  them,  when  there  is  need  for  many 
to  solve  the  problem,  although  each  alone  would  suffice  provided 
it  could  be  worked  out  by  itself),  in  order  to  completely  free  one  of 
the  unknowns  from  its  shackles.  For,  unfortunately,  a  solution 
cannot  be  obtained  from  a  single  equation,  unless  one  of  the  terms 
is  free  from  shackles ;  and  if  this  term  appears  oftener,  then  not 
unless  it  is  freed  at  least  once.  Thus  there  may  be  a  great  number 

47  This  is  quite  unintelligible  to  me  as  it  stands ;  query,  is  it  an  accurate 
transcription? 

48  This  is  tantamount  to  a  confession  by  Leibniz  that  he  cannot  explicitly 
integrate  fa2/y,  although  he  knows  that  it  is  logarithmic  or  reduces  to  the  area 
under  the  hyperbola;  for  he  has  given  this  in  the  MS.  for  Nov.  11. 


288 


THE  MONIST. 


of  inadequate  equations  to  be  found ;  and  we  have  to  examine 
which  of  them  are  in  some  way  independent  of  the  others,  i.  e., 
such  as  cannot  be  derived  from  one  another  by  a  simple  manipula- 
tion; for  instance,  the  sum  of  all  the  AP's  and  the  sum  of  all  the 
AE's. 

A  new  kind  of  Trigonometry  of-  indivisibles,  by  the  help  of 
ordinates  that  are  not  parallel  but  converge. 

Let  B  be  a  fixed  point ;  let  BDC  be  a  very  narrow  triangle  stand- 
ing upon  a  curve ;  let  DE  be  the  perpendicular  to  BC ;  from  the  point 
B  let  BA,  perpendicular  to  BC  or  parallel  to  DE,  be  drawn  to  meet 
the  tangent  AHDC,  and  let  BH  be  the  perpendicular  to  the  tangent 
DC  produced. 


Then  the  triangles  CED,  CHB,  BHA  are  similar;  hence  we 
have  BH/CE  =  HA/DE  =  B  A/CD,  and  therefore  BH,  DE  =  CE,  HA, 
and  BH,CD  =  CE,BH.  Hence  it  follows  that  the  sum  of  the  tri- 
angles or  the  area  of  the  figure  is  equal  to  the  products  of  the  AB's 
into  the  CE's,  or  the  differences  of  the  ED's  and  lastly  AH,CD  = 
DE,  BH.49 

Further,  CH/CE  =  HB/DE  =  CB/CD;  hence,  again,  CH,  DE  = 
CE,HB,  and  HB,CD  =  DE,CB;  i.  e.,  the  area  of  the  triangle,  as  is 
in  itself  evident,  is  equal  to  itself.  Lastly,  CH,  CD  =  CE,  CB;  and 
this  last  result  seems  to  be  worth  noting  for  the  case  of  a  Trochoid. 

For,  if  by  the  rolling  of  a  curve  DC  on  a  fixed  plane  CA,  a 
trochoid  curve  is  described  by  the  point  B  fixed  in  DC,  and  it  is 
given  that  the  ordinate  of  the  trochoid  drawn  to  the  fixed  plane  CA 

48  There  are  several  errors  in  the  letters  in  this  paragraph,  which  are 
probably  due  to  transcription;  thus,  an  E  for  a  (?  badly  written)  B,  an  H 
for  an  A,  etc.,  would  be  quite  an  easily-imagined  error,  provided  the  work  was 
not  verified  during  transcription. 


THE  MANUSCRIPTS  OF  LEIBNIZ.  289 

is  BH,  then  the  sum  of  the  intercepts  CH  applied  to  DC  will  be  equal 
to  the  sum  of  the  CB's  applied  to  their  own  differences.  Now  if 
any  ordinates  are  applied  to  their  own  differences,  the  same  thing 
is  always  produced  as  in  the  case  where  we  try  to  find  the  moment 
of  the  differences  about  the  axis,  which  is  the  same  as  the  moment 
when  we  take  the  sum  of  each,  or  the  maximum  ordinate,  into  the 


\ 


distance  of  its  center  of  gravity  from  the  axis,  i.  e.,  its  middle  point, 
that  is  to  say  into  half  itself.  Finally  this  is  equal  to  half  the  square 
on  the  maximum  ordinate.  Therefore  we  can  always  obtain  the 
sum  of  all  the  rectangles  BC,  CE,  which  is  always  equal  to  half 
the  square  on  BC,  or  to  the  sum  of  all  the  BP's  applied  to  the  axis 
in  F,  where  CP  is  the  normal  to  the  curve  DC. 

§  X. 

Leibniz  now  directs  his  attention  to  the  direct  method 
of  tangents,  and  proceeds  to  generalize  the  methods  of 
Descartes.  Is  it  only  a  coincidence  that  Barrow  uses  this 
method  regularly,  the  curve  that  he  is  especially  partial  to 
being  the  rectangular  hyperbola?  Weissenborn  suggests 
the  same  coincidence  occurs  with  respect  to  the  method  of 
Newton,  who  uses  analytical  approximations;  but  if  there 
is  anything  in  either  of  these  suggestions.  I  think  that  the 
Harrovian  idea,  which  is  purely  for  the  construction  of 
tangents,  is  much  nearer  to  that  of  Leibniz  in  this  manu- 
script than  is  the  Newtonian. 

However  this  may  be,  Leibniz  is  at  last  beginning  to 
consider  the  point  as  to  the  method  by  which  the  principle 
of  Sluse  is  obtained.  He  ascribes  it  to  a  development  of  the 
method  of  Descartes;  but  in  this  connection  I  cannot  get 
out  of  my  head  the  suggestion  raised  by  Barrow's  use  of 
the  first  person  plural,  "frequently  used  by  us,"  in  the 


29O  THE  MONIST. 

midst  of  a  passage  that  is  written,  contrary  to  his  usual 
custom,  in  the  first  person  singular  throughout,  where  he 
describes  the  differential  triangle  and  the  "a  and  e"  method. 
I  consider  that  Sluse  has  enunciated  a  working  rule  for 
tangents,  which  he  has  generalized  by  observation  of  the 
results  obtained  by  the  use  of  the  "a  and  e"  method;  and 
that  this  method  had  been  circulated  by  Barrow  some  time 
before  the  publication  of  the  Lectiones  Geometricae,  al- 
though I  confess  that  I  have  not  found  any  record  of  this, 
nor  any  distinct  evidence  of  a  correspondence  between 
Barrow  and  Sluse ;  but  there  is  more  than  a  suggestion  of 
this  in  the  fact  that  Sluse's  article  was  published  in  the 
Phil.  Trans,  for  1672. 

It  seems  more  than  strange  to  me  that  there  should  be 
such  a  prolific  crop  of  differential  calculus  methods  within 
a  couple  of  years  of  the  work  of  Barrow  in  all  sorts  of 
places,  raised  by  many  different  people,  and  that  none  of 
them  allude  to  the  general  seed-merchant,  as  I  consider 
Barrow  to  have  been. 

22  Nov.  1675. 

Methodi  tangentium  directae  compendium  calculi,  dum  jam 
inventis  aliarum  curvarum  tangentibus  utimur.  Quaedam 
et  de  inversa  methodo. 

[Compendium  of  the  calculus  of  the  direct  method  of  tangents, 
together  with  its  use  for  finding  tangents  to  other  curves. 
Also  some  observations  on  the  inverse  method.] 

In  that  which  I  wrote  on  Nov.  21,  I  noted  down  those  things 
which  came  to  my  mind  concerning  the  method  of  tangents.  Re- 
turning to  the  subject,  let  ACCR  and  QCCS  be  two  curves  that  cut 
one  another  in  one,  two,  or  more  points  C,  C;  let  AB(B)  be  the 
axis;  let  AB  =  jr  be  the  ordinates,  and  BC  =  3/  the  abscissae;  then 
we  shall  have  two  equations  to  the  two  lines,  each  in  terms  of  these 
two  principal  unknowns.  Now  if  these  two  equations  have  equal 
roots,  or  the  equations  have  equal  values,  then  the  lines  will  touch 
one  another.  Instead  of  the  line  QCCS,  Descartes  chooses  the  arc 
of  a  circle  VCCD,  whose  center  is  P,  so  that  PC  is  the  least  of  all 


THE  MANUSCRIPTS  OF  LEIBNIZ. 


291 


the  lines  that  can  be  drawn  from  the  point  P.  It  will  come  to  the 
same  thing,  and  often  more  simply,  if  we  take  not  the  arc  of  a 
circle  but  the  tangent  line  TC(C),  that  is  the  greatest  of  all  those 
that  can  be  drawn  from  a  given  point  T  to  the  curve.  Let  TA  =  b, 
AE  =  £,  be  assumed  as  given;  required  to  find  AB,  BC.  The  two 
equations  are,  the  one  for  the  curve  AC(C),  namely,  ax-  +  cy2  +  etc. 
=  0,  and  the  other  to  the  straight  HneTC(C)  which,  on  account  of  the 
relation  TA/AE  =  TB/BC.  will  beb/e=(b±x)/y  or  ±x=(b/e}y-b 


or  y=±    e 

Thus  the  value  of  either  one  or  other  of  the  unknowns  can  always 
be  obtained  explicitly,  and  thus  can  be  worked  out  immediately 
without  raising  the  degree  of  the  equation  of  the  given  curve 
AC(C)  ;  and  then  at  once  we  shall  obtain  an  equation  giving  the 
unknown  that  alone  remains,  so  that  we  may  determine  the  condition 
for  equal  roots.  Doubtless  this  is  the  principle  of  Sluse's  method. 
If  however  the  arc  of  the  circle  whose  center  is  P  is  used, 
following  Descartes,  then  the  new  equation,  for  the  circle,  will  be 
as  follows:  let  the  radius  PC  =  .y,  and  PB  =  z/-x,  and  we  have 


s2  =  y-  +  v2  +  A'2  -  2vx.  Hence  it  is  clear  that  we  have  the  choice  of 
either  a  circle  or  a  straight  line ;  and  when,  in  the  equation  to  the 
given  curve,  only  an  even  power  of  y  appears  (as  can  always  be 
made  to  happen  in  the  case  of  the  conies),  then  it  will  be  more  con- 
venient to  use  equations  to  circles ;  for  thus,  by  the  help  of  the  two 
values  of  y2,  the  unknown  x  can  be  immediately  worked  out;  but, 
in  general  for  all  equations  to  curves  expressed  by  a  rational  rela- 
tion, the  method  of  the  straight  line  may  be  usefully  employed. 

Hence  I  go  on  to  say  that  not  only  can  a  straight  line  or  a 
circle,  but  any  curve  you  please,  chosen  at  random,  be  taken,  so 
long  as  the  method  for  drawing  tangents  to  the  assumed  curve  is 


292  THE  MONIST. 

known ;  for  thus,  by  the  help  of  it,  the  equations  for  the  tangents 
to  the  given  curve  can  be  found.  The  employment  of  this  method 
will  yield  elegant  geometrical  results  that  are  remarkable  for  the 
manner  in  which  long  calculation  is  either  avoided  or  shortened, 
and  also  the  demonstrations  and  constructions.  For  in  this  way 
we  proceed  from  easy  curves  to  more  difficult  cases,  and  an  equa- 
tion to  a  curve  being  supposed  known,  it  is  always  possible  to  choose 
an  equation  to  some  other  curve  whose  tangents  are  known,  by  the 
help  of  which  one  of  the  unknowns  can  be  worked  out  very  easily. 

Thus,  if  it  is  given  that  hy2  +  y3  =  cxs  +  dx*  +  ex  +  f  is  the  equa- 
tion to  a  curve  of  which  the  tangents  are  required,  assume  a  curve 
of  which  the  equation  is  hy2  +  yz  =  gx  +  q,  for  that  of  which  the 
tangents  are  already  known ;  eliminating  y,  we  have  an  equation 
such  as  gx  +  q  =  cx3  +  dx2  +  ex  +  f.  This  can  be  determined  for  two 
equal  roots,  either  by  Descartes's  method  of  comparisons,  or  Hudde's 
by  means  of  an  arithmetical  progression ;  and  thus  by  working  out 
the  value  of  x,  the  value  of  either  g  or  q  may  be  found ;  and  one  of 
the  two  letters  q  or  g  can  be  chosen  arbitrarily.50  Hence,  a  way  of 
describing  that  other  curve  that  touches  the  given  curve  is  obtained ; 
now,  when  this  is  described,  let  the  tangent  be  drawn  at  the  point 
which  is  common  to  it  and  the  proposed  curve,  which  tangents  we 
have  supposed  to  be  already  known;  then  this  tangent  will  touch 
the  given  curve. 

I  think  that,  in  general,  the  calculation  will  be  possible  by  this 
method  of  assuming  a  second  curve,  as  we  have  done  in  this  case, 
which  evidently  works  out  one  of  the  unknowns.  Hence  I  fully 
believe  that  we  shall  derive  an  elegant  calculus  for  a  new  rule  of 
tangents,  which  in  addition  may  be  better  than  that  of  Sluse,  in  that 
it  evidently  works  out  immediately  one  of  the  two  unknowns,  a  thing 
that  the  method  of  Sluse  did  not  do.  Now  this  very  general  and 
extensive  power  of  assuming  any  curve  at  will  makes  it  possible, 
I  am  almost  sure,  to  reduce  any  problem  to  the  inverse  method  of 
tangents  or  to  quadratures.  Indeed  let  any  property  of  the  tan- 
gents to  a  curve  be  given,  and  let  the  relation  between  the  ordinates 

50  The  method  of  Hudde  appears  to  be  similar  in  principle  to  that  of  Sluse, 
while  that  of  Descartes  was  the  construction  of  the  derived  function  by  assum- 
ing roots,  forming  the  sum  of  the  quotients  of  the  function  divided  by  each  of 
the  assumed  root-factors  in  turn,  and  comparison  with  the  original  function. 
Both  therefore  reduce  to  finding  the  common  measure  of  the  equation  to  the 
curve  (where  the  right-hand  side  is  zero)  and  the  differential  of  it. 

Leibniz,  however,  strange  to  say,  does  not  note  that  by  taking  one  of  his 
arbitrary  constants,  q,  equal  to  f,  the  equation  has  its  degree  lowered  in  the 
particular  case  he  has  chosen. 


THE  MANUSCRIPTS  OF  LEIBNIZ.  293 

and  the  abscissae  be  required.  Then  an  equation  can  be  derived, 
which  will  contain  the  principal  unknowns,  x,  y,  and  always  two 
others  as  incidentals,  such  as  s  and  v,  or  b  and  e,  or  the  like ;  now, 
as  the  equation  contains  the  property  of  the  tangents,  by  which  s 
and  b  may  be  expressed  so  as  to  have  a  relation  to  the  tangents, 
assume  in  this  case  any  new  curve  chosen  arbitrarily,  and  then  s 
and  v  will  also  have  some  known  relation  to  this  curve.  By  means 
of  the  equation  to  the  arbitrarily  chosen  curve,  we  shall  be  able 
to  replace  the  given  property  of  tangents  in  favor  of  the  curve  re- 
quired, namely,  by  removing  one  or  other  of  the  unknowns ;  and 
by  thus  reducing  the  problem  to  such  a  state  the  inverse  calculation 
will  come  out  the  more  easily. 

The  whole  thing,  then,  comes  to  this;  that,  being  given  the 
property  of  the  tangents  of  any  figure,  we  examine  the  relations 
which  these  tangents  have  to  some  other  figure  that  is  assumed  as 
given,  and  thus  the  ordinates  or  the  tangents  to  it  are  known.  The 
method  will  also  serve  for  quadratures  of  figures,  deducing  them 
one  from  another ;  but  there  is  need  of  an  example  to  make  things 
of  this  sort  more  evident ;  for  indeed  it  is  a  matter  of  most  subtle 
intricacy. 

The  manuscripts  mentioned  above  seem  to  be  all  that 
were  found  by  Gerhardt  belonging  to  the  period  1673-5. 
I  feel  that  it  is  a  great  pity  that  they  were  not  given  in 
full,  or  at  least  a  little  more  fully.  For  instance,  Gerhardt 
mentions  that  Leibniz  in  the  MS.  of  August  1673  con- 
structs the  so-called  characteristic  triangle,  but  does  not 
give  Leibniz's  figure  in  connection.  This  figure  should 
have  been  given;  for  the  figure  given  in  October  1674  is 
not  the  characteristic  triangle  as  given  by  Leibniz  in  the 
"postscript"  (§1),  or  the  Historia  (§11),  but  it  is  the 
Pascal  diagram  (assuming  that  the  figure  given  by  Cantor 
is  the  correct  one).  It  would  be  useful  to  know  the  date 
at  which  Leibniz  drops  the  Pascal  diagram  in  favor  of 
one  or  other  of  the  Barrow  diagrams. 

It  is  to  be  noticed  at  this  date  that  Leibniz  uses  one 
infinitesimal  only,  and  verifies  that  the  method  of  Des- 
cartes comes  out  correctly  in  the  simple  case  of  the  parab- 


294  THE  MONIST. 

ola;  but  he  is  not  satisfied  with  the  generality  of  the 
method  of  neglecting  the  vanishing  quantities. 

Again,  the  second  manuscript  of  October  1674  appears 
to  be  immensely  important;  especially  as  it  contains  the 
groundwork  of  some  of  the  later  manuscripts.  Judging 
by  the  little  that  is  given  of  it,  it  would  seem  to  be  most 
desirable  that  fuller  extracts,  at  least,  should  have  been 
given.  It  is  a  matter  for  remark  that  this  manuscript  is 
a  long  essay  on  series.  Can  this  possibly  have  had  any- 
thing to  do  with  the  fact  that  it  is  not  given  in  full? 


(TO  BE  CONTINUED.) 

J.  M.  CHILD. 


DERBY,  ENGLAND. 


CRITICISMS  AND  DISCUSSIONS. 
MECHANISM  AND  THE  PROBLEM  OF  FREEDOM. 


Men's  opinions  are  far  more  commonly  the  result  of  the  gen- 
eral presuppositions  and  prejudices  of  the  age  in  which  they  live 
than  the  outcome  of  a  rational  process.  Thus  men  believe  whatever 
fits  in  with  their  general  view  of  life  and  dismiss  without  a  hearing 
anything  which  conflicts  with  it.  In  this  age  of  science  the  scientist 
has  become  the  arbiter  of  all  questions,  and  his  view  is  commonly 
accepted  as  authoritative.  Hence  problems  which  he  refuses  to 
examine,  e.  g.,  the  question  of  the  existence  of  ghosts,  are  at  once 
relegated  to  the  realm  of  superstition.  Now  there  is  some  danger 
of  freedom  being  placed  among  such  problems.  An  indication  of 
this  is  found  in  the  following  words  of  Haeckel,  which  represent 
the  attitude  of  many  contemporary  scientists  and  psychologists 
toward  the  question  of  freedom:  "The  great  struggle  between  the 
determinist  and  the  >indeterminist,  between  the  opponent  and  the 
sustainer  of  the  freedom  of  the  will,  has  ended  to-day,  after  more  than 
two  thousand  years,  completely  in  favor  of  the  determinist.  The 
human  will  has  no  more  freedom  than  that  of  the  higher  animals, 
from  which  it  differs  only  in  degree,  not  in  kind ....  We  now  know 
that  each  act  of  the  will  is  as  fatally  determined  by  the  organization 
of  the  individual  and  as  dependent  on  the  momentary  condition  of 
his  environment  as  every  other  psychic  activity."1  This  view  has 
won  its  way  by  its  scientific  prestige,  and  has  been  eagerly  accepted 
by  many  who  have  never  examined  the  evidence  for  freedom,  but 
who  nevertheless  smile  indulgently  at  those  who  are  still  so  benighted 
as  to  believe  in  it.  Therefore,  in  view  of  the  great  popularity  and 
influence  that  Haeckel  has  enjoyed  it  seems  profitable  to  examine 
the  question  of  the  relation  of  mechanism  and  freedom  with  ref- 
erence to  his  specific  teaching. 

1  The  Riddle  of  the  Universe,  p.  130f. 


296  THE  MONIST. 

There  are  two  main  principles  on  which  Haeckel  bases  his  sys- 
tem: the  doctrine  of  evolution,  and  the  "law  of  substance."  The 
doctrine  of  evolution  furnishes  the  principal  evidence  for  his  denial 
of  a  spiritual  principle  in  man  and  together  with  the  "law  of  sub- 
stance" leads  to  his  mechanistic  determinism.  We  shall  therefore 
first  consider  the  question  whether  the  existence  of  a  spiritual  prin- 
ciple is  precluded  by  evolution  or  by  any  other  arguments  suggested 
by  Haeckel.  We  shall  then  proceed  to  the  question  of  the  univer- 
sality of  the  "law  of  substance"  and  to  the  problem  of  its  relation 
to  mechanism,  and  finally  examine  briefly  the  adequacy  of  mechan- 
ism itself  as  a  philosophic  explanation  of  the  universe. 

ii. 

Haeckel  never  tires  of  citing  facts  in  support  of  the  doctrine 
of  evolution.  Since  this  doctrine  in  some  form  or  other  is  now 
almost  universally  accepted  as  true  we  need  not  stop  for  an  instant 
to  inquire  concerning  the  adequacy  of  this  evidence.  The  only 
question  for  us  to  consider  is  whether  the  doctrine  of  evolution 
inevitably  leads  to  the  reduction  of  mind  to  matter.  Put  very 
simply  Haeckel's  argument  for  this  conclusion  is  that  since  man 
developed  from  the  lowest  forms  of  life  there  is  no  reason  to  at- 
tribute to  him  a  separate  immaterial  or  spiritual  principle  not  found 
in  the  forms  of  life  from  which  he  originated.  Many  objections 
to  this  argument  at  once  suggest  themselves  to  the  thoughtful  reader. 
In  the  first  place  it  takes  for  granted  the  old  scholastic  idea  of  rigid 
continuity,  according  to  which  nothing  new  can  ever  arise.  Now 
there  are  grave  difficulties  in  this  view,  but  even  waiving  these  for 
the  moment,  Haeckel's  conclusion  by  no  means  follows.  Rather, 
the  doctrine  of  continuity,  if  strictly  held,  would  force  him  to  read 
into  the  life  of  the  lowest  organism  all  the  complex  processes  and 
meanings  which  have  been  evolved  in  the  highest  forms  of  life. 
For  if  evolution  were  rigidly  continuous  the  very  fact  that  certain 
phenomena,  such  as  sensation  and  will,  have  developed  in  the  later 
stages  of  the  evolutionary  process  would  show  that  these  phenomena 
were  implicit  in  the  earlier  forms.  Thus  Haeckel  would  be  com- 
pelled to  understand  the  protozoon  in  the  light  of  man,  rather  than 
to  reduce  man  to  the  level  of  the  protozoon.  He  indeed  seems  some- 
times to  do  this,  and  with  an  extraordinary  anthropomorphism 
bestows  elementary  will  and  elementary  emotion  upon  even  inani- 
mate matter.2  If  he  held  consistently  to  this  view  his  final  system 

2  Cf .  infra,  p.  304. 


CRITICISMS  AND  DISCUSSIONS.  297 

would  be  in  the  nature  of  a  theological  hylozoism  rather  than  a 
strictly  mechanistic  determinism.  However,  this  reading  of  con- 
tinuity, this  attribution  of  man's  processes  to  the  lower  forms  of 
life,  is  misleading,  as  it  involves  what  Baldwin  calls  "the  fallacy 
of  the  implicit."  As  a  matter  of  fact,  if  progress  is  genuine,  new 
processes  and  new  meanings  must  arise  which  cannot  be  interpreted 
in  terms  of  the  lower  stages.  Thus  even  though  life  has  arisen 
from  the  inanimate,  and  consciousness  from  the  unconscious,  yet 
they  involve  meanings  and  processes  which  cannot  be  expressed  in 
terms  of  the  stages  from  which  they  arose.  It  is  fallacious  either 
to  deny  these  new  meanings  and  attempt  to  reduce  them  to  earlier 
stages,  or  to  read  them  back  as  implicit  in  the  earlier  stages.  Hence 
the  doctrine  of  evolution  in  no  wise  militates  against  the  spiritual 
nature  of  man.  Rather  it  leads  us  to  expect  man's  nature  to  be 
higher  or  more  developed  than  the  merely  physical  or  the  merely 
biological. 

In  addition  to  the  argument  drawn  from  evolution,  Haeckel 
adduces  several  other  considerations  in  support  of  his  denial  of  the 
spiritual  nature  of  man.  He  brings  forward  the  evidence  of  ex- 
periments which  have  shown  that  various  functions  of  the  soul, 
such  as  speech  and  sense  images,  are  connected  with  definite  areas 
of  the  cortex  of  the  brain  and  disappear  when  these  areas  are 
diseased  or  destroyed.3  Again  he  calls  our  attention  to  the  close 
connection  between  man's  higher  cerebral  functions  and  purely  phy- 
siological processes — a  connection  especially  plain  in  the  case  of 
emotions.*  He  also  emphasizes  facts  concerning  the  individual's 
development  which,  in  his  opinion,  indicate  that  the  soul  originates, 
grows,  and  decays  with  the  body.5  Finally  he  points  out  that  we 
never  find  a  single  instance  of  a  spiritual  principle  unconnected  with 
a  physical  substrate.6 

Now  although  all  the  above  arguments  show  clearly  that  there 
is  some  relation  between  mind  and  body,  yet  they  do  not  succeed 
in  reducing  mind  to  body.  The  facts  can  be  read  as  easily  the 
other  way.  As  a  matter  of  fact,  we  are  as  conscious  of  the  influence 
of  mind  on  body  as  of  body  on  mind.  It  is  true  that  illness  or 
various  physical  causes  affect  man's  mental  processes,  but  it  is  also 
true  that  man's  mental  processes  affect  his  physical  condition.  In- 

3  Cf.  Last  Words  on  Evolution,  98f ;  The  Riddle  of  the  Universe,  p.  204. 

4  Cf.  The  Riddle  of  the  Universe,  pp.  127,  204. 

5  Cf.  The  Riddle  of  the  Universe,  Chap.  VIII. 
«  Cf.  Ibid.,  pp.  90,  91. 


298  THE   MONIST. 

deed  every  act  of  will  is  evidence  of  the  power  of  mind  over  matter. 
Now  to  this  Haeckel  might  retort:  "What  you  call  will  is  merely 
a  certain  functioning  of  a  physiological  organism.  I  can  even  dis- 
close to  you,  with  my  microscope,  the  minute  structures  in  the  brain 
by  which  willing  takes  place."  But  even  if  this  last  contention 
were  granted  it  would  not  prove  Haeckel's  point.  The  brain  with 
its  various  structures  may  be  the  instrument  of  mind's  expres- 
sion without  being  the  cause  of  mind.7  Moreover  the  very  facts 
of  pathology  which  are  cited  by  Haeckel  to  show  the  dependence 
of  mind  on  matter  are  used  by  Bergson  to  prove  that  mind  cannot 
be  located  in  the  brain  nor  determined  by  it.8  Furthermore  in  this 
controversy  concerning  the  relation  of  mind  and  body,  the  idealist 
can  always  go  back  to  Berkeley's  position  and  retort,  "The  brain, 
the  nervous  system,  etc.,  to  which  you  attempt  to  reduce  mind,  are 
known  only  as  ideas  of  mind,  and  cannot  be  proved  to  exist  apart 
from  mind." 

A  further  weakness  in  Haeckel's  arguments  is  that  they  often 
betray  a  total  misunderstanding  of  his  opponents'  position.  They 
are  all  directed  against  the  existence  of  a  separate,  immaterial  sub- 
stance or  soul.  Most  idealists,  however,  regard  the  soul  as  activity 
or  functioning,  rather  than  as  substance.  They  do  not  insist  on 
the  separateness  of  the  psychic  principle  or  on  the  existence  of  any 
disembodied  spirit,  but  rather  on  the  fact  that  man's  activity  cannot 
be  explained  in  purely  physical  or  physiological  terms.  Suppose 
it  be  granted  to  Haeckel  that  the  soul  is  but  the  "sum  total  of  phy- 
siological functions,"  yet  the  problem  of  the  activity  of  the  soul  is 
not  thereby  solved.  Consciousness  is  a  fundamental  fact  of  ex- 
perience, and  it  cannot  be  explained  by  being  set  aside  or  labeled 
an  epiphenomenon.  Therefore  the  materialist  must  explain  not  only 
how  the  body  reacts,  but  how  it  is  conscious,  how  it  thinks,  evalu- 
ates, loves,  struggles,  and  sacrifices.  It  is  indeed  questionable 
whether  this  activity  can  be  interpreted  in  purely  biological  or  phy- 
siological terms.  The  so-called  body  becomes  equivalent  to  the 
mind  and  demands  the  same  sort  of  an  explanation. 

We  conclude  then  from  this  discussion  that  Haeckel's  reduction 
of  the  psychical  to  the  physical  is  not  valid,  and  we  turn  to  an 


7  Cf.  Schiller,  Riddles  of  the  Sphinx,  pp.  293ff ;  Bergson,  Matter  and  Mem- 
ory,  PP-  299ff;  James,  Human  Immortality,  pp.  7-29. 

8  Cf.  Bergson,  Matter  and  Memory,  Chapter  II. 


CRITICISMS  AND  DISCUSSIONS.  299 

examination  of  the  "law  of  substance" — the  second  main  support 
of  Haeckel's  system. 

in. 

The  "law  of  substance"  is  a  combination  of  the  well-known 
scientific  laws  of  the  "conservation  of  matter"  and  "the  conservation 
of  energy."  According  to  Haeckel,  these  laws  are  but  two  aspects 
of  one  great  cosmic  law,  since  they  relate  to  the  two  inseparable 
attributes  of  substance.  In  passing  it  may  be  noted  that  little  is 
gained  by  this  combination  of  the  two  laws,  since  Haeckel's  unknown 
substance  is  incapable  of  showing  concretely  the  relation  between 
matter  and  energy.  Haeckel  regards  this  law  as  the  one  great 
eternal  cosmic  law.  On  what  evidence  then  can  he  base  its  validity  ? 

The  evidence  for  the  law  adduced  by  Haeckel  lies  in  the  realm 
of  scientific  experiment.  Thus  the  law  of  "conservation  of  energy" 
rests  on  the  fact  that  many  experiments  have  shown  that  when  one 
form  of  energy  is  changed  into  another,  it  may  be  reconverted  into 
the  original  form  of  energy  with  only  a  slight  loss  due  to  the  escape 
of  part  of  the  energy  into  an  unavailable  form.  Similarly  the  law 
of  the  "conservation  of  matter"  rests  on  experiments  which  have 
demonstrated  that  the  weight  of  a  substance  does  not  change 
throughout  a  series  of  chemical  transformations.  Moreover  no  ex- 
periments have  given  any  indication  of  the  creation  or  destruction 
of  matter  or  of  energy,  and  the  generalization  of  a  great  number 
of  phenomena  under  these  laws  has  been  indeed  a  great  achievement 
of  science.  Yet  it  is  one  thing  to  regard  these  laws  as  useful  gen- 
eralizations for  the  purposes  of  science,  and  quite  another  to  erect 
them  into  ontological  and  absolutely  universal  laws.  Against  this 
latter  proceeding,  which  is  that  of  Haeckel,  an  emphatic  protest 
must  be  made.  There  are  three  grounds  for  this  protest:  (1)  the 
laws  have  never  been  proved  to  hold  exactly  in  any  field;  (2)  the 
fact  that  the  laws  appear  to  hold  in  one  or  two  fields  is  no  justi- 
fication for  the  assertion  that  they  must  hold  in  all  fields;  (3)  ex- 
perience can  never  prove  the  absolute  universality  of  any  law. 

In  the  first  place,  it  is  manifestly  impossible  to  prove  that  the 
laws  hold  exactly  in  any  field,  since  the  inaccuracy  of  scientific  instru- 
ments is  such  that  small  differences  might  pass  unnoted.  Further- 
more there  are  always  extraneous  circumstances  which  must  be 
taken  into  account  in  an  appraisal  of  the  results  of  an  experiment. 
A  scientific  result  is  always  an  approximation,  and  the  scientific 


3OO  THE  MONIST. 

law  states  what  would  take  place  under  ideal  circumstances  rather 
than  what  occurs  in  any  concrete  situation. 

In  the  second  place,  the  fact  that  the  laws  appear  to  hold 
true  within  certain  fields  of  our  experience  does  not  show  that 
they  must  hold  in  all  fields.  Thus  the  demonstration  of  the  laws  in 
the  case  of  physical  and  chemical  changes  would  furnish  no  proof 
of  their  applicability  to  the  relation  between  the  physical  and  the 
psychical.  It  is  at  this  point  that  Haeckel's  assertion  of  the  uni- 
versality of  the  law  depends  upon  his  reduction  of  the  psychical 
to  the  physical.  Since  this  is  not  valid,  he  is  not  entitled  without 
more  ado  to  extend  the  application  of  the  law  to  the  psychical 
realm.  The  application  of  the  law  here  must  rest  upon  experi- 
ments showing  that  a  certain  amount  of  physical  energy  can  be 
transformed  into  a  definite  amount  of  psychical  energy  and  re- 
converted into  the  original  amount  of  physical  energy.  Manifest 
difficulties  stand  in  the  way  of  such  experimentation,  but  until 
something  of  the  sort  is  carried  out  there  is  little  significance  in 
speaking  of  the  psychical  life  as  a  form  of  energy.  To  do  so 
merely  covers  up  the  fact  of  our  ignorance  concerning  the  relation 
between  the  psychical  and  the  physical.  Now  apparently  Haeckel 
himself  is  aware  of  some  of  the  difficulties  in  the  way  of  regarding 
the  psychic  as  a  form  of  energy  since  in  his  last  work,  contrary  to 
many  of  his  previous  assertions,9  he  explicitly  teaches  that  the 
psychic  is  a  separate  attribute  of  substance,  coordinate  with  matter 
and  energy.10  If  this  be  admitted,  however,  the  psychic  must  de- 
mand its  own  law  of  the  "conservation  of  the  psychic,"  if  it  is  to 
come  under  the  "law  of  substance."  In  any  case,  the  important 
point  for  our  purpose  is  that  until  the  law  is  proved  to  be  valid  in 
the  psychical  field  it  furnishes  no  ground  for  a  denial  of  freedom. 

Our  last  objection  needs  no  justification,  as  it  is  a  philosophic 
commonplace  that  laws  resting  on  experience  can  be  universalized 
only  by  means  of  the  supposition  of  the  uniformity  of  nature.  This 
uniformity,  however,  cannot  be  proved  by  experience  without  the 
assumption  of  its  own  existence  in  the  attempted  proof.  Thus  the 
observation  that  the  laws  apparently  hold  in  a  comparatively  few 
instances  within  the  narrow  range  of  our  experience  is  no  proof 
that  they  have  always  held  and  will  always  hold  throughout  the 
length  and  breadth  of  the  universe. 

We  have  seen  reason  to  question  the  dogmatic  assertion  of  the 

9  Cf.  The  Riddle  of  the  Universe,  p.  220;  Anthropogenic,  p.  941. 

10  Cf.  Die  Lebenswunder,  p.  185. 


CRITICISMS  AND  DISCUSSIONS.  3OI 

universality  of  the  "law  of  substance."  Yet  if  it  be  admitted  for 
the  sake  of  argument  that  the  law  is  universal  and  necessary,  it  by 
no  means  follows  that  this  law  alone  gives  an  adequate  account  of 
reality  and  a  solution  of  all  its  riddles.  The  law  is  an  abstraction ; 
it  is  purely  quantitative,  and  as  such  leaves  out  of  account  the 
qualitative  aspects  of  the  universe.  Thus  although  the  amount  of 
matter  and  energy  in  the  universe  remain  constant,  changes  in  their 
form  or  in  their  combination  bring  about  new  qualities  not  reducible 
to  mere  quantity.  Take,  for  example,  the  case  of  a  chemist  who 
mixes  together  two  elements  in  a  new  combination.  Their  weight, 
as  a  measure  of  their  quantity,  remains  the  same,  but  this  quantita- 
tive equality  in  no  wise  explains  or  describes  the  new  odor,  the  new 
color,  or  other  new  properties  possessed  by  the  compound.  These 
qualitative  aspects,  however,  are  certainly  part  of  reality,  although 
they  cannot  be  described  by  the  law  of  substance  nor  comprehended 
in  a  system  which  uses  this  law  as  the  solution  of  all  its  problems. 

The  "law  of  substance"  is  for  Haeckel  but  a  necessary  conse- 
quence of  mechanical  causation.  In  fact  the  two  for  him  are  iden- 
tical.11 Yet  the  relationship  does  not  appear  to  be  as  simple  as  he 
would  have  us  believe.  The  law  of  substance,  he  tells  us,  is  a 
consequence  of  mechanical  causation,  yet  his  proof  of  the  latter 
rests  largely  on  his  supposed  proof  of  the  universality  of  the  former. 
Now  the  law  of  mechanical  causation,  involving  the  equivalence  of 
past  and  present,  might  lead  naturally,  though  perhaps  not  inevitably, 
to  the  "law  of  substance."  On  the  other  hand  the  "law  of  substance" 
does  not  necessarily  involve  mechanical  causation.  It  does  indeed 
preclude  spontaneity,  but  it  would  be  as  compatible  with  teleology 
as  with  mechanism,  since  it  says  nothing  concerning  the  origin  of 
changes  in  matter  or  energy.  The  amount  of  energy  and  matter  in 
the  universe  might  remain  constant  if  their  changes  were  due  to  a 
desire  for  a  future  state  as  well  as  if  they  were  due  to  a  past  stim- 
ulus. Thus  even  the  universality  of  the  "law  of  substance"  would 
not  prove  the  universality  of  mechanism.  The  latter  theory  must 
stand  on  its  own  feet  and  be  accepted  or  rejected  on  its  own  merits. 

IV. 

Haeckel  declares  that  mechanical  causation  explains  all  phe- 
nomena. To  quote  his  own  words:  "The  great  abstract  law  of 
mechanical  causality,  of  which  our  cosmological  law — the  law  of 
substance — is  but  another  and  a  concrete  expression,  now  rules  the 

"  Cf.  The  Riddle  of  the  Universe,  pp.  215,  366. 


3<D2  THE  MONIST. 

entire  universe  as  it  does  the  mind  of  man;  it  is  the  steady  im- 
movable pole-star  whose  clear  light  falls  on  our  path  through  the 
dark  labyrinth  of  the  countless  separate  phenomena."12  "The  monism 
of  the  cosmos  which  we  establish  thereon  proclaims  the  absolute 
dominion  of  'the  great  eternal  iron  laws'  throughout  the  universe. 
It  thus  shatters  at  the  same  time  the  three  central  dogmas  of  the 
dualistic  philosophy — the  personality  of  God,  the  immortality  of  the 
soul,  and  the  freedom  of  the  will."13 

Before  accepting  Haeckel's  conclusion  concerning  freedom  the 
adequacy  of  mechanism  itself  must  be  examined.  Of  the  many 
objections  which  might  be  made,  and  which  have  been  made,  to 
universal  mechanism,  we  shall  confine  ourselves  to  the  following: 
(1)  the  universality  of  mechanism  cannot  be  proved;  (2)  the  uni- 
versality of  mechanical  causation  would  not,  as  Haeckel  would  have 
us  believe,  necessarily  preclude  purpose  and  rational  or  ethical  free- 
dom; (3)  mechanism  by  itself  fails  to  give  a  satisfactory  account 
of  experience  as  we  actually  know  it. 

We  contend  that  mechanism  cannot  be  proved.  Experience 
cannot  show  that  mechanical  causation  is  universal  and  necessary, 
and  reason  does  not  disclose  any  logical  necessity  for  insisting  that 
every  aspect  of  reality  shall  be  explained  by  reference  to  the  past. 
On  the  contrary,  the  concept  of  mechanical  causation  is  full  of 
difficulties  which  force  the  mind  beyond  it.14  The  universality  of 
mechanical  causation  is  indeed  a  methodological  postulate  of  sci- 
ence, but  not  necessarily  a  universal  principle  of  reality.  Haeckel 
makes  many  dogmatic  assertions  to  the  effect  that  mechanism  is 
universal,  and  that  even  the  will  is  absolutely  bound  by  causal  law. 
Thus  the  will  is,  he  declares,  the  necessary  outcome  of  heredity 
and  environment.  Yet  obviously  he  cannot  prove  that  such  is  the 
case.  He  cannot  prove  that  A  did  a  certain  act  because  A  had  a 
certain  heredity  and  a  certain  environment,  and  that  A  could  not 
have  done  anything  else.  Indeed,  in  the  case  of  human  activities 
so  many  complex  conditions  occur  that  it  is  practically  impossible 
to  isolate  any  set  of  conditions  in  such  a  way  as  to  establish  a 
uniform  series  of  cause  and  effect.  Hence  the  establishment  of 
causal  connection  (quite  apart  from  the  question  of  its  universality 
and  necessity)  is  in  such  cases  a  task  for  the  future,  rather  than 

12  The  Riddle  of  the  Universe,  p.  366. 

13  Ibid.,  p.  381. 

14  For  a  careful  analysis  of  causation  cf.  Taylor,  Metaphysics,  pp.  158ff ; 
Ward,  Realm  of  Ends,  pp.  273ff ;  Bergson,  Time  and  Free  Will,  pp.  199-221. 


CRITICISMS  AND  DISCUSSIONS.  303 

an  accomplished  fact.  The  possibility  therefore  remains  that  the 
mental  life  may  resist  such  causal  treatment.  An  indication  in  this 
direction  is  found  in  the  comparative  lack  of  success  of  psychology 
in  the  use  of  scientific  methods  found  fruitful  in  other  fields. 

In  the  second  place,  even  though  mechanism  were  proved  to  be 
universal,  this  would  by  no  means  preclude  the  possibility  of  pur- 
pose, of  value,  and  of  rational  or  ethical  freedom.  Haeckel's  abso- 
lute denial  of  all  distinctions  of  value  is  evident  in  the  following 
quotations :  "As  our  mother  earth  is  a  mere  speck  in  the  sunbeam  in 
the  illimitable  universe,  so  man  himself  is  but  a  tiny  grain  of  proto- 
plasm in  the  perishable  framework  of  organic  nature."15  "Our  own 
'human  nature,'  which  exalted  itself  into  an  image  of  God  in  its 
anthropistic  illusion,  sinks  to  the  level  of  a  placental  mammal,  which 
has  no  more  value  for  the  universe  at  large  than  the  ant,  the  fly  of  a 
summer's  day,  the  microscopic  infusorium,  or  the  smallest  bacillus. 
Humanity  is  but  a  transitory  phase  of  the  evolution  of  an  eternal 
substance,  a  particular  phenomenal  form  of  matter  and  energy,  the 
true  proportion  of  which  we  soon  perceive  when  we  set  it  on  the 
background  of  infinite  space  and  eternal  time."16  From  Haeckel's 
point  of  view,  indeed,  neither  man  nor  the  bacillus  can  have  any 
value  for  the  "universe  at  large,"  since  there  is,  in  his  opinion,  no 
purpose  whatever  in  the  universe.  All  is  but  the  result  of  blind 
forces,  and  even  the  progress  of  evolution  is  of  no  value  to  the 
universe.17  Now  this  denial  of  value  is  explained  by  the  fact  that 
Haeckel  always  associates  teleology  with  a  separate  immaterial 
principle.  He  regards  it  as  an  interruption  of  mechanical  cau- 
sation, and  so  feels  that  it  is  incompatible  with  monism.  Whether 
or  not  an  absolute  monism  of  any  sort  is  compatible  with  dis- 
tinctions of  value  and  with  freedom,  at  least  it  is  plain  that  the 
denial  of  teleology  does  not  necessarily  follow  from  the  establish- 
ment of  mechanical  causation.  Mechanism,  as  many  teleologists 
tell  us,  may  be  the  instrument  of  purpose.  Far  from  being  an- 
tagonistic to  teleology  it  alone  makes  teleology  possible.  Without 
it  purpose  would  be  impotent.  For  example — to  take  an  analogy 
from  human  life — man  can  utilize  natural  processes  for  the  carry- 
ing out  of  his  purposes  only  in  so  far  as  he  can  rely  upon  their 
mechanical  uniformity.  Even  a  machine  is  an  embodiment  of  pur- 
pose. It  \yorks  in  a  mechanical  way,  but  its  construction  can  be 

15  The  Riddle  of  the  Universe,  p.  14. 
18  The  Riddle  of  the  Universe,  p.  244. 
17  Cf  Ibid.,  Chap.  XIII. 


3O4  THE   MONIST. 

explained  only  in  terms  of  purpose.  From  this  point  of  view, 
teleology  is  not  an  external  principle  opposed  to  mechanism,  but 
rather  is  immanent  in  all  natural  processes,  and  includes  and  tran- 
scends their  merely  mechanical  aspects.  The  processes  of  the  uni- 
verse are  describable  in  terms  of  mechanical  causation,  but  these 
series  of  mechanical  changes  are  what  they  are  by  reference  to  their 
value  for  the  whole. 

Again  we  object  to  mechanism  taken  as  a  sole  explanation  of 
the  universe  on  the  ground  that  it  fails  to  take  into  account  many 
facts  of  experience.  Although  supposed  to  be  the  direct  outcome 
of  an  acceptance  of  evolution,  mechanism  has  been  unable  to  give 
a  satisfactory  explanation  of  evolution  itself.  Furthermore  mech- 
anism cannot  explain  the  existence  of  values,  purposes,  and  ideals, 
and  many  other  aspects  of  reality. 

Bergson,  perhaps  better  than  any  one  else,  has  succeeded  in 
proving  the  first  point.  In  his  careful  examination  of  theories  of 
evolution,  he  shows  how  mechanism  is  forced  to  take  refuge  in 
a  miracle  to  account  either  for  the  successive  production  and  preser- 
vation of  millions  of  minute  variations  in  the  same  direction,  or  for 
the  complementary  changes  of  the  various  parts  of  an  organ  neces- 
sary for  the  preservation  and  improvement  of  its  functioning. 
Moreover  this  same  miracle  must  be  repeated  innumerable  times  as 
the  same  change  has  taken  place  in  many  different  lines  of  evolu- 
tion.18 Furthermore,  in  every  explanation  of  evolution,  terms  such 
as  adaptation  and  struggle  for  existence  occur,  but  these  are  not 
mechanistic  terms,  since  they  imply  purpose,  ends,  value.  The  mech- 
anist holds  that  all  achievements  of  evolution  are  merely  results 
of  external  and  internal  forces,  which  are  absolutely  blind.  Yet  if 
such  is  the  case,  why  is  the  organism  said  to  struggle  for  existence  ? 
Haeckel  himself,  indeed,  often  finds  a  place  for  the  action  of  internal 
forces  and  declares  that  the  movement  of  molecules  is  due  to  an 
inner  will.  "Even  the  atom  is  not  without  a  rudimentary  form  of 
sensation  and  will,  or,  as  it  is  better  expressed,  of  feeling  and  in- 
clination— that  is,  a  universal  'soul'  of  the  simplest  character."19 
The  term  "inclination"  suits  Haeckel's  purpose  by  its  vagueness,  but 
if  it  is  at  all  comparable  to  will  it  implies  a  reaching  for  the  future 
which  is  not  explicable  as  merely  the  result  of  a  previous  force. 
To  do  justice  to  this  "inclination"  Haeckel  would  be  forced  beyond 
his  rigid  determinism. 

18  Cf.  Bergson,  Creative  Evolution,  pp.  62-76. 

19  The  Riddle  of  the  Universe,  p.  225. 


CRITICISMS  AND  DISCUSSIONS.  305 

The  discussion  of  the  inadequacy  of  mechanism  as  an  account 
of  evolution  has  led  directly  to  our  second  criticism :  that  mechanism 
fails  to  do  justice  to  the  existence  of  values  and  purposes  which  are 
present  not  alone  in  our  inner  experience  but  which  find  an  outer 
embodiment  in  the  great  achievements  of  civilization.20  Surely  the 
painting  of  a  great  picture,  the  writing  of  a  drama,  or  the  founding 
of  a  college  cannot  be  accounted  for  as  the  result  of  purely  natural 
forces.  Now  the  mechanist,  of  course,  does  not  attempt  to  deny 
the  presence  and  power  of  ideals  in  human  life.  His  contention  is 
simply  that  these  ideals  themselves  are  the  result  of  purely  mechan- 
ical forces.  Will,  Haeckel  says,  is  absolutely  determined.21  Thus, 
according  to  Haeckel,  the  psychologist  can  trace  the  behavior  of 
the  self  to  causes  in  preceding  conditions  much  as  the  physicist 
traces  causal  connections  between  the  motions  of  stones.  Haeckel, 
however,  overlooks  the  fact  that  at  this  point  we  happen  to  be  in  a 
peculiarly  favored  position.  We  can  see  the  action  from  within  as 
well  as  from  without,  and  as  we  do  so  we  discover  a  process  of 
determination  differing  profoundly  from  the  mode  of  determination 
described  by  the  scientist.  In  our  own  case  our  action  cannot  be 
understood  apart  from  our  ideal.  This  ideal,  although  due  to  pre- 
ceding conditions  of  one  sort  or  another,  does  not  act  upon  us  as 
an  external  compelling  force,  but  influences  us  through  the  appeal 
it  makes  to  our  own  interests.  It  is  an  ideal  for  us  because  we 
ourselves  select  it,  and  not  because  it  is  forced  upon  us  by  any  ex- 
ternal force.  But  this  process  of  the  selection  of  an  ideal,  or  of 
evaluation,  is  distinct  from  any  process  found  in  the  purely  physical 
world  and  is  not  describable  in  mechanistic  terms.22 

Haeckel  himself  grows  eloquent  over  the  ideals  of  the  good, 
the  true,  and  the  beautiful,  and  urges  us  to  put  these  ideals  before 
any  false  ideals  promulgated  by  superstition.  Such  exhortations, 
however,  have  apparently  little  place  in  an  absolutely  mechanistic 
scheme  where  each  self  is  absolutely  determined  by  his  heredity 
and  environment.  Haeckel  becomes  indignant  over  what  he  regards 
as  superstitions,  yet  he  should  recall  that  all  superstitions  as  well 
as  the  despised  dualistic  philosophy  are,  on  his  scheme,  natural 
products  and  therefore  as  necessary  as  his  own  monistic  utterances. 

20  Cf.  Bosanquet,  The  Value  and  Destiny  of  the  Individual,  pp.  109ff. 

21  Cf.  The  Riddle  of  the  Universe,  p.  131 ;  History  of  Creation,  Vol.  I,  p. 
237. 

22  For  a  clear  description  of  the  distinction  between  personal  and  mechan- 
ical determination,  cf.  Ward,  Realm  of  Ends,  pp.  179ff. 


306  THE  MONIST. 

Furthermore  the  ideals  of  the  good,  the  true,  and  the  beautiful 
must  be,  for  Haeckel,  purely  human  ideals,  since  no  values  exist 
for  the  universe.  But  if  man  himself  has  as  little  value  as  Haeckel 
gives  him,  it  is  strange  that  he  should  regard  human  ideals  as  worthy 
of  reverence  and  worship. 

A  final  word  must  be  added  to  our  criticism  of  mechanism. 
The  theory  of  mechanism  itself  is  not,  as  Haeckel  must  believe,  a 
purely  natural  product.  It  is  due  to  the  organizing  activity  of  man's 
intelligence  and  could  not  exist  without  it.  Haeckel  regards  this 
unifying  and  critical  faculty  of  man  as  due  to  the  "concatenation 
of  presentations."23  Yet  the  mere  concatenation  of  presentations 
could  never  of  itself  lead  to  the  criticism  and  combination  necessary 
to  bind  together  these  various  sensations  under  the  law  of  causa- 
tion. This  unifying  of  experience  demands,  as  Eucken  has  so  clearly 
shown,  that  man  be  able  to  separate  himself  from  the  chain  of  na- 
ture in  order  to  combine  and  order  the  presentations  that  come  to 
him.  Hence  the  formulation  of  the  theory  of  mechanism  is  a  fact 
which  mechanism  itself  fails  to  explain,  and  the  very  existence  of 
the  theory  is  evidence  of  its  own  inadequacy  as  a  final  explanation  of 
all  facts  in  the  universe. 

Our  examination  of  Haeckel's  philosophy  has  shown  the  lack 
of  cogency  of  his  denial  of  freedom.  While  this  in  itself  furnishes 
no  evidence  for  the  reality  of  freedom,  it  at  least  frees  us  from  many 
objections  that  are  commonly  raised  against  it.  It  indicates  that  the 
problem  cannot  be  disposed  of  in  so  summary  a  manner  by  science, 
and  thus  affords  ground  for  those  who  in  the  twentieth  century, 
in  spite  of  Haeckel's  dictum,  maintain  the  possibility  of  freedom. 

GERTRUDE  CARMAN  BUSSEY. 
GOUCHER  COLLEGE,  BALTIMORE,  MD. 


DETERMINISM  OF  FREE  WILL. 

WITH  REFERENCE  TO  THE  PRECEDING  ARTICLE. 

There  is  a  strange  confusion  about  mechanicalism  and  freedom 
of  will  which  seems  to  have  been  constructed  by  our  theological 
school  of  educators  on  the  basis  of  a  misinterpretation  of  philo- 
sophical thought,  and  errors  thus  derived  are  still  perpetuated. 

The  idea  of  the  will  is  perhaps  the  fundamental  conception  of 

23  Cf.  The  Riddle  of  the  Universe,  pp.  121f. 


CRITICISMS  AND  DISCUSSIONS.  307 

ethics,  and  an  important  item  for  moral  purposes  is  the  freedom  of 
an  acting  person.  But  "free  will"  is  nothing  mysterious  nor  in- 
credible ;  it  is  that  condition  of  a  will  which  is  not  hindered  by  com- 
pulsion. He  is  free  who  acts  on  his  own  account,  according  to  his 
own  character,  and  is  not  interfered  with  by  external  circumstances 
which  would  make  it  impossible  for  him  to  act  as  he  wishes.  A  man 
under  compulsion  is  not  responsible  for  his  action;  for  his  act  is 
the  act  of  some  one  else,  or  is  due  to  the  circumstances  which  force 
him  against  his  own  will.  The  external  circumstances  may  be  ever 
so  indirect  and  may  be  reducible  to  fear.  A  man  threatened  by  the 
consequences  of  the  results  of  his  act  is  no  freer  than  a  man  who 
is  directly  forced  into  acting  contrary  to  his  will  by  facing  the  re- 
volver of  a  highway  robber.  If  an  act  is  committed  because  the 
acting  person  wishes  the  act  and  also  willingly  accepts  all  of  its 
consequences,  it  is  and  ought  to  be  considered  an  act  of  free  will, 
and  there  is  scarcely  any  thinker  who  would  not  admit  this  definition 
of  free  will. 

Is  there  any  one  who  denies  that  the  act  of  a  free  will,  as  here 
defined,  is  as  much  determined  as  any  other  event  in  this  world  in 
which  we  live?  If  the  free  act  of  a  man  is  really  the  result  of  de- 
liberation and  if  it  is  performed  according  to  the  nature  of  the 
actor's  character,  the  result  of  this  decision  will  be  as  necessary  as 
the  act  of  an  unfree  man  who  acts  under  compulsion  according  to 
motives  of  fear  or  any  external  force.  Determinism  is  a  general 
feature  of  the  world  which  expresses  the  truth  that  the  law  of 
causation  remains  unbroken.  According  to  the  law  of  causation, 
everything  is  determined,  even  the  act  of  a  free  man. 

Yet  there  are,  or  rather  have  been,  some  theologians  who  believe 
in  free  will,  not  as  free  will  necessarily  must  be,  viz.,  an  unhampered 
will,  but  as  a  carte  blanche  or  tabula  rasa,  a  cause  that  is  not  caused, 
or  as  a  determinant  which  on  its  part  is  undetermined,  which  is 
free  in  the  sense  that  it  is  unformed,  or  a  factor  that  is  somehow 
an  exception  to  causation  and  not  the  product  of  the  efficacy  of 
causation.  They  think  that  a  man  is  not  responsible  if  his  actions 
are  determined  or  determinable  and  can  be  predicted,  just  as  in 
moving  pictures  only  such  consequences  will  happen  as  are  on  the 
films,  and  the  man  who  knows  the  film  would  naturally  and  neces- 
sarily be  able  to  tell  what  is  going  to  happen  in  the  next  moment. 
What  an  undetermined  will  is  or  would  be,  has  never  as  yet  been 
clearly  described;  it  is  only  declared  to  be  an  exception  to  the  law 


308  THE  MONIST. 

of  causality,  and  being  undetermined  seems  to  be  as  much  a  mere 
chance  product  as  the  haphazard  cast  of  dies,  in  which  case  of 
course  the  actor  could  no  longer  be  regarded  as  responsible  for  a 
deed  not  determined  by  himself. 

The  truth  is  that  if  we  were  omniscient  we  could  predict  the 
history  of  the  world  from  step  to  step  just  as  the  theatrical  man- 
ager of  the  movies  knows  the  next  act  if  he  knows  the  film  that  is 
to  project  it  on  the  screen.  If  I  know  all  the  characters  of  the 
acting  persons,  I  will  be  able  to  predict  the  outcome  of  their  activ- 
ity under  definite  conditions,  and  there  can  be  no  quibbling  about  it. 

We  must  not  identify  necessity  and  compulsion.  Everything 
is  determined ;  and  all  acts  are  determined  with  necessity,  even  the 
free  acts  of  a  free  man.  Further  it  would  be  wrong  to  say  that 
man  is  compelled  to  act  according  to  factors  which  are  none  of  his 
making,  if  he  necessarily  acts  according  to  his  will. 

It  is  true  that  there  are  factors  which  have  preceded  him; 
among  them  there  are  factors  such  as  have  determined  his  char- 
acter. He  has  been  determined  and  his  will  has  been  given  him. 
In  this  sense  it  is  claimed  that  he  is  as  unfree  as  any  slave  who  is 
not  his  own  master.  But  is  that  not  a  wrong  conclusion  that  here 
too  identifies  necessity  with  compulsion?  It  is  necessary  that  a 
man  should  act  according  to  his  character  if  he  is  not  under  com- 
pulsion. The  acts  of  a  free  man  are  necessary  because  his  will 
necessarily  and  naturally  follows  the  impulses  of  his  own  character. 
To  say  that  we  are  slaves  because  we  follow  necessarily  our  own 
instincts  is  simply  an  illogical  distortion  of  facts.  The  truth  is  that 
in  doing  what  we  will  we  obey  the  behest  of  those  factors  which 
shaped  our  will.  However,  granting  that  our  will  is  not  of  our  own 
making,  we  will  be  obliged  to  confess  that  we  are  the  continuation 
of  those  factors  which  make  us ;  or  in  other  words,  our  ancestors 
whose  will  we  incorporate  are  ourselves  in  a  former  generation. 
Thus  we  ought  to  recognize  openly  and  unhesitatingly  that  the 
whole  development  of  the  world  is  not  a  piecing  together  of  inde- 
pendent individuals,  but  that  we  are  mere  fragments  of  a  continuous 
whole,  we  are  pieces  of  a  prolonged  history  of  one  and  the  same 
aspiration  which  may  be  modified,  improved,  or  even  on  the  other 
hand  weakened  and  debased.  Former  generations  have  made  us 
of  the  present  age,  and  future  generations  will  be  as  much  the 
product  of  the  present  generation  as  we  are  of  the  past.  Thus 
if  we  speak  of  having  been  made  by  prior  factors  we  must  recognize 


CRITICISMS  AND  DISCUSSIONS.  309 

that  the  factors  that  made  us  are  our  own  existence,  as  we  existed 
in  former  days, — yet  the  truth  remains  that  a  free  will  is  definitely 
determined.  A  free  will  which  acts  in  an  unhampered  way  is  as 
much  determined  as  any  will  which  suffers  violence  or  acts  under 
compulsion. 

Miss  Bussey  has  taken  up  Haeckel  and  criticizes  him  for  de- 
nying freedom  of  will  where  he  stands  up  for  determinism.  I  do 
not  think  that  Professor  Haeckel  will  take  up  the  cudgel  and  defend 
himself.  On  the  other  hand  I  grant  that  Professor  Haeckel  is  an 
enthusiastic  defender  of  the  monistic  world-conception  for  which 
he  demands  a  strict  and  universal  application  of  the  mechanistic 
theory  to  all  events  of  existence.  I  will  not  deny  that  Professor 
Haeckel  sometimes  accepts  views  which  I  myself  would  not  endorse. 
For  instance  he  identifies  God  with  matter  and  energy  while  I  would 
look  upon  God  in  contrast  to  matter  and  energy,  as  a  religious 
formulation  of  the  world  order  which  is  the  ultimate  raison  d'etre 
of  natural  law  throughout  the  sphere  of  existence,  including  also 
the  natural  law  that  governs  human  society  and  is  the  basis  of  the 
rules  of  conduct.  But  this  is  a  point  which  could  easily  be  recon- 
structed or  altered,  for  Professor  Haeckel  himself  would  scarcely 
object  to  it. 

In  order  to  understand  Haeckel  one  ought  to  interpret  his 
writings  in  the  spirit  in  which  he  has  written  them,  and  ought  not 
imply  mistakes  which  are  rather  incidental  points,  such  as  Miss 
Bussey  criticizes. 

Miss  Bussey  in  criticizing  Professor  Haeckel  should  consider 
that  he  rejects  the  theory  of  free  will  because  he  understands  by  free 
will  the  theological  conception  of  an  undetermined  will,  viz.,  that 
kind  of  a  free  will  that  does  not  exist,  because  it  is  a  self-contradic- 
tory notion,  an  impossible  and  foolish  conception  of  a  misguided 
brain.  If  he  rejects  it  he  does  so  only  in  the  sense  in  which  theo- 
logians have  misrepresented  freedom  of  will  as  being  exempt  from 
the  law  of  causation.  And  in  doing  so  he  is  certainly  right  in  the 
face  of  Miss  Bussey  or  any  one  who  believes  in  a  freedom  of  that 
kind,  proclaiming  that  it  is  independent  of  causation. 

There  is  no  need  of  entering  into  the  details  of  Miss  Bussey 's 
discussion.  Any  of  our  readers  who  knows  Haeckel  will  be  able 
to  form  his  own  judgment.  Only  a  few  points  shall  be  mentioned 
here. 

The  universe  has  certainly  to  be  explained  from  the  highest 


3IO  THE  MONIST. 

product  its  development  achieves  and  not  from  its  lowest  beginnings. 
It  is  man  that  gives  us  the  key  to  the  appearance  of  the  moner, 
while  the  moner  will  not  be  able  to  tell  what  its  evolution  will  bring 
out  in  the  end.  On  the  other  hand  we  have  not  solved  the  problem 
unless  we  trace  the  development  of  a  rational  being  step  by  step 
in  a  mechanistic  fashion  of  cause  and  effect.  To  deny  it  would 
mean  to  abolish  science  in  spots.  I  prefer  to  keep  my  trust  in 
science,  for  science  to  me  is  God's  revelation.  The  most  important 
step  for  instance  is  the  development  of  reason,  and  it  has  been  ex- 
plained in  a  mechanistic  sense  by  Ludwig  Noire  when  he  shows  how 
the  origin  of  language  has  produced  reason  and  not  the  reverse ;  or, 
to  express  his  principle  in  a  popular  way,  "We  think  because  we 
speak"  and  not  "we  speak  because  we  think."  The  mechan- 
ical mechanism  of  speech  came  first,  and  it  was  the  mechanism  of 
logic  and  grammar  which  has  enabled  us  to  think. 

It  is  not  a  fault  of  Haeckel's  if  he  holds  the  view  that  man 
explains  the  nature  and  significance  of  the  moner.  It  proves  that 
he  is  not  onesided.  His  claim  is  but  the  natural  consequence  of  a 
consideration  of  evolution. 

The  law  of  the  conservation  of  matter  and  energy  is  an  a  priori 
law,  which  in  its  general'  meaning  is  similar  to  mathematical  postu- 
lates. It  is  a  demand  of  science  and  need  not  be  proved  in  detail. 
It  is  a  pre-supposition  just  as  much  as  is  the  law  of  causation 
which  the  scientist  assumes  when  he  investigates  natural  phenom- 
ena. That  there  is  a  purpose  in  the  universe  is  a  proposition  which 
would  involve  a  belief  that  the  universe  as  a  whole  is  to  be  under- 
stood as  an  individual  personal  being  after  the  fashion  of  a  man. 
It  would  involve  an  anthropomorphic  conception  of  God,  and  I 
doubt  whether  even  among  our  theologians  there  are  now  many 
bold  enough  to  take  such  a  position.  This,  however,  does  not  ex- 
clude that  the  universe  in  its  processes  follows  a  definite  direction, 
a  claim  which  is  proved  by  the  facts  of  evolution  and  is  probably 
not  denied  by  either  a  theistic  or  atheistic  interpretation  of  the 
word. 

Why  the  formulation  of  the  theory  of  mechanicalism  should 
be  a  fact  which  mechanicalism  itself  fails  to  explain  is  unintelligible, 
and  why  its  own  existence  should  be  evidence  of  its  own  inadequacy 
is  hard  to  understand,  unless  the  notion  of  mechanicalism  be  nar- 
rowed to  a  limited  field  which  does  not  include  the  entire  construe- 


CRITICISMS  AND  DISCUSSIONS.  3!  I 

tion  of  mechanicalism  and  its  internal  interrelations,  such  as  for 
instance  the  interrelations  of  logical  rules  and  conditions. 

We  may  be  able  to  uphold  the  theory  of  free  will  but  we  shall 
certainly  not  be  able  to  deny  the  principle  of  determinism,  and  this 
is  a  blessing  for  the  ethicist  who  preaches  morality  and  claims  that 
the  freedom  of  will  is  essential  for  it,  because  if  free  will  were 
indeed  an  exception  to  the  law  of  causation  and  the  will  were  unde- 
termined and  not  changeable  by  education  but  remained  a  tabula 
rasa  in  spite  of  all  attempts  to  change  and  improve  it,  or  make  it 
definite  in  the  right  direction,  what  would  be  the  use  of  wasting  our 
energies  in  promoting  the  welfare  of  mankind  and  eliminating  evil 
influences?  Let  us  be  glad  that  determinism  is  true,  for  otherwise 
there  would  be  no  science,  and  principles  of  conduct  would  be  a 
meaningless  play  of  a  misguided  and  erring  imagination. 

Haeckel  apparently  commits  a  very  grave  mistake.  His  opin- 
ions are  "the  result  of  the  general  presuppositions  and  prejudices  of 
the  age."  He  and  many  others  "believe  whatever  fits  in  with  their 
view  of  life  and  dismiss  without  a  hearing  anything  which  conflicts 
with  it."  Miss  Bussey  claims  that  "in  this  age  of  science  the  scien- 
tist has  become  the  arbiter  of  all  questions,  and  his  view  is  com- 
monly accepted  as  authoritative."  In  other  words,  we  expect  that 
science  shall  solve  our  problems,  and  we  are  prejudiced  enough  to 
bow  down  before  science  and  accept  its  verdict.  Haeckel  for  in- 
stance is  so  prejudiced  that  he  believes  in  the  universality  of  natural 
laws,  and,  says  Miss  Bussey,  "It  is  a  philosophic  commonplace  that 
laws  resting  on  experience  can  be  universalized  only  by  means  of 
the  supposition  of  the  uniformity  of  nature."  It  is  a  pity  that 
Haeckel  follows  this  fallacy  and  accepts  the  uniformity  of  nature, 
but  the  worst  is  that  I  too  plead  guilty.  I  believe  not  only  in  his 
"supposition  of  the  uniformity  of  nature,"  but  also  in  science  with 
all  that  it  implies,  especially  determinism  which  demands  the  de- 
terminedness  of  everything,  even  the  determinedness  of  an  unham- 
pered and,  in  this  sense,  free  will.  I  can  not  help  it.  I  am  in  the 
same  predicament  as  Professor  Haeckel.  May  God  have  mercy  on 
our  souls!  EDITOR. 

THE  BELIEF  IN  GOD  AND  IMMORTALITY. 

Professor  James  H.  Leuba,  professor  psychology  and  peda- 
gogy in  Bryn  Mawr  College,  has  undertaken  to  write  a  book  on 
The  Belief  in  God  and  Immortality.  It  is  not  a  proof  or  disproof 


312  THE  MONIST. 

of  the  doctrines  essential  in  all  positive  religious  creeds  but  a  study 
of  psychological  statistics  as  to  frequency  and  distribution  of  be- 
liefs in  a  personal  God  and  a  personal  immortality,  and  he  finds 
that  upon  the  whole  in  each  group  investigated  as  to  their  religious 
beliefs,  the  more  distinguished  fraction  includes  by  far  the  smaller 
numbers  of  believers. 

Professor  Leuba's  work  is  divided  into  three  parts.  The  first 
part  enters  into  a  discussion  of  the  characteristics  of  a  belief  in  a 
continuation  after  death.  He  begins  with  the  savage's  idea  of  soul 
and  ghost,  setting  forth  in  his  second  chapter  the  origin  of  the 
ghost  idea,  the  appearance  of  ghosts  in  dreams  and  visions.  He 
distinguishes  from  the  belief  in  soul-ghosts  the  belief  in  immor- 
tality which  he  regards  as  late  in  the  development  of  mankind. 
The  fourth  chapter  is  devoted  to  "The  Origin  of  the  Modern  Con- 
ception of  Immortality,"  beginning  with  a  "translation  to  a  land  of 
immortality."  The  fifth  chapter  enters  into  metaphysics,  the  deduc- 
tions of  which  however  are  regarded  as  insufficient. 

In  later  days  more  scientific  methods  have  been  used  by  relying 
on  physical  and  psychical  manifestations  and  on  the  historical  facts 
on  which  the  resurrection  of  Christ  is  taught. 

In  Part  II  the  belief  in  the  personality  of  God  and  immortal- 
ity is  made  an  object  of  statistical  study,  first  (Chapter  VII)  among 
the  students  of  American  colleges.  In  this  it  has  become  necessary 
to  make  a  distinction  between  the  personal  and  impersonal  con- 
ceptions of  God.  The  eighth  chapter  is  devoted  to  an  investigation 
of  the  belief  in  immortality,  including  a  comparison  of  the  changes 
taking  place  during  college  years.  Here  follows  a  detailed  investi- 
gation (introduced  first  by  the  causes  of  the  failure  to  answer  and 
the  interpretation  of  the  questionnaire)  of  the  beliefs  held  by 
the  scientists,  the  historians,  the  sociologists,  the  psychologists,  and 
the  philosophers,  concluding  with  a  comparison  of  the  signed  and 
unsigned  answers.  He  comments  on  the  results  of  his  investigation 
thus: 

"The  essential  problem  facing  organized  Christianity  is  con- 
stituted by  the  wide-spread  rejection  of  its  two  fundamental  dog- 
mas— a  rejection  apparently  destined  to  extend  parallel  with  the 
diffusion  of  knowledge  and  the  moral  qualities  that  make  for  emi- 
nence in  scholarly  pursuits." 

The  third  part  which  might  be  considered  as  independent  of 
the  first  two  is  devoted  to  the  question  of  the  utility  of  the  belief 


CRITICISMS  AND  DISCUSSIONS.  313 

in  personal  immortality  and  a  personal  God.  Professor  Leuba 
asks  the  question,  "Is  humanity  better  off  with  than  without  that 
belief  (in  a  personal  God  and  a  personal  immortality)  ?  He  answers: 
"The  utility  of  the  belief  in  immortality  to  civilized  nations  is  much 
more  limited  than  is  commonly  supposed ....  we  may  even  be  brought 
to  conclude  that  its  disappearance  from  among  the  most  civilized 
nations  would  be,  on  the  whole,  a  gain." 

It  is  noteworthy  that  his  results  show  that  the  desire  for  im- 
mortality and  the  usefulness  of  the  belief  is  rather  disappearing 
with  an  increase  of  intelligence.  There  is  an  increasing  tendency 
to  disclaim  any  desire  for  immortality.  This  is  in  strong  contrast 
to  the  supposition  formerly  quite  common  that  even  disbelievers 
yearn  for  immortality,  but  among  the  answers  received  to  a  ques- 
tionnaire Professor  Leuba  finds  even  a  relatively  considerable  num- 
ber who  abhor  the  idea  of  an  endless  continuation  and  he  quotes  a 
number  of  instances.  For  instance  a  woman  thirty  years  of  age 
declares  that  she  has  always  felt  death  to  be  better  than  all  else, 
anticipating  it  as  the  best  thing  life  has  to  offer;  and  concluding 
with  the  sentence  that  death  itself  is  a  consummation  devoutly  to 
be  wished. 

Another  letter  is  quoted  as  stating,  "I  feel  a  great  dread  of  the 
possibility  of  having  to  live  forever,  or  even  again,"  and  Professor 
Leuba  quotes  from  Swinburne's  poem  "The  Garden  of  Proserpina" 
the  poet's  hope  of  annihilation,  where  he  says: 

"Then  star  nor  sun  shall  waken, 

Nor  any  change,  of  light ; 
Nor  sound  of  waters  shaken, 

Nor  any  sound  or  sight; 
Nor  wintry  leaves  nor  vernal, 
Nor  days  nor  things  diurnal; 
Only  the  sleep  eternal 

In  an  eternal  night." 

John  Addington  Symonds  echoes  the  same  ideas  in  prose.  He 
says: 

"Until  that  immortality  of  the  individual  is  irrefragably  dem- 
onstrated, the  sweet,  the  immeasurably  precious  hope  of  ending 
with  this  life,  the  ache  and  languor  of  existence,  remains  open  to 
burdened  human  personalities." 

The  greater  stimulus  for  a  desire  for  immortality  comes  in 
cases  of  the  death  of  beloved  persons,  and  the  most  impressive 
instance  of  this  kind  is  quoted  by  Professor  Leuba  in  the  case  of 


314  THE  MONIST. 

a  widow  writing  to  her  friend,  the  famous  Professor  Schleier- 
macher.  Quoting  from  Schleiermacher's  Leben  as  quoted  by  James 
Martineau  in  A  Study  of  Religion,  Vol.  II,  page  337: 

"O  Schleier,  in  the  midst  of  my  sorrow  there  are  yet  blessed 
moments  when  I  vividly  feel  what  a  love  ours  was,  and  that  surely 
this  love  is  eternal,  and  it  is  impossible  that  God  can  destroy  it; 
for  God  himself  is  love.  I  bear  this  life  while  nature  will;  for  I 
have  still  work  to  do  for  the  children,  his  and  mine ;  but  O  God ! 
with  what  longings,  what  foreshadowings  of  unutterable  blessedness, 
do  I  gaze  across  into  that  world  where  he  lives!  What  joy  for  me 
to  die! 

"Schleier,  shall  I  not  find  him  again?  O  my  God!  I  implore 
you,  Schleier,  by  all  that  is  dear  to  God  and  sacred,  give  me,  if 
you  can,  the  certain  assurance  of  finding  and  knowing  him  again. 
Tell  me  your  inmost  faith  on  this,  dear  Schleier;  Oh!  if  it  fails,  I 
am  undone.  It  is  for  this  that  I  live,  for  this  that  I  submissively  and 
quietly  endure :  this  is  the  one  only  outlook  that  sheds  a  light  on  my 
dark  life, — to  find  him  again,  to  live  for  him  again.  O  God!  he 
cannot  be  destroyed !" 

In  commenting  that  the  psychological  state  might  have  been 
quite  different  in  Schleiermacher's  friend  if  she  had  remarried. 
Professor  Leuba  says :  "In  that  occurrence  her  former  yearnings 
for  another  life  might  have  been  replaced  by  dread  of  the  time 
when  she  would  be  face  to  face  with  two  husbands." 

Perhaps  the  most  dignified  expression  of  an  impersonal  im- 
mortality has  been  expressed  by  George  Eliot  in  her  "Choir  In- 
visible," but  the  main  and  classical  instance  is  the  orthodox  Bud- 
dhist faith,  and  Professor  Leuba  quotes  at  length  the  text  from 
Buddhist  scriptures  as  translated  by  Henry  Clarke  Warren,  where 
Buddha  insists  on  not  being  born  again  and  that  the  present  life 
is  his  final  entry  into  Nirvana.  It  reads  thus: 

"And  being,  O  priests,  myself  subject  to  birth,  I  perceived  the 
wretchedness  of  what  is  subject  to  birth,  and  craving  the  incompar- 
able security  of  a  Nirvana  free  from  birth,  I  attained  the  incom- 
parable security  of  a  Nirvana  free  from  birth;  myself  subject  to 
old  age, ....  disease, ....  death, ....  sorrow, ....  corruption,  I  per- 
ceived the  wretchedness  of  what  is  subject  to  corruption,  and,  crav- 
ing the  incomparable  security  of  a  Nirvana  free  from  corruption,  I 
attained  the  incomparable  security  of  a  Nirvana  free  from  corrup- 
tion. And  the  knowledge  and  the  insight  sprang  up  within  me,  'My 


CRITICISMS  AND  DISCUSSIONS.  31$ 

deliverance  is  unshakable;  this  is  my  last  existence;  no  more  shall 
I  be  born  again.'  And  it  occurred  to  me,  O  priests,  as  follows : 

"  'This  doctrine  to  which  I  have  attained  is  profound,  recondite, 
and  difficult  of  comprehension,  good,  excellent,  and  not  reached  by 
mere  reasoning,  subtile,  and  intelligible  only  to  the  wise.  Mankind, 
on  the  other  hand,  is  captivated,  entranced,  held  spell-bound  by  its 
lusts ;  and  forasmuch  as  mankind  is  captivated,  entranced,  held 
spell-bound  by  its  lusts,  it  is  hard  for  them ....  to  understand  how 
all  the  constituents  of  being  may  be  made  to  subside,  all  the  sub- 
strata of  being  be  relinquished,  and  desire  be  made  to  vanish,  and 
absence  of  passion,  cessation,  and  Nirvana  be  attained.' " 

It  is  peculiar  that  among  scientists  there  was  one  who  clung 
with  great  insistence  to  the  belief  in  immortality,  and  this  is  no  less 
an  authority  than  the  great  biologist,  Henri  Pasteur,  and  he  kept  his 
religious  faith  and  science  in  two  different  departments  of  his 
mind.  He  says: 

"My  philosophy  is  of  the  heart  and  not  of  the  mind,  and  I 
give  myself  up,  for  instance,  to  those  feelings  about  eternity  which 
come  naturally  at  the  bedside  of  a  cherished  child  drawing  its  last 
breath. 

"There  are  two  men  in  each  one  of  us:  the  scientist,  he  who 
starts  with  a  clear  field  and  desires  to  rise  to  the  knowledge  of 
Nature  through  observation,  experimentation,  and  reasoning;  and 
the  man  of  sentiment,  the  man  of  belief,  the  man  who  mourns  his 
dead  children  and  who  cannot,  alas,  prove  that  he  will  see  them 
again,  but  who  believes  that  he  will,  and  lives  in  that  hope;.... 
the  man  who  feels  that  the  force  that  is  within  him  cannot  die." 

Professor  Leuba  adds  the  following  comment  on  Pasteur: 

"I  may  remark  incidentally  upon  the  off-hand  manner  in  which 
Pasteur  divides  life  into  two  spheres,  that  of  science  and  that  of 
feeling,  and  apparently  finds  no  use  for  logic  and  reason  in  the 
latter.  This  is  a  shocking  example  of  a  dangerous  practice  which, 
when  carried  to  its  logical  consequence,  would  permit  one  to  believe 
whatever  he  pleases.  When  I  attempt  to  understand  this  attitude 
in  a  distinguished  man  of  science,  I  can  only  conjecture  that  he 
treated  religion  as  something  primarily  intended  to  comfort  anyway, 
anyhow." 

Professor  Leuba's  book  does  not  decide  the  question  of  the 
acceptability  or  unacceptability  of  the  belief  itself,  but  is  simply  a 
statistical  investigation  and  for  that  reason  possesses  virtue  for 


316  THE  MONIST. 

theists  as  well  as  unbelievers  in  helping  to  find  out  the  psychological 
state  of  things  as  it  happens  to  be  in  our  present  generation,  and 
from  that  standpoint  the  book  will  retain  its  virtue  whatever  be  the 
position  of  the  reader. 


SIR  OLIVER  LODGE  ON  LIFE  AFTER  DEATH. 

Sir  Oliver  has  always  been  a  believer  in  mediumistic  experience 
and  in  the  spirit  existence  of  man  in  the  other  world,  and  in  spite 
of  his  knowledge  of  physics  he  has  taken  a  broad  stand  by  coming 
out  squarely  and  unreservedly  in  showing  his  faith.  Details  of  such 
an  expression  might  be  amusing  if  it  were  not  actually  sad  to  see 
a  man  of  his  significance  stooping  to  views  which  otherwise  prevail 
only  in  the  circles  of  half -educated  people.  His  son  Raymond  died 
at  the  front  in  Flanders  on  September  14,  1915,  and  the  bereaved 
father  has  published  a  book1  containing  a  summary  of  his  own 
philosophical  views  and  a  record  of  communications  received  from 
Raymond  since  his  death. 

From  this  we  learn  that  Raymond  woke  up  in  the  other  world 
and  got  accustomed  to  his  new  surroundings.  There  are  seven 
spheres  all  above  the  earth  and  turning  around  with  the  earth,  but 
there  is  no  consecutive  night  and  day.  It  is  always  daylight  except 
when  one  desires  darkness ;  then  night  spreads  according  to  one's 
wishes.  Raymond  resides  in  a  house  which  appears  to  be  made 
of  brick,  and  spirit  houses  form  streets  in  which  the  spirits  walk 
and  move.  People  who  have  lost  arms  or  legs  develop  new  ones 
as  if  by  a  kind  of  natural  recuperation,  so  he  tells  his  parents  that 
he  has  replaced  a  tooth,  and  comrades  of  his  who  had  lost  arms  or 
other  limbs  are  restored  to  their  original  natural  shape,  but  this  res- 
toration is  not  quite  simple  and  there  is  a  kind  of  spirit-doctors 
who  help  with  their  restoration.  There  is  a  special  difficulty  in 
restoring  the  spiritual  body  if  the  material  body  has  been  destroyed 
before  its  regeneration  in  the  spirit  world,  so  Raymond  gives  a 
definite  warning  that  dead  bodies  should  not  be  cremated  before 
father  has  published  a  book  containing  a  summary  of  his  own 
they  have  been  restored  in  the  spirit  plane  of  life. 

The  seven  spheres  which  are  built  around  the  earthly  plane 
seem  to  revolve  with  it  at  different  rates  of  speed,  so  that  the  first 
sphere  is  not  revolving  at  the  same  rate  as  the  second,  third,  fourth, 
fifth,  sixth  and  seventh  spheres.  Greater  circumference  makes  the 


CRITICISMS  AND  DISCUSSIONS.  317 

revolution  more  rapid  and  this  increase  of  rotation  has  an  actual 
effect  on  the  atmospheric  conditions  prevailing  in  different  spheres. 
When  asked  for  details  about  the  nature  of  the  other  world  Ray- 
mond said : 

"What  I  am  worrying  about  is  how  it  is  all  made  and  of  what  it 
is  composed.  I  have  not  found  out  yet,  but  I  have  a  theory.  It  is 
not  an  original  idea  of  mine.  I  was  helped  to  it  by  words  dropped 
here  and  there.  People  who  think  everything  is  created  by  thought 
are  wrong.  I  imagined  for  a  little  while  that  one's  thoughts  over 
here  formed  the  buildings  and  flowers  and  trees  and  solid  ground ; 
but  there  is  something  more  than  that. 

"There  is  something  always  rising  from  the  earth — something 
chemical  in  form.  As  it  rises  to  ours  it  goes  through  various 
changes  and  solidifies  here.  I  feel  sure  it  is  something  given  off 
from  the  earth  that  makes  the  solid  trees,  flowers,  etc 

"All  the  decay  that  goes  on  on  the  earth  is  not  lost.  It  doesn't 
just  form  manure  or  dust.  Certain  vegetable  and  decayed  tissue 
does  form  manure  for  a  time,  but  it  gives  off  an  essence  or  a  gas 
which  ascends  and  which  becomes  what  you  call  a  'smell.'  Every- 
thing dead  has  a  smell,  if  you  notice ;  and  I  know  now  that  the  smell 
is  of  actual  use,  because  it  is  from  that  smell  that  we  are  able  to 
produce  duplicates  of  whatever  form  it  had  before  it  became  a  smell. 
Even  old  wood  has  a  smell  different  from  new  wood ;  you  may  have 
to  have  a  keen  nose  to  detect  it  on  the  earth  plane. 

"Old  rags,  cloth  decaying  and  going  rotten,  all  have  smells. 
Different  kinds  of  cloth  give  off  different  smells.  You  can  under- 
stand how  all  this  interests  me.  Apparently,  so  far  as  I  can  gather, 
the  rotting  wool  appears  to  be  used  for  making  things  like  tweeds 
on  our  side.  But  I  know  that  I  am  jumping;  I'm  guessing  at  it. 
My  suit,  I  expect,  was  made  from  decayed  worsted  on  your  side. 

"Some  people  here  won't  grasp  this  even  yet — about  the  material 
cause  of  all  these  things.  They  go  talking  about  spiritual  robes 
made  of  light,  built  by  thoughts  on  the  earth  plane.  I  don't  believe 
it.  They  go  about  thinking  that  it  is  a  thought  robe  they're  wearing, 
resulting  from  the  spiritual  life  they  led ;  and  when  we  try  to  tell 
them  it  is  manufactured  out  of  materials  they  don't  believe  it.  They 
say,  'No,  no;  it's  a  robe  of  light  and  brightness  which  I  manufac- 
tured by  thought.'  So  we  just  leave  it.  But  I  don't  say  that  they 
don't  get  robes  quicker  when  they  have  led  spiritual  lives  down 


318  THE  MONIST. 

there;  I  think  they  do,  and  that's  what  makes  them  think  that  they 
made  the  robes  by  their  lives. 

"You  know  flowers  how  they  decay.  We  have  got  flowers 
here ;  your  decayed  flowers  flower  again  with  us — beautiful  flowers." 

They  have  not  only  spirit  doctors  but  also  manufacturers  and 
can  provide  you  with  materials  if  you  so  desire.  Raymond  himself 
does  not  smoke,  but  a  friend  of  his,  a  great  smoker  on  the  earth 
plane,  demanded  cigars  and  he  got  them,  but  only  about  five;  and 
the  things  given  him  looked  like  cigars,  but  after  smoking  about 
five  cigars  he  no  longer  cared  for  more.  He  changed  his  habit  and 
got  accustomed  to  a  more  spiritual  mode  of  life. 

Colors  have  their  significance,  and  different  colors  have  dif- 
ferent effects  upon  the  character  of  the  spirits. 

"There's  plenty  of  flowers  growing  here,  you  will  be  glad  to 
hear.  But  we  don't  cut  them  here.  They  don't  die  and  grow  again ; 
they  seem  to  renew  themselves.  Just  like  people,  they  are  there  all 
the  time  renewing  their  spirit  bodies.  The  higher  the  sphere  he 
went  to,  the  lighter  the  bodies  seemed  to  be — he  means  the  fairer, 
lighter  in  color.  He's  got  an  idea  that  the  reason  why  people  have 
drawn  angels  with  long  fair  hair  and  very  fair  complexions  is  that 
they  have  been  inspired  by  somebody  from  very  high  spheres." 

The  information  Professor  Lodge  publishes  was  received  from 
the  medium  Mrs.  Leonard  through  her  "control"  known  as  "Fedo." 

Incidentally  we  find  a  personal  remark  put  in  brackets  and  in 
italics  of  which  Sir  Oliver  is  apparently  the  authority.  It  reads: 
"A  good  deal  of  this  struck  me  as  nonsense,  as  if  Peda  has  picked 
it  up  from  some  sitter." 

Mediums  have  said  much  nonsense  in  print  as  well  as  in  private 
seances,  and  the  spirits  of  dead  people  have  distinguished  themselves 
by  silly  utterances ;  but  the  recent  story  of  Raymond's  communica- 
tions rather  excels  all  prior  tales  of  mediumistic  lore  in  the  silliness 
of  its  revelations.  But  the  saddest  part  of  it  consists  in  the  fact 
that  a  great  scientist,  no  less  a  one  than  Sir  Oliver  Lodge,  has  pub- 
lished the  book  and  so  stands  sponsor  for  it. 

Sir  Oliver  Lodge  is  a  scientist  who  has  done  much  creditable 
work  and  has  written  a  number  of  books  which  exhibit  keen  thought 
and  a  good  grasp  of  his  subject,  his  specialty  being  physics.  The 
books  he  has  written  are  as  folows: 

Elementary  Mechanics;  Modern  Views  of  Electricity;  Pioneers 
of  Science;  Signalling  Without  Wires;  Lightning  Conductors  and 


CRITICISMS  AND  DISCUSSIONS.  319 

Lightning  Guards;  School  Teaching  and  School  Reform;  Mathe- 
matics for  Parents  and  Teachers;  Life  and  Matter;  Electrons; 
Modern  Views  of  Matter;  The  Substance  of  Faith;  Man  and  the 
Universe;  The  Ether  of  Space;  The  Survival  of  Man;  Parent  and 
Child;  Reason  and  Belief;  and  Modern  Problems. 


CURRENT  PERIODICALS. 

In  Vol.  XIV  (1915)  of  the  fifth  series  of  the  Atti  of  the  Royal 
Academy  of  the  Lincei  at  Rome  is  a  publication  in  full  of  the  treatise 
De  corporibus  regularibus  of  Pietro  Franceschi  or  Delia  Francesca 
which  was  found  in  1912  in  the  Vatican  Library  by  G.  Mancini.  To 
this  is  prefixed  a  learned  dissertation  by  Mancini  to  show  that  this 
treatise  was  pilfered  by  Luca  Pacioli  in  his  work  on  mensuration,  the 
Divina  proportione ;  and  a  report  by  Gino  Loria  on  Mancini's  memoir. 

*  *       * 

The  articles  of  greatest  interest  to  philosophical  mathematicians 
in  recent  numbers  of  Vol.  XVII  (1916)  of  the  Transactions  of  the 
American  Mathematical  Society  are  as  follows.  In  the  number  for 
April,  Robert  L.  Moore  gives  three  systems  of  axioms  for  plane 
analysis  situs — the  non-metrical  part  of  the  theory  of  plane  sets  of 
points,  including  the  theory  of  plane  curves ;  Charles  N.  Haskins 
writes  on  the  measurable  bounds  and  the  distribution  of  functional 
values  of  "summable"  functions — which  here  means  functions  which 
are  integrable  in  the  generalized  sense  of  Lebesgue;  and  Dunham 
Jackson  proves  in  another  way  an  important  theorem  of  Haskins. 
In  the  number  for  July,  L.  L.  Silverman  discusses  the  generalization 
of  the  notion  of  the  summability  of  a  series  to  the  limit  of  a  function 
of  a  continuous  variable ;  G.  H.  Hardy  develops  a  new  and  powerful 
method  for  the  discussion  of  Weierstrass's  continuous  function 
which  is  not  differentiable,  and  allied  questions;  and  William  F. 
Osgood,  to  show  that  a  theorem  of  Weierstrass  for  analytic  func- 
tions of  n  complex  variables  is  true  for  other  "spaces"  than  that  of 
analysis,  lays  down  a  general  definition  of  "infinite  regions,"  which 
includes  the  cases  of  projective  geometry,  the  geometry  of  inversion, 
the  geometry  of  the  space  of  analysis,  and  so  on. 

*  *       * 

In  the  Bulletin  of  the  American  Mathematical  Society  for  June, 
1916,  Dr.  A.  Bernstein  reduces  the  number  of  postulates  which 


32O  THE  MONIST. 

Huntington  gave  in  1904  for  Boole's  algebra  of  logic  from  ten  to 
eight,  and  that  of  postulated  special  elements  from  three  ("zero", 
the  "whole,"  and  the  "negative")  to  one  (the  "negative").  An 
interesting  and  valuable  address  delivered  before  the  University  of 
Chicago  by  Prof.  Edward  B.  Van  Vleck  on  "Current  Tendencies 
of  Mathematical  Research"  is  printed  in  the  October  number. 


The  number  of  the  Revue  de  metaphysique  et  de  morale  for 
May,  1916,  contains  a  long  and  important  article  by  A.  N.  White- 
head  on  the  relationist  theory  of  space.  This  theory  is  developed 
for  a  great  part  by  help  of  the  symbols  of  the  author  and  Russell's 
work.  The  other  articles  in  this  number  are  by  F.  Colonna  d'Istria 
(religion  according  to  Cabanis),  Leon  Brunschvicg  (the  relations 
of  the  intellectual  and  the  moral  conscience),  R.  Hubert  (the  Car- 
tesian theory  of  enumeration :  on  the  fourth  Rule  of  the  Discours) , 
and  Georges  Guy-Grand  (impartiality  and  neutrality).  In  the  July 
number  of  the  Revue  Lionel  Dauriac  writes  on  contingence  and 
category,  and  tries  to  decide  whether  Kant  was  right  or  wrong  in 
not  separating  the  necessary  and  the  a  priori.  Gaston  Milhaud  dis- 
cusses the  famous  mystical  crisis  through  which  Descartes  passed  in 
1619.  Henri  Dufumier  maintains  that  the  algebra  of  classes  in  logic 
only  takes  a  systematic  form  if  we  consider  it  as  a  generalization 
of  the  mathematical  theory  of  aggregates.  F.  Buisson  explains 
"the  true  meaning  of  the  sacred  union."  Finally,  there  is  a  necrol- 
ogy of  Victor  Delbos  (1862-1916). 


In  the  eighteenth  volume  (1916)  of  Prof.  Gina  Loria's  quar- 
terly Bollettino  di  bibliografia  e  storia  delle  scienze  matematiche , 
the  most  interesting  articles  in  the  first  two  numbers  (April  and 
June)  seem  to  be:  J.  H.  Graf's  collection  of  the  correspondence 
between  Ludwig  Schlafli  and  some  of  his  Italian  mathematical  con- 
temporaries (pp.  21-35,  49-64)  ;  and  G.  Vivanti's  review  of  the 
late  Julius  Konig's  Neue  Grundlagen  der  Logik,  Arithmetik  und 
Mengenlehre  of  1914  (pp.  37-39). 


VOL.  XXVII  JULY,  1917  NO.  3 


THE  MONIST 


THE  ELECTRONIC  THEORY  OF  MATTER.1 

• 

"Wic  Alles  sich  zum  Ganzen  webt ! 
Eins  in  dem  Andern  wirkt  und  lebt." 

— Goethe. 

THE  subject  of  the  considerations  that  follow  is  pro- 
posed as  the  sixth  under  the  division  of  physics  in 
the  published  program  of  this  congress.  Unquestionably 
the  proposal  was  most  timely  and  fortunate,  for  no  theme 
of  purely  scientific  content  is  more  important  or  more  cen- 
tral on  the  stage  of  interest  or  more  worthy  of  the  atten- 
tion of  the  assembled  savants  of  two  continents.  Surely 
it  is  eminently  appropriate  that  the  New  World  should 
foster  the  New  Knowledge,  should  master  it,  acclaim  it, 
proclaim  it,  and  advance  it. 

The  most  obvious  criticism  upon  any  attempt  to  treat 
this  theme  on  the  present  occasion  would  seem  to  be  that 
the  barrel  is  too  large  for  the  hoop.  So  far  and  wide 
reaching  is  the  new  doctrine  of  matter,  so  interpenetrative 
of  so  many  remote  and  alien  disciplines,  that  any  half-way 
adequate  presentation  of  even  the  most  near-lying  con- 
siderations would  necessarily  pass  swiftly  beyond  the 
largest  bounds  to  be  assigned  this  paper  and  easily  ex- 
pand into  a  stately  volume. 

1  This  paper,  read  (in  Spanish")  at  the  First  Pan-American  Scientific 
Congress  (Santiago  de  Chile,  Dec.  25,  1908  to  January  5.  1909),  has  appeared 
thus  far  only  in  the  Trabajos  del  Cuarto  Congreso  Cicntifico  (i°  Pan- Ameri- 
cano), Vol.  V,  pp.  1-22,  Santiago  de  Chile,  1910. 


322  THE   MONIST. 

\Vc  must  begin  then  with  renunciation.  The  attempt 
can  not  be  made  to  detail  but  only  to  suggest  some  of  the 
proofs  (which  are  manifold  and  decisive)  of  the  actual 
existence  of  the  corpuscle,  sub-atom,  or  electron,  as  the 
uniform  elementary  constituent  of  the  visible  universe  is 
variously  named.  The  isolated  independent  subsistence  of 
this  corpuscle  is  the  central  revelation  of  the  new  knowl- 
edge. Tt  was  first  discovered  many  years  ago,  and  pro- 
claimed to  the  world  as  the  fourth  or  radiant  state  of 
matter  by  Sir  William  Crookes,  after  whom  the  vacuum 
tubes  in  which  the  green  phosphorescence  accompanying 
the  passage  of  an  electric  current  were  and  still  are  named. 
That  something  called  cathode  rays  emerged  from  the 
cathode  or  negative  pole  and  moved  in  right  lines,  was 
proved  by  the  shadow  cast  by  an  interposed  mica  cross. 
The  English  declared  these  rays  were  particles,  shot  out 
from  the  cathode  (pole)  against  the  inside  walls  of  the 
tube ;  but  the  Germans  held  it  was  only  ether  waves  stirred 
up  at  the  pole  and  propagated  rectilinearly.  That  the  Eng- 
lish were  right  was  shown  conclusively  by  subjecting  the 
rays  first  to  magnetic  and  then  to  electric  attraction,  where- 
by it  appeared  that  they  behave  in  all  ways  precisely  as 
minute  particles  laden  with  negative  electricity.  Amazing 
is  the  control  which  the  experimenter  exhibits  over  these 
flying  hosts  of  electric  atoms;  by  deft  manipulation  of  his 
infinitely  fine  magnetic  or  electric  fingers  he  may  turn  the 
stream  of  corpuscles  as  he  will  and  even  bend  it  into  a 
spiral  or  into  a  circular  hoop  far  more  supplely  than  one 
might  bend  the  superfinest  Damascus  blade.  But  incon- 
ceivably more  delicate  still  is  the  touch  of  the  mathe- 
matical reason,  whereby  even  the  individual  electron  is 
caught  in  its  flight  and  forced  to  tell  the  secret  of  its  speed. 
For  one  may  subject  the  flying  particles  simultaneously 
to  opposite  electric  and  magnetic  influences  by  immersing 
them  in  two  coexistent  and  mutually  annulling  fields  of 


THE  ELECTRONIC  THEORY  OF  MATTER.  323 

force,  so  that  they  fly  undisturbed  straight  from  the  nega- 
tive to  the  positive  pole.  These  two  self-destroying  forces 
are  H  ev  and  c  X,  whence  v  =  X/H,  whereby  v  the  veloc- 
ity of  the  corpuscle  is  known  when  we  know  H  the  magnetic 
and  X  the  electric  force,  both  of  which  are  readily  meas- 
ured. This  velocity  increases  with  the  exhaustion  of  the 
tube  from  eight  thousand  up  to  one  hundred  thousand  kilo- 
meters per  second,  which  is  many  thousand  times  the  mean 
speed  even  of  hydrogen  molecules  at  the  highest  tempera- 
ture ever  yet  attained. 

But  far  more  wonderful  and  incomparably  more  im- 
portant than  this  determination  of  a  variable  velocity  is 
the  determination  of  a  constant,  the  most  fundamental  yet 
discovered  in  nature.  Science  itself  may  be  defined  as  the 
eternal  search  for  invariants  amid  the  eternal  flux  of  var- 
iants, and  this  astounding  constant  of  which  I  am  about  to 
speak  reminds  us  indeed  of  Plato's  unwavering  axis  of  the 
universe  turning  forever  in  the  lap  of  Necessity.  In  the 
equation  Hcv  =  X?,  the  symbol  c  denotes  the  negative 
electric  charge  borne  by  the  individual  corpuscle.  If  we  take 
away  the  magnetic  force,  leaving  only  the  electric,  this  latter 
will  bend  the  path  of  the  flying  corpuscle  as  gravity  bends 
the  path  of  the  level-aimed  cannon  ball  into  a  parabola. 
Now  Galilei  has  taught  us  the  formula  for  the  amount  (s} 
of  the  bend  or  the  fall  in  the  time  t ;  it  is  s  =  V>at2  where  a 
is  the  acceleration  in  question.  Here  the  acceleration  is 
the  force  X^  divided  by  the  mass  in  of  the  corpuscle;  the 
time  is  the  tube  length  /  divided  by  the  velocity  7';  and  the 
distance  s  is  the  descent  of  the  green  spot  at  the  end  of  the 
tube; 

hence  ^  +*  V^(c/m} .  (lz/vz\  whence  c/m  *=  (2i^//2X), 
where  all  on  the  right  side  is  known.  Hereby  is  determined 
this  ratio  of  the  electric  charge  to  the  mass  of  the  flying 
corpuscle,  and  this  ratio  is  found  to  be  everywhere  the 


324  THE   MONIST. 

same  (unless  indeed  the  velocity  of  the  corpuscle,  of  which 
it  is  in  strictness  a  function,  approaches  that  of  light). 

The  value  of  this  remarkable  constant  (for  all  ordinary 
velocities)  is  in  the  accepted  C.  G.  S.  system  no  less  than 
17,000,000  ( i  .7Xio7).  Why  is  it  so  large ?  Is  it  because 
the  charge  e  is  so  great,  or  because  the  mass  m  is  so  small? 
This  question  can  be  answered  and  has  been  answered  by 
the  exquisitely  beautiful  experiments  of  the  two  Wilsons 
(C.  T.  R.  and  H.  A.)  on  the  formation  of  clouds  by  con- 
densation of  vapor  around  nuclei.  Not  only  does  the  water 
collect  around  particles  of  dust  but  also  around  any  par- 
ticles charged  with  electricity:  nay  more,  it  refuses  to 
collect  except  around  nuclei  until  the  vapor  reaches  eight- 
fold saturation.  Now  it  has  been  found  possible  to  free  a 
cylinder  of  air  from  dust,  and  supersaturate  it  with  vapor, 
and  then  to  form  in  it  suddenly  a  dense  cloud  by  electri- 
fying its  particles  with  radiations  from  radium  or  still 
better  by  charging  its  individual  molecules  with  electrons 
shot  out  from  a  metal  plate  played  on  by  ultra-violet  light. 
By  attracting  electrically  these  drops  coagulated  around 
these  molecules  one  may  suspend  them  in  the  air  of  the 
cylinder  like  balloons  or  make  them  fall  as  slowly  as  one 
will,  so  that  their  velocity  of  fall  may  be  measured;  and 
Stokes  has  deduced  the  formula  for  this  velocity,  v  - 
2/g.ga2/\i  where  g  is  the  known  acceleration  of  gravity 
and  \i  the  known  coefficient  of  viscosity ;  hence  a,  the  radius, 
and  thence  the  size  of  the  drop  is  found;  and  hence  by 
measuring  the  amount  of  watery  vapor  deposited  one  finds 
the  number  of  the  drops  and  so  can  count  the  number  of 
electrified  molecules,  that  is  the  number  of  corpuscles,  since 
each  molecule  has  but  one  negative  electron.  Plainly,  if 
by  one  chance  in  a  trillion  two  corpuscles  should  light  on 
one  molecule,  their  mutual  repulsion  would  dislodge  them 
instantly. 

By  electrometric  methods  one  may  find  the  total  charge 


THE  ELECTRONIC  THEORY  OF  MATTER.  325 

of  electricity  on  the  total  water,  the  sum  of  the  drops,  and 
dividing  this  by  the  number  of  drops  or  corpuscles  one  finds 
the  charge  c  on  each,  and  then  on  dividing  this  by  the  con- 
stant ratio  there  results  the  mass  m  of  each  corpuscle. 
These  numbers  turn  out  to  be  appalling  in  their  minuteness. 
The  charge  e  equals  3io/io12  of  an  electrostatic  unit,  or 
one  one-hundred-trillionth  (icr-20)  of  an  electromagnetic 
unit,  and  is  the  long  well-known  approximate  value  of  the 
charge  borne  by  an  atom  of  hydrogen  in  the  electrolysis  of 
dilute  solutions.  The  mass  in  of  the  corpuscle  proves  to  be 
six  hundred  quintillionths  of  a  gramme  (6  X  IO—28),  a 
degree  of  parvitude  far  beyond  the  utmost  stretch  of  the 
imagination.  The  same  may  indeed  be  said  of  the  atom 
of  hydrogen,  but  the  mass  of  this  atom  is  1700  times2  the 
mass  of  the  corpuscle.  Hitherto  this  hydrogen  atom  has 
been  conceived  as  standing  on  the  remotest  confines  of 
matter,  but  the  new  knowledge  shows  us  a  still  lower  world 
of  corpuscles,  nearly  2000  times  smaller. 

At  this  point  it  may  be  proper  to  enter  a  caution.  It 
is  almost  universal  to  speak  of  this  corpuscle  as  of  invari- 
able mass  bearing  an  invariable  charge  of  negative  elec- 
tricity, and  the  calculations  and  experiments  do  indeed 
yield  results  uniform  within  the  limits  of  error.  But  we 
must  remember  that  these  experiments  and  calculations 
have  always  treated  and  apparently  must  always  treat  not 
the  individual  corpuscle  but  millions  on  millions  of  cor- 
puscles and  atoms.  The  results  then  were  only  averages 
of  countless  numbers  of  individuals,  and  the  constancy  of 
such  an  average  implies  nothing  at  all  as  to  the  constancy 
or  inconstancy  of  the  individual,  just  as  the  comparative 
steadiness  of  the  rates  of  birth,  death,  marriage,  homicide 
and  the  like,  even  in  a  population  of  a  few  millions,  implies 

-  Later  determinations  raise  this  number  to  1830  or  even  1872.  M.  Perrm's 
experiments  on  "visible  molecules"  indicate  that  the  mass  of  a  hydrogen  atom 
is  1.63  quadrillionths  (1.63/1024)  of  a  gram.  Hence  the  mass  of  an  electron 
would  be  0.8/1027  gram. 


326  THE  MONIST. 

nothing  whatever  as  to  the  rate  in  any  particular  family. 
For  all  we  know  the  range  of  individuality  among  atoms 
and  corpuscles  may  be  quite  as  great  as  among  suns  or 
planets  or  men,  and  this  we  must  say  even  in  face  of  the 
famous  dictum  of  Maxwell,  that  atoms  of  any  one  sub- 
stance have  all  the  marks  of  manufactured  articles,  being 
all  exactly  alike. 

Returning  from  this  digression  we  must  now  ask  what 
is  the  mass  and  what  is  the  charge  of  electricity  of  the 
corpuscle?  It  is  precisely  here  that  the  new  knowledge 
calls  for  the  profoundest  transformation  of  our  concep- 
tions, for  it  derives  the  phenomenon  of  mass  in  the  cor- 
puscle solely  from  the  motion  of  the  flying  charge  of  nega- 
tive electricity.  We  all  know  that  work  is  needed  to  start 
a  body  in  motion,  as  a  car  even  on  a  perfectly  smooth  track. 
For  any  particular  body  having  a  particular  velocity  v, 
the  amount  of  work  necessary  is  perfectly  definite,  namely, 
V^  multiplied  by  a  constant,  M,  called  the  mass  of  the 
body.  We  say  the  kinetic  energy  imparted  is  HMf2.  This 
supposes  the  motion  is  in  a  vacuum,  which  is  never  the 
case;  in  practice  the  motion  is  always  in  some  fluid,  as 
water  or  air.  Then  we  all  know  that  more  work  is  needed, 
according  to  velocity.  One  fans  oneself  gently  with  ease, 
but  violently  only  with  great  effort.  In  fact,  the  fluid  is 
also  set  in  motion  as  well  as  the  body,  and  this  calls  for 
energy  or  work.  How  much  fluid  is  dragged  or  pressed 
along  with  the  body  will  depend  on  the  body's  size,  shape 
and  speed  and  on  the  density  of  the  fluid.  Some  of  the  sim- 
plest cases  have  been  studied.  Sir  George  Gabriel  Stokes 
has  found  that  in  case  of  a  sphere  the  work  done  on  the  fluid 
is  %M'  V2,  where  M'  is  the  mass  of  a  volume  of  the  fluid 
half  as  large  as  the  sphere  (shown  by  Green,  1833),  so  that 
the  total  energy  imparted  is  %(M  +  M')V2-.  Now  all 
motions  take  place  in  the  all-pervading  ether.  It  follows 
then  that,  if  the  ether  itself  has  mass,  when  put  into  move- 


THE  ELECTRONIC  THEORY  OF  MATTER.  327 

ment  it  must  absorb  energy  or  require  work,  and  bence 
that  some  perhaps  infinitesimal  part  of  the  mass  of  a 
moving  body  even  in  a  vacuum  must  be  due  to  the  swirl 
set  up  in  the  ether.  In  the  case  of  the  moving  corpuscle 
the  analogy  is  not  absolutely  perfect,  but  exact  enough  to 
make  intelligible  the  statement  that  if  a  conducting  sphere 
of  radius  r,  having  a  charge  e  and  mass  ;//,  be  set  moving 
with  velocity  v,  the  energy  developed  in  the  ethereal  mag- 
netic field  has  been  proved  to  be  Vzk(e/r}  .z>2,  so  that  the 
total  work  done  is  V»[m  -f-  %k(c/r)]v2,  and  the  ordinary 
mass  m  is  thus  increased  by  %k(e2/r),  which  stands  for  the 
inertia  overcome  in  the  ether.  (This  k  is  a  factor  due  to 
the  crowding  together  of  the  lines  of  force  into  a  plane 
through  the  sphere  center,  and  perpendicular  to  the  mo- 
tion, and  increases  rapidly  as  the  velocity  becomes  great.) 
Since  e  is  extremely  small,  this  increase  is  wholly  imper- 
ceptible in  case  of  aft  single  bodies  subject  to  our  senses 
or  experiment.  But  for  the  corpuscle,  when  r  becomes 
inconceivably  small,  this  so-called  electric  mass  assumes 
important  proportions,  yea,  it  accounts  for  absolutely  all 
the  mass  of  the  corpuscle,  which  must  have  this  electric 
mass  and  need  have  no  other  at  all.  For  Kaufmann  has 
measured  the  value  of  e/m  (or  m/e)  for  the  various  cor- 
puscles emitted  with  various  velocities  by  radium ;  and  J.  J. 
Thomson  has  calculated  k  for  these  velocities.  It  turns 
out  that  the  calculated  relative  increase  (due  to  rising 
velocity)  in  the  electrical  mass  is  constantly  equal  to  the 
observed  relative  increase  in  the  whole  mass,  whence  one 
must  conclude  that  the  electrical  mass  is  the  whole  mass, 
for  if  there  were  any  ordinary  non-electrical  mass,  however 
small,  it  would  certainly  not  thus  increase  apparently  with 
the  increasing  velocity.  The  mass  of  the  corpuscle  is  thus 
not  located,  at  least  in  any  appreciable  degree,  in  the  cor- 
puscle itself,  but  only  in  the  universal  ether  around  it. 
Imagine  a  sphere  surface  perfectly  rigid  but  absolutely 


328  THE   MONIST. 

void,  empty  even  of  ether  itself,  a  mere  round  hole  in 
universal  ether.  If  set  in  motion  this  hollow  sphere  would 
gather  mass  as  it  gathered  velocity,  but  the  mass  would 
not  be  inside,  it  would  be  wholly  outside,  inwrought  in  the 
universal  eddy  set  up  in  the  infinite  ether.  In  this  sense 
the  mass  of  the  moving  hollow  sphere  would  be  coextensive 
with  the  whole  space  filled  by  ether,  and  in  this  sense  we 
may  say  the  same  of  the  mass  of  a  flying  corpuscle:  it 
reaches  throughout  the  world.  We  may  imagine  it  as  a 
mere  needle-point  from  which  Faraday  tubes  of  force  radi- 
ate to  the  utmost  stars.  But  since  the  ether  bound  by  the 
tubes  varies  as  the  squared  density  of  the  tubes,  and  hence 
varies  inversely  as  the  fourth  power  of  the  distance  from  the 
sphere  center,  it  follows  that  the  corpuscle  mass  is  after  all 
highly  concentrated  round  the  corpuscle  core.  For  an  easy 
reckoning  shows  that  the  corpuscle  radius  r  is  only  about 
five  millionths  of  the  molecule  radius,  which  is  commonly 
taken  as  the  hundred  millionth  of  a  centimeter ;  that  is,  of 
course,  in  order  of  magnitude.  Hence  from  the  surface  of 
a  molecule  to  the  surface  of  a  corpuscle  at  its  center  the 
mass  intensity  would  increase  more  than  a  trillionfold. 

It  follows  that  one  can  no  longer  affirm  with  perfect 
rigor  the  principle  of  the  conservation  of  mass,  for  the 
masses  vary  constantly  with  the  velocities  of  the  corpuscles. 
But  to  our  gross  senses  even  when  armed  with  the  most 
delicate  instruments  these  variations  might  forever  remain 
imperceptible.  However,  Heydweiler  claims  to  have  actu- 
ally detected  a  difference  in  the  joint  weights  of  water  and 
copper  sulphate  before  solution  and  after,  and  Wallace 
holds  that  the  mass  of  water  is  changed  by  freezing — 
highly  interesting  results,  that  await  confirmation.  But  it 
must  not  be  supposed  that  the  notion  of  mass  itself  is 
hereby  eliminated  or  e^en  reduced  to  greater  simplicity. 
For  all  these  results  assume  to  start  with  the  assumption 
that  the  ether  itself  has  mass.  Calling  then  to  one's  help 


THE  ELECTRONIC  THEORY  OF  MATTER.  329 

• 

the  Faraday  image  of  tubes  of  force  and  still  more  the 
hydrodynamics  of  vortex  rings,  one  may  deduce  from 
ether-mass  the  mass  of  all  material  bodies;  but  mass  itself 
adheres  along  with  time  and  space  inexpugnably  in  our 
reasonings. 

Corpuscles  therefore  are;  atoms  also  are;  how  then 
shall  we  think  them  related?  As  the  corpuscle  mass  is 
only  Vnoo  of  the  hydrogen  atom  mass,  shall  we  think  this 
smallest  atom  as  compounded  of  1700  corpuscles?  There 
are  many  reasons  against  such  a  doctrine,  reasons  that  lead 
one  to  think  the  number  of  corpuscles  in  the  atom  as  always 
small.  But  shall  we  think  the  atom  as  in  any  case  com- 
posed of  corpuscles?  There  seems  to  be  no  escape  from 
such  a  conception,  which  lies  directly  across  the  path  of 
thought.  For  many  experiments  that  cannot  be  mentioned 
here  show  that  corpuscles  are  all-pervasive.  Metals  heated 
pour  them  forth,  as  do  all  other  substances  hot,  and  some 
shoot  them  out  even  when  cold,  as  rubidium,  and  at  fearful 
speed,  as  all  radio-active  substances;  yea,  if  we  had  in- 
struments fine  enough  we  might  detect  them  in  every  sub- 
stance, and  everywhere  maintaining  the  constant  ratio  e/m 
inviolate.  Moreover,  that  the  corpuscle  is  closely  related  to 
the  atom  is  clearly  seen  in  the  fact  already  mentioned  that 
the  corpuscle's  and  the  hydrogen-atom's  charges  of  elec- 
tricity are  the  same. 

Before  trying  to  construct  imaginatively  the  atom  out  of 
corpuscles  we  must  recall  that  there  are  rays  (as  Goldstein's 
Canals trahlen)  of  positive  as  well  as  of  negative  electricity, 
that  are  deflected  by  a  magnet  oppositely  to  the  negative 
cathode  rays.  For  them  the  ratio  e/m  is  not  constant 
and  never  exceeds  10,000,  which  is  also  its  value  for  hydro- 
gen ions  in  electrolysis  of  dilute  solutions.  It  is  natural  to 
figure  thus  the  positive  corpuscle  as  a  sphere  of  positive 
electrification,  about  the  size  of  an  atom.  Of  course  such 
a  sphere  implies  an  equal  and  balancing  amount  of  negative 


33O  THE   MONIST. 

• 

electricity,  and  this  we  imagine  distributed  throughout  the 
sphere  as  equal  corpuscles  or  units  of  negative  electricity. 
Since  the  atom  is  permanent,  this  distribution  of  negative 
electrons  in  the  positive  sphere  must  form  a  system  in  stable 
equilibrium,  and  the  question  arises,  what  arrangement  of 
the  electrons  would  constitute  such  a  stable  system?  The 
problem  is  mathematico-mechanical,  and  its  general  solu- 
tion lies  beyond  the  range  of  our  present  powers  of  analysis ; 
but  if  we  propose  the  problem  not  for  space  but  for  the 
plane,  we  may  solve  it  and  get  a  system  of  answers  quanti- 
tatively different  but  formally  analogous  to  those  that  must 
be  rendered  for  threefold  space. 

At  this  point  theory  and  experiment  have  joined  hands 
in  a  most  friendly  fashion.  As  early  as  1881  the  present 
Cavendish  professor  of  physics  at  Cambridge,  stimulated 
by  the  brilliant  experiments  of  Crookes,  in  a  long  neglected 
but  now  classical  paper  in  the  Philosophical  Transactions, 
discussed  the  motion  of  a  charged  sphere  in  an  electric 
field,  thereby  laying  the  foundations  of  the  doctrine  of  elec- 
tric mass.  Twenty-three  years  later  ( 1904)  he  advanced 
to  the  discussion  of  the  equilibrium  of  a  system  of  negative 
electric  charges  abandoned  to  their  mutual  attractions  in 
a  positively  electrified  shell.*  He  showed  that  the  configura- 
tion of  planar  equilibrium  is  (in  general)  a  regular  poly- 
gon concentric  with  the  sphere,  but  for  six  particles  the  equi- 
librium is  unstable,  one  particle  will  break  ranks  and  rush 
to  the  center,  while  the  other  five  form  a  regular  pentagon. 
Similarly  if  there  be  seven,  eight  or  nine;  if  there  be  ten, 
three  will  form  an  inner  equilateral  triangle,  and  so  on  up 
to  seventeen,  when  one  of  the  inner  ring  will  again  break 
ranks  and  fly  to  the  center,  leaving  an  inner  ring  of  five, 
and  an  outer  ring  of  eleven.  (Four  corpuscles  cannot  be  in 
planar  equilibrium  at  rest,  but  only  when  the  four  are  in 
rapid  rotation.  At  rest  they  are  at  the  corners  of  a  regular 

*  For  an  additional  note  see  page  480. 


THE  ELECTRONIC  THEORY  OF  MATTER.  33! 

tetrahedron  whose  edge  equals  the  radius  of  the  sphere. 
Six  corpuscles  are  balanced  at  six  corners  of  a  regular 
octahedron.)  When  the  number  reaches  thirty-two,  the 
three-ring  system  becomes  unstable,  again  a  particle 
seeks  the  center,  leaving  an  inner  ring  of  five,  a  middle 
ring  of  eleven,  an  outer  ring  of  fifteen.  Looking  at  it 
another  way  one  may  ask  how  many  must  be  put  inside  a 
ring  of  n  to  make  it  stable.  The  answers  are:  for  5,  o;  for 
6,  7,  8,  each  I ;  for  9,  2;  for  12,  8;  for  13,  10;  for  15,  15; 
for  20, 39 ;  for  30, 101 ;  for  40, 232.  You  see  how  rapidly  (as 
the  cube  of  n)  the  inside  corpuscles  multiply  as  the  outer 
ring  increases  in  number.  The  structure  must  be  compact, 
densely  peopled  toward  the  center,  not  hollow  like  a  shell. 
The  whole  scheme  of  numbers  has  been  worked  out  by 
J.  J,  Thomson  (Philosophical  Magazine,  1904)  mathe- 
matically, and  a  beautiful  experiment  first  made  for  an- 
other purpose  by  the  American  Mayer,  afterwards  under 
other  forms  by  Wood  and  Monckmann,  confirms  his  re- 
sults. On  a  water  surface  any  number  of  small  equally 
magnetized  needles  are  made  to  float  by  being  thrust 
through  cork  discs,  only  like  poles  being  above  the  water. 
These  repel  each  other  like  the  negative  electric  corpuscles. 
The  sphere  of  positive  electrification  is  represented  by  a 
magnet  hung  above  the  water,  the  opposite  pole  pointing 
downward.  This  holds  the  magnets  in  groups  in  stable 
equilibrium,  and  the  arrangements  of  the  magnets  actually 
observed  agree  excellently  with  the  arrangement  required 
and  predicted  by  mathematical  analysis.  Provisionally  then, 
we  may  proceed  on  the  working  hypothesis  that  the  atom  is 
a  system  of  corpuscles  composed  of  a  number  of  concentric 
sub-systems  all  held  in  stable  equilibrium  by  an  enclosing 
sphere  of  positive  electrification.  This  conception  may  be 
somewhat  vague  and  may  hereafter  require  modification, 
but  it  is  far  clearer  and  more  precise  than  was  possible 
a  few  years  ago  and  constitutes  a  notable  advance  in  phys- 


332  THE  MONIST. 

ical  theory.  We  have  spoken  of  the  configuration  as  stable, 
but  this  stability  must  not  be  understood  too  strictly  nor  as 
always  equally  rigid.  Since  we  may  have  an  outer  ring  of 
five,  six,  seven  or  eight  with  only  one  at  the  center,  it  is 
plain  that  if  there  were  seven  in  equilibrium,  neither  an 
increase  nor  a  decrease  of  just  one  would  disturb  the  system 
much;  but  if  there  were  sixteen  the  arrangement  would 
be  one  of  two  rings,  eleven  in  the  outer,  five  in  the  inner ; 
take  away  one  and  the  arrangement  persists  with  only  ten 
outside;  but  add  one  and  the  two-ring  system  is  no  longer 
stable,  a  particle  goes  to  the  center,  the  rings  remain  un- 
changed, a  three-ring  system  is  formed.  A  better  though 
more  complex  example  is  afforded  by  the  group  of  eleven, 
ranging  from  58  to  68  corpuscles  (Thomson) ;  all  these 
have  five  rings,  thus,  in  order: 

19     20     20     20     20     20     20     20     20     20     21 
16     16     16     16     17     17     17     17     17     17     17 

13     13     13     13     J3     13     13     H     14     J5     15 
8      8      8      9      9     10     10     10     id     10     10 

_2_j2_3_3^3^3jl^_5_j$_5 
58    59    60    61     62    63     64    65     66    67    68 

Here  we  see  that  if  a  corpuscle  be  injected  into  the  58- 
system  it  produces  the  least  possible  disturbance,  place  is 
made  for  it  in  the  outmost  ring  and  the  others  remain  as 
they  were,  a  59-system  results.  But  if  a  corpuscle  be  added 
to  this  it  can  find  no  place  in  any  ring  but  the  central;  it 
must  find  its  own  way  to  the  center  ring  or  else  dislodge 
a  corpuscle  from  the  outer  ring,  which  corpuscle  will  then 
dislodge  another  from  the  next  and  so  on  till  a  corpuscle  is 
finally  dislodged  into  the  innermost  ring  from  the  one  next 
to  it.  Still  another  corpuscle  may  be  injected  and  another 
and  another,  profoundly  altering  the  original  arrange- 
ment but  preserving  the  outer  ring  unbroken. 


THE  ELECTRONIC  THEORY  OF  MATTER.  333 

So  it  goes  on,  the  center  becoming  denser  and  the  outer 
ring  more  stable  till  the  total  number  67  is  reached,  having 
the  arrangement  20,  17,  15,  10,  5.  Here  the  central  mass, 
though  it  may  still  make  room  for  another  corpuscle,  is 
very  steady  and  stubborn,  so  that  now  when  another  cor- 
puscle is  injected,  the  outer  ring  yields,  absorbs  it  and  now 
has  21.  Accordingly  this  67-system  is  like  that  of  58, 
most  stable,  changing  least  from  its  original  form.  The 
group  of  arrangements  from  59  to  67  corpuscles  forms  a 
series  all  having  twenty  in  the  outer  row,  the  stability  of 
each  system  increasing  up  to  the  last,  after  which  a  new 
group  begins  with  similar  properties  but  only  eight  mem- 
bers. Now  these  corpuscles  are  units  of  negative  elec- 
tricity. As  the  number  of  these  inside  the  atom  increases, 
the  outer  ring  remaining  the  same,  the  stability,  measured 
by  Q,  the  work  necessary  to  disperse  all  the  units  infinitely 
apart,  increases;  the  more  inside  the  more  firmly  the  out- 
side ring  is  held.  Hence  the  59-system  will  be  least  stable, 
a  unit  would  easily  fly  off  leaving  only  the  preceding  58- 
system.  If  wre  suppose  this  59-system  neutral,  on  losing 
this  negative  unit  it  becomes  comparatively  at  least  electro- 
positive; in  fact  the  most  strongly  electro-positive  of  this 
series  of  arrangements.  The  following  members  must  lose 
more  and  more  negative  units  in  order  to  become  electro- 
positive as  this  59-system.  The  67-system  is  charged  with 
the  utmost  negative  excess  and  so  is  most  electro-negative 
or  least  electro-positive.  The  outer  ring  of  20  will  in  fact 
admit  no  more  negative  units  inside,  but  on  still  further 
addition  a  new  outer  ring  of  21  is  formed  and  a  new 
series  begins  with  a  highly  electro-positive  system  of  68 
and  again  runs  down  to  an  electro-negative  system  of  77, 
in  each  of  which  the  outer  ring  is  21.  Herewith  then  we 
attain  a  new  notion  of  valency.  For  the  59-system  has 
only  just  sufficient  corpuscles  inside  to  maintain  its  outer 
ring  of  20;  the  fifty-ninth  in  the  system,  the  twentieth  in 


334  THE  MONIST. 

the  ring,  might  easily  break  away  leaving  only  19  outside 
and  the  atom  positive  from  the  loss  of  the  negative  unit. 
But  it  could  not  remain  positive  for  it  would  draw  to 
it  another  corpuscle  and  so  restore  its  ring  of  twenty, 
and  this  process  might  be  repeated.  But  as  many  as  8 
negative  corpuscles  might  be  injected  into  the  ring  of  20, 
raising  the  total  number  to  67;  hence  we  may  say  this 
59-system  corresponds  to  an  atom  of  valency  8  for  a  nega- 
tive unit  charge  and  of  valency  o  for  a  positive  charge, 
since  it  could  not  assume  permanently  the  positive  unit 
charge.  Consider  next  the  67-system.  It  is  impossible  to 
drive  a  single  negative  unit  within  the  2O-ring ;  if  one 
collide  with  the  atom  it  stops  in  the  outer  ring  making  21, 
but  this  ring  is  very  unstable  and  would  easily  lose  this 
electric  unit ;  hence  we  may  say  this  system  has  no  negative 
valency  or  a  negative  valency  equal  to  o.  However,  this 
same  67-system  might  lose  one,  two  or  three.  .  .  .or  even 
eight  atoms,  reducing  its  negative,  t.  e.,  raising  its  positive, 
charge,  though  with  harder  and  harder  work ;  it  could  not 
lose  more  without  changing  its  outer  ring  and  passing  into 
another  series.  Hence  we  may  say  that  its  positive  val- 
ency is  8,  just  as  its  negative  valency  is  o.  It  is  needless 
to  dilate  upon  the  intermediate  members.  Similar  con- 
siderations show  that  we  may  arrange  them  thus: 
No.  of  cor- 
puscles .  .  59  60  6 1  62  63  64  65  66  67  68 

\  -ho  +i  +  2  +3  +4  --3  —2  —i  — o 
Valenc    .      _g  _ 


Electro-positive  Electro-negative 

Such  series  are  actually  found  among  known  chemical  ele- 
ments. Such  are  helium,  lithium,  barium,  boron,  carbon, 
nitrogen,  oxygen,  fluorin,  neon,  and  neon,  sodium,  mag- 
nesium, aluminum,  silicon,  phosphorus,  sulphur,  chlorin, 
argon.  Of  course  it  is  not  meant  for  a  moment  that  any 


THE  ELECTRONIC  THEORY  OF  MATTER.  335 

such  planar  arrangement  is  actually  present  in  chemical 
atoms ;  their  corpuscles  must  be  arranged  in  tridimensional 
space ;  but  it  can  hardly  be  doubted  that  relations  analogous 
to  the  foregoing,  only  more  complicated,  must  characterize 
spatial  as  well  as  planar  arrangements.  If  we  call  the 
tendency  of  a  system  to  shed  a  corpuscle  corpuscular  pres- 
sure (outward},  then  it  appears  that  this  pressure  changes 
abruptly  at  the  end  of  each  series ;  thus  at  58  the  pressure 
is  very  low,  at  59  very  high ;  at  67  low ;  at  68  high :  it  falls 
through  the  electro-positives  down  to  and  through  the 
electro-negatives.  We  might  then  define  positive  valency 
of  an  electro-positive  (or  negative)  as  the  greatest  number 
of  corpuscules  it  can  lose  without  abrupt  fall  in  corpuscular 
pressure;  the  negative  valency  of  the  electro-negative  is 
the  number  it  can  gain  without  sudden  rise  of  corpuscular 
pressure.  Upon  these  definitions  and  conceptions  has  been 
erected  a  most  plausible  theory  of  chemical  combination, 
into  which  we  cannot  enter.  But  one  other  aspect  may  not 
be  passed  by  in  silence.  While  no  one  affirms  that  the 
planar  forms  of  equilibrium  are  the  actual  forms  assumed 
by  corpuscles  in  atoms,  it  seems  hardly  possible  that  they 
are  not  similar  thereto,  similar  enough  to  allow  a  most 
important  conclusion.  These  forms  are  arranged  in  series, 
and  the  members  of  these  series  bear  striking  resemblances. 
There  are  rings  outside  of  rings,  and  rings  outside  of  these, 
and  so  on.  Thus  around  a  central  one  there  is  a  ring  of  6, 
giving  7;  and  around  this  a  ring  of  n,  yielding  18;  and 
around  this  a  ring  of  15  making  33 ;  and  around  this  a  ring 
of  1 8,  making  51  in  all ;  and  around  this  a  ring  of  21,  which 
makes  72;  and  around  this  still  another  of  24,  or  a  total 
of  96. 

It  seems  impossible  that  atoms  consisting  of  these 
or  any  such  systems  of  corpuscles  should  not  have  many 
likenesses  in  property.  We  are  reminded  of  a  determi- 
nant of  a  definite  form,  whose  degree  is  raised  by  border- 


336  THE   MONIST. 

ing  it  successively  by  parallel  lines  above  and  below, 
on  the  right  and  left.  The  general  properties  of  the  de- 
terminants remain  the  same.  If  then  all  possible  forms 
of  equilibrium  should  actually  be  realized  as  chemical  ele- 
ments, of  necessity  these  elements  would  fall  into  series 
and  in  fact  a  twofold  series  which  might  be  expressed  by 
a  vertical  and  horizontal  alignment,  the  elements  in  the 
vertical  rows  being  alike  in  their  centraj  rings  but  differing 
in  the  number  of  rings;  those  in  a  horizontal  row  being 
alike  in  the  number  of  rings  and  in  their  outer  ring  but 
different  in  their  inner  rings.  Herewith  then  we  are  lifted 
up  to  what  is  commonly  regarded  as  the  apex  of  chemical 
theory,  the  periodic  law  of  Mendelyeev,  which  thus  ap- 
pears not  as  an  empirical  observation,  however  great 
or  important,  but  as  an  inexorable  necessity  of  the  me- 
chanical laws  of  configuration  and  equilibrium.  It  is 
most  remarkable  also  that  herewith  the  law  is  explained 
not  only  in  its  rigor  where  it  is  rigorous,  but  also- in  its 
laxity  where  it  is  lax.  For  there  is  no  necessity  that  all 
the  possible  forms  of  equilibrium  should  be  actualized; 
there  might  very  well  be  gaps,  even  considerable  gaps,  in 
both  the  vertical  and  horizontal  series.  In  that  case  some 
gap,  say  in  a  vertical  row,  might  have  next  to  it  some 
actual  form  in  a  near-lying  parallel  line,  which  would  thus 
present  not  an  exact  but  only  an  approximate  repetition 
of  property. 

Thus  the  electronic  theory  of  matter  yields  not  only  a 
vivid  idea  of  the  necessary  existence  of  a  double  system 
of  valences  among  atoms,  and  of  the  probabilities  and 
nature  of  chemical  combinations,  but  also  yields  deduc- 
tively in  a  highly  acceptable  form  the  confessedly  highest 
and  most  significant  induction  yet  reached  in  chemical 
theory.  This  conception  of  the  atom  not  as  an  infinitesimal 
grain,  strong  in  solid  singleness,  as  Lucretius  fancied  it, 
nor  yet  as  a  vortex-filament  in  an  incompressible  friction- 


THE  ELECTRONIC  THEORY  OF  MATTER.  337 

less  ether,  so  sleek  and  slippery  as  to  wriggle  out  from 
under  the  edge  of  the  keenest  knife  and  sharpest  scissors, 
as  Helmholtz  and  Kelvin  conceived  it,  but  as  a  highly 
organized  community  with  members  held  together  in  unity 
in  stable  equilibrium,  not  at  rest  but  in  a  system  of  com- 
plicated movements  of  inconceivable  velocity,  whose  very 
swiftness  itself  contributes  indispensably  to  the  stability 
of  the  configuration  (see  p.  480), — this  conception  not  only 
imparts  new  grandeur  to  physics  but  aligns  it  on  the  one 
hand  with  astronomy,  on  the  other  with  biology  and  even 
anthropology.  For  we  are  all  familiar  in  general  terms 
with  the  planetary  system  and  also  with  Bode's  law,  a 
special  case,  it  would  seem,  of  some  principle  of  balancing, 
such  as  reaches  from  the  atom  to  the  constellation,  from 
the  star  dust  of  a  nebula  to  the  most  complex  organization 
of  human  society.  Indeed  the  principle  of  natural  selection 
would  seem  to  be  hardly  less  operative  in  the  world  of 
brute  matter  than  in  the  world  of  life. 

More  specifically,  however,  the  electronic  theory  casts 
a  broad  beam  of  light  on  some  long  outstanding  enigmas 
of  astronomy.  The  motion  of  comets,  presenting  in  the 
vast  sweep  of  their  tails  an  apparent  repulsion  from  the 
sun,  finds  in  this  theory  a  long  desiderated  explanation. 
At  this  point  science  has  to  thank  a  large  number  of  widely 
separated  conceptions  and  personalities.  It  was  the  British 
Maxwell  who  as  early  as  1873  confirmed  the  suspicion  of 
the  German  Euler  (1746),  that  ethereal  undulations  must 
produce  a  longitudinal  pressure  along  the  ray  of  heat,  and 
three  years  later  the  Italian  Bartoli  reached  a  similar,  more 
general  conclusion  by  a  wholly  different  path.  The  mathe- 
matical prophecy  declared  this  pressure  to  be  E(i  -(-  r}/v, 
where  E  is  the  energy  and  v  the  velocity  of  light,  and  r 
the  coefficient  of  reflection  of  the  receiving  medium.  But 
this  phenomenon  is  so  extremely  subtile  as  long  to  have 
eluded  the  keenest  observation,  though  the  unerring  finger 


338  THE   MONIST. 

of  mathematics  was  pointed  at  it  steadily  for  twenty-eight 
years. 

At  length  (1901)  the  Russian  Lebedev  succeeded  in 
detecting  and  even  in  measuring  it.  Two  years  later  the 
Americans  Nichols  and  Hull  not  only  repeated  Lebedev's 
experiment  with  far  higher  precision,  but  showed  decisively 
that  the  observed  value  of  the  repulsion  agreed  within  the 
limits  of  error  with  that  foretold  by  the  English  clairvoyant 
Maxwell.  Of  course  this  repulsive  push  is  inconceivably 
minute,  and  on  even  a  very  small  sphere  it  would  be  im- 
perceptible in  comparison  with  the  extremely  feeble  attrac- 
tion of  gravity.  However,  the  latter  decreases  as  the  mass 
or  as  the  cubed  radius,  while  the  repulsion  decreases  as  the 
surface  or  as  the  squared  radius,  of  a  spherical  particle. 
No  matter  then  how  much  greater  the  attraction  than  the 
repulsion  on  any  given  sphere,  as  the  radius  decreases  the 
repulsion  must  finally  gain  the  upper  hand,  the  particle 
sufficiently  minute  must  be  repelled  by  the  light  away  from 
the  sun.  A  particle  of  earth  o.ooooi  of  an  inch  in  diameter 
would  hang  balanced  between  the  push  and  the  pull;  if 
larger  it  would  fall,  if  smaller  it  would  rise  and  fly  away 
before  the  thrust  of  the  light.  Now  we  need  not  indeed 
have  recourse  to  electric  corpuscles  to  find  particles  much 
below  this  critical  magnitude,  and  the  phenomenon  of  com- 
etary  tails  blown  backward  from  the  sun  with  inconceiv- 
able velocity,  as  by  the  breath  of  a  god,  is  readily  intelli- 
gible as  resulting  from  the  now  demonstrated  pressure  of 
light. 

Into  the  details  of  this  matter  it  is  impossible  to  enter 
here ;  suffice  it  that  the  illustrious  Swedish  physicist,  Svante 
Arrhenius,  has  subjected  the  whole  subject  to  rigorous 
calculation  which  has  been  in  the  main  verified,  at  certain 
points  amended,  by  Schwarzschild,  so  that  the  enigma  of 
cometary  motion  may  now  be  regarded  as  solved.  It  is 
found  in  fact  that  the  maximum  direct  pressure  at  the  sur- 


THE  ELECTRONIC  THEORY  OF  MATTER.  339 

face  of  the  sun  is  2%  mg.  per  square  centimeter.  While 
gravitation  sets  an  upper  limit  to  the  diameter  of  the  re- 
pelled particle  (0.0015  mm.)  the  diffraction  of  light  sets 
a  lower,  namely,  about  o.i  the  wave  length  of  the  light  in 
question.  Only  particles  whose  diameters  lie  between  these 
limits  can  be  repelled,  all  others  are  attracted.  The  great- 
est theoretic  repulsion  is  19  times  the  attraction  of  gravi- 
tation on  a  particle  of  water;  but  the  heterogeneity  of 
solar  light  reduces  this  by  nearly  half,  leaving  as  maximum 
only  the  tenfold  of  gravitation  on  a  water  sphere  of 
0.00016  mm.  diameter.  Since  molecules  fall  far  below  this 
size,  it  appears  that  Maxwell's  law  does  not  apply  to  gases. 
But  here  again  the  corpuscle  vindicates  its  great  import- 
ance in  cosmogonic  theory.  For  the  gases  near  the  sun 
must  be  at  least  partially  ionized,  since  its  light  is  rich  in 
ultraviolet  rays  and  these  provoke  the  radiation  of  ions. 
But  these  ionized  gases,  as  proved  by  the  Wilsons,  are 
readily  beclouded  by  the  vapor  condensing  around  the  ions 
or  corpuscles.  The  drops  thus  formed  must  be  repelled  by 
the  light  pressure,  or  if  too  heavy  must  fall  toward  the 
sun  and  bear  away  with  them  the  negative  electricity,  leav- 
ing the  gas  positively  laden.  Hence  the  great  part  played 
by  electricity  in  cometary  phenomena. 

While  the  second  and  third  of  Bredichin's  groups  of 
comets  are  easily  understood  as  composed  of  hydrocarbon 
particles,  the  first  group  shows  repulsion  nineteen  times  as 
great  as  gravity,  and  the  comets  of  Rooerdam  and  Swift 
(1893  II  and  1892  I)  even  37  and  40.5  times  as  great. 
Such  repulsion  would  require  a  specific  gravity  only  one- 
tenth  that  of  water,  but  it  may  well  be  that  hydrocarbon 
spherelets  subjected  to  expulsion  of  hydrogen  under  intense 
solar  heat  may  be  turned  into  sponge-like  carbon  pellets 
of  the  levity  required. 

The  combined  conceptions  of  the  forward  thrust  of 
light  and  the  universal  radiation  of  corpuscles  give  us  a 


34O  THE   MONIST. 

widely  imaginative  vision  of  the  heavens  above  us.  The 
corona  of  the  sun  with  its  greenish  pearly  light  is  no  longer 
a  mystery.  We  think  of  it  as  composed  of  particles  near 
the  critical  sizes  and  hence  held  suspended  in  the  sky,  like 
the  coffin  of  Mohammed,  between  the  pull  of  gravitation 
and  the  push  of  the  light,  two  forces  obeying  the  same  law 
of  inverse  squares.  Its  tenuity  transcends  all  conceiving, 
it  being  five  million  times  thinner  than  the  head  of  the 
comet,  and  its  whole  mass  if  of  coal  would  be  burned  upon 
earth  in  a  week.  Nevertheless  it  is  something,  and  it 
assures  us  that  an  endless  drizzle  of  still  finer  dust  is 
steadily  poured  out  by  the  sun  into  surrounding  space.  In 
perhaps  six  thousand  billion  years  the  sun  might  dissipate 
itself  entirely.  Unless  time  be  finite  it  would  follow  that 
at  this  rate  all  the  suns  in  the  universe  would  have  dis- 
solved unending  ages  ago  and  vanished  like  the  baseless 
fabric  of  a  dream.  Meantime,  however,  there  has  been 
integration  proceeding  pari  passu  with  disintegration.  Since 
the  earth  in  flight  round  the  sun  sweeps  up  about  twenty 
thousand  tons  of  meteoritic  matter  yearly,  it  is  easy  to 
reckon  that  the  sun  must  catch  some  three  hundred  milliard 
tons  in  the  storm  that  drives  forever  about  him.  These 
meteorites  are  in  fact  the  building  stones  of  the  suns.  Of 
what  are  they  themselves  built?  According  to  Norden- 
skjold,  they  are  woven  together  of  metallic  atom  on  atom, 
the  universal  floating  dust  of  dissolved  or  dissolving  worlds. 
An  awful,  a  tremendous  cycle! 

As  the  sun  thus  inspires  and  expires  matter  like  some 
stupendous  lung,  similarly  it  respires  electricity.  The  cloud 
particles  repelled  by  light-pressure  bear  away  negative 
electric  nuclei  and  leave  a  positive  charge  behind  of  nearly 
250  milliard  coulombs,  enough  to  dissolve  24  tons  of  water. 
As  the  solar  dust  aggregates  into  meteorites  it  dispels  its 
negative  electrons  and  these  are  drawn  in  a  ceaseless  flood 
toward  the  sun  by  its  positive  charge.  The  sun  is  thus  at 


THE  ELECTRONIC  THEORY  OF  MATTER.  34! 

once  a  source  and  a  sink  of  electrons  streaming  into  and 
from  it,  to  and  from  the  uttermost  walls  of  the  world. 

We  look  aloft  into  the  azure  heavens  and  think  to  be- 
hold a  sky  unflecked  by  the  minutest  cloud.  But  reason 
perceives  even  there  an  eternal  whirlwind  of  cosmic  dust 
swirling  around  planets  and  stars  and  darkling  suns.  Well 
for  humanity  that  the  veil  of  this  absorbent  mist  is  flung 
abroad  over  the  whole  heavens!  Else  would  the  dome  of 
the  sky  glow  like  a  furnace,  and  the  countenance  of  crea- 
tion be  withered  and  blasted.  But  not  only  do  these  nebu- 
las and  frozen  stars  shield  the  earth  from  planetary  death ; 
they  are  the  very  guardians  of  the  universe  itself  (accor- 
ding to  Arrhenius)  against  that  heat-death  (Warmetod) 
with  whose  frightful  specter  Clausius  has  so  long  appalled 
the  stoutest  hearts  of  science  and  philosophy.  For  in  his 
second  law  of  thermodynamics  the  German  declared  that 
the  entropy  of  the  universe  tends  to  a  maximum  while  the 
energy  remains  constant.  Now  this  entropy  (of  a  body) 
is  its  total  heat  divided  by  its  absolute  temperature.  It 
measures  the  amount  of  heat  turned  into  a  body  in 
the  process  of  exchange  and  so  rendered  unavailable  for 
outward  effect.  The  one  universally  observed  case  is  that 
bodies  at  unequal  temperatures  tend  to  exchange  heat 
till  they  attain  the  same  temperature,  when  all  effective 
interaction  ceases.  Applying  this  observation  to  the  uni- 
verse, Clausius  declared  that  it  was  tending  to  such 
equilibrium,  which  would  suspend  all  effective  activity, 
and  this  universal  uniform  condition  he  named  heat-death, 
and  taught  that  it  was  inevitable,  merely  a  matter  of 
time  —  a  fearful  piece  of  reasoning,  which  it  seemed 
equally  impossible  to  accept  or  to  refute.  But  the  facts 
were  against  Clausius,  for  least  of  all  men  would  he 
deny  the  past  was  infinite,  hence  the  universe  had  had  time 
to  run  down  infinitely  often,  nevertheless  it  is  manifestly 
not  run  down  but  running  on  still.  Since  it  has  not  met 


342  THE   MONIST. 

the  Wdrmctod  in  the  infinite  past,  neither  need  it  do  so 
in  the  infinite  future.  Maxwell  imagined  an  intelligent 
demon  sorting  out  the  atoms  in  a  uniformly  heated  gas, 
by  opening  the  windows  of  an  invisible  partition  for  the 
passage  of  each  fast-moving  particle  and  shutting  them 
against  the  slow-moving  one,  thus  sifting  them  into  two 
apartments,  the  one  hot  and  therefore  able  to  work  on, 
the  other  cold.  We  know,  however,  o£  no  such  demons, 
though  it  would  be  rash  to  deny  them  all  existence.  But 
on  the  other  hand  molecules  moving  swift  enough  (more 
than  ii  kms.  per  second)  must  tear  themselves  away  from 
the  earth's  attraction,  leaving  behind  their  slower  fellows, 
thus  sorting  out  the  two  classes  after  the  fashion  of  Max- 
well's demons.  In  this  way  perhaps  the  moon  has  grad- 
ually lost  her  atmosphere.  So  too  may  the  nebulas  lose 
fast-flying  molecules  that  wander  into  their  outer  parts 
and  retain  only  the  slower  ones  and  so  remain  or  even 
become  cool.  Meantime  the  fast  flyers  would  be  caught  in 
the  widespread  atmosphere  of  some  condensing  star  and  so 
raise  still  higher  its  rising  temperature.  Whether  or  no 
this  be  the  exact  fashion  in  which  the  universal  activity 
is  maintained,  we  may  be  sure  that  it  is  maintained  in 
some  fashion,  and  the  dreadful  presage  of  universal  heat- 
death  that  has  so  long  oppressed  the  scientific  conscious- 
ness may  now  be  dismissed  as  the  nightmare  of  a  fevered 
dream. 

Still  other  riddles  of  the  heavens  have  yielded  to  the 
divination  of  the  corpuscular  theory.  Since  we  must  now 
think  of  the  sun  as  sprinkling  all  space  with  an  incessant 
shower  of  dust,  clearly  the  earth  too  must  be  thus  sprinkled. 
The  atmosphere  can  no  longer  be  thought  as  imperceptible 
beyond  100  kms.,  but  must  certainly  reach  an  average 
height  of  400  kms.  Were  it  not  for  the  electric  charges 
with  which  the  particles  are  laden,  the  amount  of  sun-dust 
that  could  reach  the  earth  could  hardly  exceed  two  hundred 


THE  ELECTRONIC  THEORY  OF  MATTER.  343 

tons  yearly,  one  one-hundredth  only  of  what  actually 
reaches  us  in  meteors  and  shooting  stars.  On  the  contrary, 
Nordenskjold  reckons  the  cosmic  dust  positively  charged 
that  reaches  the  earth  at  ten  million  tons  yearly,  and 
Chamberlain  advocates  a  planitesimal  theory  that  con- 
siders the  planets  as  mainly  built  up  of  collected  meteors. 
Be  that  as  it  may,  the  significance  of  the  negatively  laden 
particles  in  the  higher  regions  of  our  atmosphere  is  beyond 
question.  Of  the  luminous  effects  of  wide-scattered  dust 
in  the  air,  the  appalling  eruptions  of  Krakatoa  (1883)  and 
Mount  Pelee  (1902),  which  reddened  the  sky  for  months, 
have  furnished  examples.  On  a  far  grander  and  more 
benignant  scale  the  sun  powders  the  upper  air  into  un- 
earthly radiance.  Inhabitants  of  the  polar  regions  have 
the  nightly  vision  of  the  Aurora  Borealis  in  two  forms 
long  confounded  but  now  clearly  distinguished:  the  one 
a  nearly  steady  phosphorescent  gleam  swelling  in  arch  on 
arch  toward  the  apex  of  the  sky;  the  other  consisting  of 
beams,  of  fountains  of  light  spouting  their  torrents  of 
splendor  zenithward  from  the  fluttering  draperies  of  flame 
that  fringe  the  northern  horizon.  The  explanations  of 
these  two  classes  are  of  course  not  quite  the  same,  and  in 
their  detail  one  must  distinguish  between  the  maximal  and 
minimal  years  of  sunspots ;  but  in  general  one  may  say  that 
the  negatively  laden  particles  shot  out  in  perpetual  tempest 
from  the  sun  must  beat  upon  the  atmospheric  envelop  of 
the  earth,  which  is  speeding  on  as  through  a  driving  rain. 
It  is  the  equatorial  regions  that  are  full  exposed  to  this 
tempest.  Great  however  as  is  the  velocity  of  the  particles, 
they  do  not  in  general  pierce  through  this  envelop,  but 
are  caught  as  it  were  in  the  net  of  the  lines  of  the  earth's 
magnetic  force.  Round  these  they  spin  in  descending  con- 
verging spirals  toward  the  poles.  On  impinging  upon 
strata  of  air  as  dense  as  in  a  vacuum  tube  they  act  as  in 
such  tubes  and  exhaust  their  energy  in  the  fitful  gleams 


344  THE  MONIST. 

peculiar  to  cathode  rays.  In  case  however  of  a  maximal 
year  of  sunspots  the  velocity  of  the  projected  particles  is 
enormously  increased  and  they  drive  on  nearly  in  straight 
lines  in  spite  of  the  suction  of  the  lines  of  force  both  in  the 
sun's  and  even  in  the  earth's  magnetic  field.  Hence  the 
Aurora  may  in  these  years  descend  even  toward  tropical 
regions.  It  is  at  once  perceived  that  the  relation  between 
the  two  phenomena  of  sunspots  and  polar  light  must  be 
intimate  and  so  it  is,  but  into  this  there  is  no  time  to  enter. 

Still  another  astronomical  puzzle,  not  however  of  polar 
but  of  tropical  observation,  seems  now  in  fair  way  toward 
solution.  The  inhabitants  of  even  higher  latitudes  may  be- 
hold morning  and  evening  at  the  equinoxes  a  cone  of  light 
rising  from  the  horizon  and  called  zodiacal  from  its  con- 
nection with  the  zodiac.  It  is  certainly  a  luminous  cloud 
of  particles  glittering  in  the  sunlight.  Some  have  thought 
it  to  be  the  last  remnant  of  nebular  dust  still  circling  the 
sun  like  a  ring.  But  according  to  Arrhenius  both  the 
zodiacal  light  itself  and  its  still  more  perplexing  Gegen- 
schein  are  due  to  the  same  corpuscular  storm  that  pours 
steadily  out  from  the  sun  and  down  upon  it  from  all  sur- 
rounding space.  Surely  enough  has  now  been  said  to  show 
that  the  corpuscular  theory  has  the  widest-reaching  astro- 
nomic and  cosmogonic  bearings. 

May  it  not  also  have  biologic  significance?  The  prob- 
lem of  the  origin  of  species  on  our  planet  has  gone  through 
a  variety  of  phases.  Linnaeus  held  that  the  Infinite  Ens 
had  in  the  beginning  created  just  so  many  species,  which 
remained  unchanged  down  to  our  own  day.  This  rigid 
conception  was  shaken  by  Lamarck  and  others  a  century 
ago,  but  again  restored  to  acceptance  by  the  great  author- 
ity of  Cuvier.  Finally  it  went  down  forever  before  the 
researches  of  Darwin  and  the  school  of  evolution.  More 
recently  De  Vries  has  detected  species  in  the  very  act  of 
transvolution,  not  however  through  the  gradual  accumu- 


THE  ELECTRONIC  THEORY  OF  MATTER.  345 

lation  of  infinitesimal  variations,  but  by  finite  leaps  or  mu- 
tations establishing  a  new  "species  in  a  single  generation. 
Meantime  the  deeper  problem  of  the  origin  of  life  on  our 
planet  has  not  been  advanced  toward  solution,  unless  the 
successive  recognition  of  one  proposed  solution  after  an- 
other as  unsatisfactory  may  be  said  to  be  advancing. 
Thinkers  of  the  highest  rank  still  cling  to  the  notion  of 
"spontaneous  generation,"  under  the  compulsion  of  such 
reasoning  as  this :  there  was  a  time,  no  matter  how  remote, 
when  there  was  no  life  on  the  earth;  now  there  is  life; 
therefore  sometime  between  now  and  then  life  began  to 
be.  Others  however  spy  out  another  possibility,  namely, 
that  life  was  imported  from  some  other  planet.  The  illus- 
trious Kelvin  insisted  that  the  maxim  omne  vivum  e  vivo 
was  as  sure  as  the  law  of  gravitation  and  hence  was  driven 
to  maintain  ( 1871 )  the  hypothesis  that  life  had  been  borne 
to  our  planet  on  some  meteorite,  some  disrupted  fragment 
of  another  world.  But  the  difficulties  that  embarrass  the 
development  of  such  a  notion  seem  quite  unconquerable. 
More  acceptable  is  the  modern  idea  of  panspermia,  hinted 
by  Richter  as  early  as  1865.  In  a  word,  this  doctrine  holds 
that  the  germs  of  life  are  scattered  as  spores  throughout 
all  the  deepest  abysses  of  space,  that  they  are  driven  by 
light-pressure  along  the  sunbeams  from  planet  to  planet, 
from  system  to  system,  on  journeys  that  may  last  for  thou- 
sands or  ten  thousands  of  years.  The  intense  cold  of  the 
interstellar  spaces  need  not  chill  them  to  death  since  they 
are  unaffected  by  a  bath  of  liquid  hydrogen  ( — 252°  C). 
They  need  not  dry  out,  they  need  not  suffer  any  destructive 
chemical  change,  as  oxidation,  on  their  solitary  flight  from 
world  to  world,  since  evaporation  and  chemical  processes 
are  suspended  in  that  Lethean  flood.  They  need  not  be 
too  large  for  the  fingers  of  the  light  to  push  before  it,  since 
they  have  been  discovered  having  diameters  between  0.0002 
and  0.0003  mrn.,  and  are  doubtless  often  much  smaller, 


346  THE    MONIST. 

magnitudes  well  within  the  grasp  of  the  sunbeam.  But  how 
could  they  be  lifted  up  into  the  higher  regions  of  the  plan- 
etary atmosphere,  and  there  surrendered  to  the  propulsion 
of  the  waves  of  ether?  Currents  of  air  might  lift  them 
a  hundred  kilometers  from  the  earth,  but  could  never  re- 
lease them  from  the  atmospheric  envelope.  Here  again 
one  must  invoke  the  omnipotence  of  the  electron.  The 
negative-laden  sundust  that  kindles  the  Aurora  fires  in 
the  upper  air  must  also  beat  upon  these  spores  and  charge 
them  with  electricity.  So  charged  they  must  powerfully 
repel  each  other  in  every  direction  and  some  would  be 
launched  outward  into  the  depths  of  ether  and  there  seized 
and  sped  onward  by  the  impulsion  of  the  light.  Is  the 
electric  field  strong  enough  for  this  action?  Assuredly. 
An  electric  field  of  two  hundred  volts  per  meter  would 
suffice.  Such  are  familiar  near  the  earth's  surface,  and  far 
intenser  ones  must  prevail  in  the  regions  of  polar  light. 

Undoubtedly  it  is  a  most  perilous  voyage  upon  which 
such  a  germ  of  life  sets  out  from  the  system  say  of  a 
Centauri  to  that  of  our  own  sun.  The  immense  majority 
of  these  ether-farers  would  almost  certainly  be  lost  and 
perish;  but  here  and  there  some  one  would  arrive  after 
nine  thousand  years  safe  at  the  utmost  borders  of  our  solar 
world  and  there  by  chance  light  upon  some  grain  of  sun- 
dust  in  the  reflection  of  the  zodiacal  light  and  be  borne 
along  therewith  sunward  within  the  planetary  ring  and 
even  into  the  atmosphere  of  some  planet,  as  our  own,  which 
would  seize  upon  it  with  gravitation  and  slowly  drag  it 
to  the  earth.  Perilous  too  would  be  the  landing  on  the 
shores  of  Time,  yet  some  lucky  sailor  would  succeed  and 
so  would  establish  a  form  of  life  upon  this  planet.  Surely 
a  most  tremendous  conception,  striking  wonder  and  awe 
through  the  hardest  heart!  Amazing  too  it  is  to  reflect 
that  this  prodigious  idea,  so  carefully  wrought  out  and 
bulwarked  at  every  point  by  the  adamantine  pillars  of 


THE  ELECTRONIC  THEORY  OF  MATTER.  347 

mathematical  calculation,  should  have  been  anticipated  in 
its  grandest  proportions  by  the  brooding  fancy  of  the  pre- 
Christian  Gnostics!  For  in  the  Naassene  scriptures  pre- 
served to  us  by  the  good  bishop  Hippolytus  in  his  Refutatio 
omnium  haeresium  we  find  creation  described  allegorically 
in  the  parable  of  the  Sower  as  the  sowing  down  of  seeds 
from  the  unportrayable  Godhead.  Moreover  in  the  deep- 
thoughted  Gnostic  Basilides  we  find  repeatedly  the  same 
idea,  and  the  new  knowledge  has  actually  adopted  his  fa- 
vorite technical  term  "Panspermia"  as  its  own,  to  express 
this  most  recent  of  astronomico-biological  ideas.  Life  then, 
at  least  in  its  germs,  is  everywhere  pulsing  and  throbbing 
throughout  the  universe  and  when  the  finger  of  time  points 
to  the  accepted  moment  the  myriad  forms  of  life  leap  into 
being  out  from  the  teeming  womb  of  ether.  We  may  also 
see  that  these  forms  are  probably  nearly  the  same,  at  least 
closely  allied  even  at  opposite  poles  of  the  Milky  Way,  and 
man  may  feel  his  blood  kinship  with  the  tenants  of  the  re- 
motest world.  It  is  impossible  then  to  repress  the  sugges- 
tion that  the  actual  forms  of  life,  being  everywhere  what 
they  are,  have  in  themselves  some  deep-lying  hitherto  un- 
suspected reason  for  being  thus  and  not  otherwise,  some 
reason  profound  as  the  properties  of  numbers  or  the  logical 
necessities  of  the  geometry  of  Euclid. 

May  one  not  even  venture  to  hint  that  the  biological 
import  of  the  corpuscular  theory  is  not  yet  exhausted? 
That  in  the  almost  endless  divisibility  of  the  atoms  into 
sub-atoms  or  electrons  there  inheres  some  undiscovered 
connection  with  the  pangenesis  required  in  theories  of 
heredity  ?  That  the  mysteries  of  Mendelism  and  Mutation 
may  yet  be  illuminated  by  flashes  of  electric  light  as  in  a 
polar  sky  at  midnight?  But  even  were  I  able,  there  is  no 
time  to  pursue  this  thought  further. 

After  all,  however,  what  do  we  know  of  electricity? 
•Lord  Kelvin  declared  that  in  his  age  he  understood  it  no 


348  THE    MONIST. 

better  than  in  his  youth.  The  colossal  theory  we  have  been 
considering  assumes  all  the  fundamental  and  hitherto  in- 
explicable laws  of  electric  action.  In  particular,  besides 
the  ether  and  its  wonderful  properties,  it  assumes  the  law 
of  inverse  squares  as  the  mode  of  all  interaction  of  elec- 
trons. Herein  of  course  it  is  perfectly  justified,  but  the 
mind  cannot  lay  the  importunate  query,  Why  does  this 
interaction  vary  precisely  as  the  inverse  square  of  the  dis- 
tance? Moreover,  since  all  this  action  takes  place  in  the 
uniform  universal  ether  whose  regions  are  distinguishable 
only  by  the  motions  that  affect  them,  the  mind  is  led  irre- 
sistibly to  the  conjecture  that  the  problem  is  ultimately 
hydrodynamical,  that  all  these  electric  phenomena  must 
some  way  be  thinkable  as  movements  in  an  all-pervading 
medium.  Hence  then  the  great  significance  of  the  vortex 
theory  developed  by  Thomson  (Kelvin)  from  the  central 
property  of  vortex  filaments  discovered  by  Helmholtz. 
Kelvin  and  his  disciples  thought  to  recognize  the  indis- 
soluble atom  in  the  indissoluble  vortex-ring  and  imagined 
that  an  exhaustive  doctrine  of  knots  would  yield  a  list  of 
atoms  or  elements.  The  new  knowledge  shows  indeed 
that  the  atom  is  not  indissoluble,  nay,  is  in  some  cases 
actually  dissolving,  but  the  vortex-ring  or  filament  is  not 
thereby  deprived  of  its  scientific  importance.  It  may  yet 
be  that  the  sub-atom  or  electron  is  essentially  a  vortex  in 
ether,  and  the  theoretic  properties  of  such  vortices  may  be 
the  observed  properties  of  electrons.  At  this  point  then 
the  question  arises :  What  is  the  relation  between  the  equa- 
tions of  electricity  and  the  equations  of  hydrodynamics? 
Or  between  the  electromagnetic  and  the  hydrodynamic 
fields?  Or,  finally,  between  the  interactions  of  electric 
units  and  of  hydrodynamic  elements?  It  seems  hard  then 
to  exaggerate  the  moment  attaching  to  such  researches  as 
those  of  the  two  Bjerknes  (father  and  son)  upon  fields  of 
force,  researches  both  experimental  and  mathematical. 


THE  ELECTRONIC  THEORY  OF  MATTER.  349 

They  have  proved  that  the  relation  in  question  is  certainly 
a  close  one,  that  nearly  all  the  elementary  actions  assumed 
in  electromagnetic  theory  may  be  surprisingly  simulated 
by  actual  experiments  on  pulses  in  a  liquid  medium.  True, 
there  presents  itself  a  queer  paradox:  the  hydrodynamic 
field  appears  not  as  the  direct  but  as  the  inverted  image 
of  the  electro-magnetic  field,  attraction  and  repulsion  in 
the  one  answering  to  repulsion  and  attraction  in  the  other. 
But  in  spite  of  this  perversion,  the  results  remain  highly 
interesting  and  point  the  way  along  which  research  must 
follow  till  something  more  satisfying  shall  be  suggested 
or  discovered. 

Meantime  it  is  no  more  important  to  see  clearly  the 
wide  range  and  immense  perspective  of  the  new  knowledge 
than  it  is  to  recognize  unequivocally  its  necessary  limita- 
tions. Even  when  we  suppose  the  hydrodynamic  analogy 
perfect,  even  were  it  possible  to  state  all  the  facts  of  the 
material  and  ethereal  worlds,  in  a  word,  of  the  world  of 
space  and  sense,  in  terms  of  motion  rotational  or  irrota- 
tional  in  a  universal  ether  of  whatever  properties,  even 
though  the  vision  of  the  Laplacian  Intelligence  should  thus 
be  actualized  on  a  scale  far  grander  than  Laplace  himself 
ever  dreamed  of,  it  would  still  remain  true  that  the  real 
problem  of  the  world  was  just  as  far  as  ever  from  solution. 
For  let  it  be  understood  once  for  all  that  this  problem  is 
not  even  to  be  stated  finally  in  terms  of  mass  and  motion, 
the  sole  concepts  available  in  the  corpuscular  or  any  other 
physical  theory.  Mass  and  motion  are  not  ultimates  in 
human  thinking,  no  physical  concepts  can  be  ultimates. 
The  supreme  all-comprehending  fact  of  the  world  is  Mind, 
Soul,  Spirit,  and  the  ultimates  of  all  thinking,  of  all  reality, 
must  be  psychical.  The  physical  world  is  an  idea,  a  sensible 
form,  which  the  mind  constructs  at  every  instant  by  the 
inherent  law  of  its  own  activity.  It  is  a  splendid,  a  glorious 
construct,  a  real  construct,  well  worthy  the  everlasting 


350  THE   MONIST. 

study  and  admiration  of  its  own  creator.  Contemplating 
this  amazing  creation  the  Spirit  beholds  its  own  image  as 
in  a  mirror,  and  it  may  even  explore  the  depths  of  its  own 
being  by  interpreting  backward  its  own  image ;  just  as  the 
mathematician  may  translate  the  analytic  (algebraic)  prop- 
erties of  his  equation  into  geometric  properties  of  the  cor- 
responding locus  and  again  may  interpret  the  geometric 
properties  of  this  locus  into  corresponding  algebraic  prop- 
erties of  the  original  equation.  But  the  equation  is  not  the 
locus,  nor  ever  can  be,  nor  would  it  cease  to  be,  nor  change 
its  properties,  if  the  geometric  construction  were  quite  im- 
possible. Speaking  then  in  allegory  one  may  declare  that 
the  psychical  world  is  a  sublime  equation  of  infinite  degree ; 
the  physical  world  is  its  majestic  geometric  locus,  its  con- 
struct in  terms  of  time  and  space,  mass  and  motion.  Be- 
tween the  two  there  subsists  or  may  subsist  some  one-to-one 
relation.  Let  us  study  the  grand  image  with  unreserved 
admiration  and  with  unflagging  zeal.  But  let  us  never  for- 
get that  after  all  it  is  only  an  image,  a  stupendous  parable. 
Let  us  never  forget  the  great  word  of  Goethe:  Alles  Ver- 
gangliche  ist  nur  ein  Gleichniss.  Otherwise  the  brightest 
achievements  of  physical  and  physiological  research  may 
prove  to  be  only  traps  for  our  unwary  feet.  Otherwise  we 
shall  surely  fall  into  the  pit  of  materialism;  we  shall  mis- 
take the  significant  for  the  significate,  we  shall  see  in  the 
whole  universe  only  an  interplay  of  corpuscles,  and  shall 
talk  with  Cabanis  of  the  brain  secreting  thought  as  the 
liver  secretes  bile.  Such  an  issue  would  be  deplorable  be- 
yond all  words,  it  would  indeed  be  a  bankruptcy  of  science 
absolutely  hopeless.  As  secondaries  electrons  are  invalu- 
able; as  primaries  they  are  absolutely  worthless.  The 
favorite  maxim  of  Sir  William  Hamilton  abides  in  full 
validity : 

"On  earth  is  nothing  great  but  man; 
In  man  is  nothing  great  but  mind." 


THE  ELECTRONIC  THEORY  OF  MATTER.  35! 

But  if  even  the  sublimest  flights  of  physical  speculation 
vouchsafe  us  no  glimpse  beyond  the  veil,  how  shall  we 
ever  lift  it?  What  gaze  of  reason  shall  ever  penetrate  its 
folds?  Who  knows  indeed  that  in  the  nature  of  the  case 
it  can  be  lifted  ?  For  my  part  I  should  be  content  to  think 
it  a  veil  of  Isis.  Is  there  not  a  kind  of  inspiration  in  the 
thought  of  revelation  after  revelation,  forever  and  ever, 
world  without  end? 

''Higher  than  your  arrows  fly, 
Deeper  than  your  plummets  fall, 
Is  the  deepest,  the  Most  High, 
Is  the  All  in  All." 

Chase  without  catch  would  indeed  be  disheartening,  but 
chase  with  nothing  more  to  catch  would  certainly  be  empty 
and  uninviting.  If  the  chase  is  to  be  eternal  and  yet  in- 
spiring, the  quarry  must  be  infinite,  the  supreme  truth  must 
be  forever  approachable  but  never  attainable!  In  the  an- 
cient myth  it  is  said  that  Egeus  concealed  from  Theseus  the 
secret  of  his  birth,  burying  the  evidence  beneath  a  stone 
by  the  seaside.  The  Father  of  heaven  and  earth  has  per- 
haps secreted  somewhere  the  proofs  of  the  origin  and  na- 
ture of  all  things ;  but  upon  the  shore  of  what  ocean,  O  man 
of  science,  has  He  rolled  the  stone  that  hides  them  ?3 

WILLIAM  BENJAMIN  SMITH. 
NEW  ORLEANS,  LOUISIANA. 

3  Adapted  from  Maurice  Guerin's  Lc  Ccntaure. 


PURPOSE  AS  SYSTEMATIC  UNITY. 

i. 

THE  present  investigation  is  undertaken  for  the  sake 
of  the  light  which  it  may  throw  on  the  problem  of 
value.  Assuming  that  value  is  a  function  of  what  may 
broadly  be  termed  "interest,"  it  becomes  imperative  to  get 
at  the  fundamental  or  generic  character  of  this  phenom- 
enon. What  is  that  attitude  or  act  or  process  which  is 
characteristic  of  living  things,  which  is  unmistakably  pres- 
ent in  the  motor-affective  consciousness  of  man,  and  which 
shades  away  through  instinct  to  the  doubtful  borderland  of 
tropism  ?  Both  the  vocabulary  and  the  grammatical  struc- 
ture of  language  provide  for  the  teleological  categories. 
"Purpose,"  "means  and  end,"  "in  order  to,"  "for  the  sake 
of,"  "with  a  view  to" — these  and  many  other  kindred  forms 
of  speech  are  evidently  applicable  to  the  same  context. 
There  is  something  in  our  world  to  which  they  serve  to  call 
attention.  What  is  it? 

I  propose  to  view  the  matter  objectively  rather  than 
introspectively.  What  we  wish  to  discover  is  the  nature 
of  the  thing,  and  not  the  nature  of  the  consciousness  of  the 
thing.  It  is  fair,  I  think,  to  apply  the  analogy  of  mechan- 
ism. One  would  not  think  of  approaching  this  latter  ques- 
tion by  an  examination  of  the  consciousness  of  mechanism. 
Similarly,  purpose  is  supposed  to  be  a  kind  of  happening 
or  chain  of  events  differing  in  its  determination  from  that 
of  mechanism.  It  may  appear  that  consciousness  is  inci- 


PURPOSE  AS  SYSTEMATIC  UNITY.  353 

dental  to  the  purposive  kind  of  determination.  But  in  that 
case  we  should  begin  with  the  process  as  a  whole  and  work 
in.  We  should  not  shut  ourselves  up  in  advance  to  the 
view  that  purpose  takes  place  only  in  the  introspective 
stream  of  consciousness.  We  cannot,  in  other  words,  de- 
termine the  role  of  consciousness  in  purpose  unless  we 
first  take  that  view  of  the  matter  in  which  both  conscious- 
ness and  its  physical  context  are  taken  into  account.  Be- 
havior or  conduct,  broadly  surveyed  in  all  the  dimensions 
that  experience  affords,  can  alone  give  us  the  proper  per- 
spective. We  want  if  possible  to  discover  what  it  is  to  be 
interested,  not  what  it  is  merely  to  feel  interested.  What 
is  implied  in  being  favorably  or  unfavorably  disposed  to 
anything?  It  may  be  that  it  all  comes  to  nothing  more  than 
a  peculiar  quality  or  arrangement  among  the  data  of  intro- 
spection, and  in  that  case  a  structural  psychology  of  feel- 
ing, will,  desire  or  ideation  will  tell  the  whole  story.  But 
such  a  conclusion  would  be  equivalent  to  an  abandonment 
of  the  widespread  notion  that  purposiveness  is  a  kind  of 
determination  of  events  differing  from  their  mechanical 
determination.  The  really  important  claim  made  in  behalf 
of  purpose  is  the  claim  that  things  happen  because  of  pur- 
pose. Are  acts  performed  on  account  of  ends  ?  Is  it  proper 
to  explain  what  takes  place  in  human  or  animal  life,  or  in 
the  course  of  nature  at  large,  by  the  categories  of  teleol- 
ogy? The  most  exhaustive  introspective  analysis  of  the 
motor-affective  consciousness  would  leave  this  question 
unanswered,  and  to  confine  ourselves  to  the  data  which 
such  analysis  affords  would  be  to  prejudge  it  unfavorably. 
It  is,  of  course,  permissible  to  suppose  that  even  though 
a  case  should  be  made  for  purpose  in  a  physical  or  cosmic 
sense,  value  should  be  limited  to  subjective  purpose.  But 
it  is  evident  that  the  question  would  then  be  merely  one  of 
terms.  If  there  are  objective  purposes  as  well  as  sub- 
jective, it  would  be  necessary  to  employ  some  term  to 


354  THE  MONIST. 

designate  the  objects  of  both  attitudes.  It  would  be  im- 
portant to  construe  subjective  purpose  as  a  species  of  this 
broader  genus,  which  would  be  accomplished  best  by  taking 
it  as  a  kind  of  valuing.  Furthermore  objective  or  dynamic 
purpose,  if  there  be  such,  would  be  far  too  important  in  its 
bearing  on  the  special  value-sciences  to  warrant  our  dis- 
regarding it  in  a  general  theory  of  value. 

II.   NEGATIVE  MEANINGS. 

In  looking  for  a  clue  to  the  meaning  of  dynamic  or  ob- 
jective interest,  we  must  free  our  minds  so  far  as  possible 
from  the  purely  negative  associations  which  the  teleological 
terms  have  acquired.  The  case  for  teleology  is  prejudiced 
by  a  suggestion  of  anti-scientific  bias,  or  of  unscientific 
laxity.  This  is  due  no  doubt  mainly  to  the  religious  or 
popular  auspices  under  which  it  has  been  advanced.  The 
teleological  hypothesis  is  often  invoked  to  satisfy  aspira- 
tions, to  flatter  human  nature  or  to  conceal  ignorance.  In 
the  present  controversy  over  vitalism  the  proofs  of  pur- 
posiveness  seem  to  consist  mainly  in  the  indictment  of 
mechanism.  Purpose  is  not  recommended  on  account  of 
its  own  success,  but  on  account  of  the  failure  of  something 
else.  When  so  invoked  it  means  little  more  than  that  un- 
known factor,  x,  which  is  needed  to  complete  the  explana- 
tion of  such  phenomena  as  growth  or  organic  equilibrium. 
It  is  not  surprising  that  vitalists  should  be  regarded  as 
impatient  scientists  who  cannot  wait  for  a  rigorous  experi- 
mental solution,  but  must  needs  invent  an  agency  ad  hoc; 
or  at  best  as  irresponsible  critics  who  remind  plodding 
science  of  its  outstanding  difficulties  without  assisting  in 
the  serious  work  of  overcoming  them. 

The  party  of  teleology  according  to  this  view  is  a  sort 
of  opposition  party  to  the  real  scientists,  who  are  sobered 
by  being  in  power.  It  is  the  function  of  this  opposition 
party  to  challenge  and  censure,  rather  than  to  legislate 


PURPOSE  AS  SYSTEMATIC   UNITY.  355 

and  administer.  It  can  afford  to  be  careless  or  premature 
because  it  is  not  in  office.  For  itself  it  has  no  policy,  but 
confines  itself  to  impeaching  those  policies  of  mechanism, 
determinism,  naturalism  and  experimentalism  which  au- 
thoritative science  is  patiently  executing.  There  is  doubt- 
less a  certain  merit  in  this  two-party  system  in  science. 
But  certainly  for  our  constructive  purposes  there  can  be  no 
virtue  in  a  conception  of  purpose  as  merely  not  something 
else.  If  the  conception  is  to  be  of  any  use  to  us  it  must 
have  a  positive  explanatory  value  of  its  own. 

Nor  is  it  at  all  necessary  to  suppose  that  purpose  is  the 
contradictory  alternative  to  some  other  hypothesis  such  as 
mechanism.  Purpose  must  doubtless  be  different  from 
mechanism  if  it  is  not  to  lose  its  identity  altogether ;  but 
that  it  should  be  incompatible  with  mechanism,  or  observed 
only  in  its  breach,  does  not  follow.  Through  being  thus 
regarded  from  the  outset  as  an  antithesis  to  an  established 
and  universally  credited  theory,  teleology  needlessly  makes 
enemies  for  itself.  There  is  certainly  no  reason  to  suppose 
in  advance  that  teleology  is  less  compatible  with  mechanism 
than  statics  with  dynamics,  or  the  atomic  theory  with  the 
electro-magnetic  theory  of  light.  There  is  certainly  an 
empirical  presumption  to  the  contrary.  A  man  who  goes 
to  his  journey's  end  in  order  to  keep  an  engagement,  does 
not  appear  to  violate  the  law  of  gravitation  in  so  doing. 
Let  us  therefore  endeavor  to  get  the  positive  sense  of  the 
teleological  type  of  explanation,  and  let  us  say  of  its  com- 
patibility or  incompatibility  with  the  mechanical  type  of 
explanation,  that  that  is  as  it  may  be. 

III.   PROVISIONAL  DEFINITION. 

We  start  with  the  popular  supposition  that  there  is  a 
peculiar  and  specific  mode  of  explanation,  which  may  cer- 
tainly be  employed  in  the  case  of  the  rational  conduct  of 
man,  which  may  probably  be  applied  to  lower  forms  of  life, 


356  THE    MONIST. 

which  may  for  speculative  reasons  be  extended  to  the  cos- 
mos as  a  whole,  and  for  which  the  name  is  "purpose." 

i.  Let  us  first  consider  a  case  of  human  conduct.  An 
off-hand  provisional  view  of  this  alleged  mode  of  explana- 
tion is  afforded  by  Socrates's  famous  allusion  to  Anaxag- 
oras  in  Plato's  "Phsedo."  Socrates,  it  will  be  remembered, 
distinguishes  two  ways  of  explaining  his  being  in  prison. 
On  the  one  hand  it  is  to  be  explained  by  reference  to  his 
bones  and  muscles.  But  this,  he  thinks,  would  be  an  in- 
appropriate explanation ;  not  untrue  to  be  sure,  since  bones 
and  muscles  do  supply  the  necessary  "conditions," — but 
not  the  sort  of  explanation  that  touches  the  real  cause  of 
a  mind's  acting.  The  second  and  preferred  explanation 
is  in  terms  of  Socrates's  purpose  of  "enduring  any  punish- 
ment which  the  law  inflicts."  A  mind,  in  other  words,  acts 
for  the  best,  according  to  its  lights.  To  explain  its  action, 
therefore,  it  is  necessary  to  discover  what  it  deems  best, 
and  then  to  construe  the  particular  act  as  an  instance  of 
that  best.  In  the  present  case  it  is  supposed  that  Socrates 
is  actuated  by  the  general  principle  of  submission  to  the 
law,  and  that  he  has  judged  his  remaining  in  prison  to  be 
what  under  the  existing  circumstances  that  principle  im- 
plies. 

Let  us  analyze  the  situation  more  carefully,  lest  we 
omit  any  essential  factor.  In  the  first  place,  there  must  be 
a  general  type  of  action,  such  as  submission  to  law,  of 
which  a  particular  act,  such  as  remaining  in  prison,  may 
be  regarded  as  an  instance.  In  the  second  place,  there  must 
be  an  agent  possessed  of  a  stable  disposition  or  tendency 
to  perform  acts  of  a  certain  class,  under  varying  circum- 
stances. The  particular  performances  will  differ  according 
to  circumstances,  but  they  must  be  consistent  in  some  re- 
spect. Then,  thirdly,  there  must  be  some  determinate  re- 
lation between  the  rule  or  type  of  action  and  the  agent's 
disposition.  But  what  is  this  determinate  relation?  The 


PURPOSE  AS  SYSTEMATIC  UNITY.  357 

simplest  alternative  is  to  suppose  that  the  rule  of  action  is 
identical  with  the  constant  or  consistent  feature  of  the  dis- 
position. Thus  we  might  suppose  that  Socrates  tended 
under  varying  circumstances  to  submit  to  the  law.  But 
this  will  not  do.  For  if  it  should  happen  that  his  remaining 
in  prison  were  as  a  matter  of  fact  not  what  the  law  re- 
quired ;  if  it  should  happen  that  there  had  been  some  error 
in  transmitting  the  commands  of  the  authorities,  or  if  it 
should  turn  out  upon  reflection  that  Socrates's  escape 
rather  than  his  passively  yielding  to  tyrannical  oppression 
was  more  in  keeping  with  his  constitutional  rights,  that 
would  not  disprove  his  purpose  to  submit  to  law.  What 
is  necessary  is  that  Socrates  should  mean  to  submit  to 
law,  or  that  he  should  think  his  act  to  be  a  case  of  sub- 
mitting to  the  law.  The  link  between  the  rule  and  the 
disposition  is  an  act  of  interpretation  or  judgment.  In 
other  words,  one  is  said  to  be  governed  by  a  purpose  M, 
when  M  is  some  generalized  form  of  action,  and  when  one 
is  disposed  consistently  to  perform  what  one  believes 
(whether  correctly  or  mistaken)  to  be  a  case  of  M. 

2.  This,  then,  appears  to  be  what  is  meant  by  purpose 
when  purposiveness  is  imputed  to  the  rational  or  reflective 
procedure  of  man.  Let  us  now  turn  to  what  common  sense 
would  regard  as  a  more  doubtful  case  of  purpose,  the  case 
of  animal  beliavior.  The  differentia  of  animal  behavior 
which  was  first  remarked,  was  the  power  of  self-motion. 
Whereas  an  inanimate  object  merely  submitted  to  motion 
imparted  to  it  from  without  by  impact,  a  living  thing 
seemed  to  be  an  original  source  of  motion.  Associated 
with  this  phenomenon  was  the  relatively  unpredictable 
character  of  the  action  of  living  organisms.  What  they 
did  was  so  far  due  to  internal  and  unobservable  factors 
that  you  could  not  rely  on  their  yielding  in  any  uniform 
way  to  the  operation  of  the  external  forces  that  you  might 
observe  or  apply.  Living  things  had  a  way  of  moving  of 


358  THE  MONIST. 

themselves,  without  any  apparent  cause  which  might  serve 
to  put  you  on  your  guard.  Hence  they  were  said  to  ex- 
hibit "spontaneity." 

It  is  still  customary  to  characterize  living  things  in  this 
way.  Biologists  describe  the  organism  as  "an  active,  self- 
assertive,  living  creature — to  some  extent  master  of  its 
fate."1  But  this  spontaneity  or  self-motion  no  longer  serves 
to  distinguish  living  from  inanimate  things,  owing  to  the 
development  of  the  science  of  energy.  We  should  now 
speak  of  this  apparent  spontaneity  as  a  release  of  stored 
energy.  The  organism  accumulates  chemical  energy  by 
the  process  of  metabolism,  and  then  discharges  it  when 
subjected  to  some  kind  of  stimulation  from  without.  When 
the  discharge  occurs  it  is  out  of  all  proportion  to  the  stimu- 
lation. Indeed  in  some  cases  there  appears  to  be  no  ex- 
ternal stimulus  at  all.  In  any  case  the  internal  factor  is  so 
much  more  important  than  the  external  factor  that  the 
latter  affords  no  safe  basis  for  prediction.  As  organisms 
become  more  elaborate  the  discharge  comes  to  depend  more 
and  more  upon  the  quantity  and  balance  of  its  stored  en- 
ergies and  less  and  less  upon  what  is  done  to  it  from 
without.  But  even  so  this  phenomenon  of  release  or  dis- 
charge does  not  differ  in  principle  from  what  happens  in 
the  case  of  combustion  or  in  the  case  of  the  action  of  high 
explosives.  If  the  behavior  of  living  things  is  spontaneous 
in  this  sense,  there  is  also  "spontaneous  combustion"  in  the 
same  sense.  In  the  one  case  as  in  the  other  we  now  suppose 
that  the  action  would  be  predictable  even  with  the  utmost 
quantitative  precision  if  we  knew  the  internal  organization 
of  the  acting  body,  as  well  as  the  character  and  intensity 
of  the  stimulus.  It  is  merely  a  question  of  the  relative 
preponderance  of  central  over  peripheral  factors. 

Hence  we  are  at  present  inclined  to  look  elsewhere  for 
the  differentia  of  life,  and  to  find  it,  not  in  the  spontaneity 

1  Thomson,  Heredity,  p.  172. 


PURPOSE  AS  SYSTEMATIC  UNITY.  359 

of  action,  but  in  its  direction  toward  something.  The  ex- 
plosion, we  say,  is  blind  and  aimless, — indifferent  to  con- 
sequences; whereas  life  is  circumspect  and  prophetic. 
Forewarned  is  forearmed.  This  is  what  we  mean  When  we 
speak  of  living  things  as  exhibiting  intelligence.  We  do 
not  credit  all  living  things  with  intelligence;  but  we  have 
no  hesitation  in  imputing  it  to  the  higher  forms  of  animal 
life,  and  the  phenomena  of  instinct  and  tropism  have  led 
to  our  imputing  at  least  a  quasi-intelligence  to  the  lower 
animals  and  even  to  plants. 

In  so  far  as  we  impute  intelligence  to  living  things, 
we  feel  the  need  of  explaining  their  action  in  a  peculiar 
way.  The  explosion  is  satisfactorily  accounted  for  as  a 
resultant  of  two  physically  existing  factors,  the  internal 
organization  of  stored  energies  and  the  external  spark  or 
trigger.  But  in  the  case  of  intelligence  it  seems  necessary 
or  at  least  appropriate  to  refer  to  the  sequel,  to  that  which 
is  merely  in  prospect  at  the  moment  when  the  action  occurs. 
Thus  a  dog  moves  rapidly  away,  or  gets  behind  some  inter- 
vening obstacle,  when  his  master  takes  down  the  whip. 
In  so  far  as  this  implies  intelligence  we  think  of  it  not  in 
terms  merely  of  existing  chemical  energy  and  the  light 
impinging  on  the  optic  nerve.  We  take  account  also  of 
what  is  going  to  happen,  namely  the  painful  beating.  We 
say  that  that  also  explains  why  the  animal  is  acting  as  he 
does.  Or  we  say  that  the  animal  is  acting  "in  order  to 
avoid"  the  beating.  But  since  the  beating  which  is  avoided 
does  not  as  a  matter  of  fact  occur,  we  are  thus  appealing 
to  a  factor  which  is  in  some  sense  merely  possible  or  hypo- 
thetical. Over  and  above  the  animal's  power  of  spontane- 
ous motion,  over  and  above  the  external  action  of  the  stim- 
ulus, there  is  some  additional  factor  which  refers  to  this 
mere  possibility  and  which  decisively  determines  the  direc- 
tion which  the  discharge  takes.  I  do  not  mean  to  assert 
that  this  third  factor  cannot  be  traced  to  the  previous  ex- 


360  THE    MONIST. 

periences  of  the  animal.  Probably  it  can ;  and  this  has  led 
comparative  psychologists  to  associate  intelligence  with 
docility,  or  the  capacity  to  "learn  by  experience."  But  that 
is  not  the  point.  However  he  may  have  come  by  it,  the 
animal  is  supposed  at  the  moment  of  action  to  possess  a 
capacity  for  prospectively  determined  action.  He  acts  not 
because  of  what  is  or  has  been  merely,  but  because  of  what 
may  be  by  virtue  of  his  action,  or  what  ivould  be  without 
his  action.  He  acts,  we  say,  from  fear  of  a  painful  whip- 
ping, or  from  hope  of  immunity.  There  is  no  way  of 
describing  either  the  fear  or  the  hope,  without  admitting 
it  to  be  the  fear  or  the  hope  of  something,  which  something 
is  not  upon  the  plane  of  past  or  present  physical  existence 
as  ordinarily  conceived. 

If,  now,  we  put  together  the  results  of  the  analysis  of 
our  two  examples  we  shall  have  a  provisional  view  of 
interested  or  purposive  action.  In  both  cases  there  is  an 
organism  with  certain  accumulated  energies  and  certain 
organized  propensities.  In  both  cases  there  is  a  specific 
external  situation  which  acts  upon  the  organism  and  lib- 
erates the  energies  and  propensities.  So  far  there  is  noth- 
ing to  distinguish  these  cases  from  such  physico-chemical 
analogies  as  I  have  cited.  But  in  both  cases  there  is  a 
third  and  differential  factor  which  constitutes  their  pur- 
posive aspect.  The  act  is  construed  by  the  agent  in  terms 
of  something  ulterior  and  non-existential.  Socrates  judges 
his  act  to  be  of  the  general  type  of  submission  to  law;  to 
the  dog  the  whip  is  a  sign  of  beating  or  pain-to-come,  and 
his  flight  is  a  response  "as  to"  pain.  In  both  cases  the 
agent  views  the  situation  whether  by  inference  or  asso- 
ciation, in  the  light  of  some  aspect  or  relation  that  tran- 
scends given  fact ;  and  his  acting  as  he  does  is  determined 
by  his  viewing  it  as  he  does. 

Jennings  has  termed  this  characteristic  of  behavior 
"reaction  to  representative  stimuli."    "The  sea  urchin.  . .  . 


ITRPOSE  AS  SYSTEMATIC  UNITY.  361 

responds  to  a  sudden  shadow  falling  upon  it  by  pointing 
its  spines  in  the  direction  from  which  the  shadow  comes. 
This  action  is  defensive,  serving  to  protect  it  from  enemies 
that  in  approaching  may  have  cast  the  shadow.  The  re- 
action is  produced  by  the  shadow,  but  it  refers,  in  its  bio- 
logical value,  to  something  behind  the  shadow. 

"In  all  these  cases  the  reaction  to  the  change  cannot 
be  considered  due  to  any  direct  injurious  or  beneficial  effect 
of  the  actual  change  itself.  The  actual  change  merely 
represents  a  possible  change  behind  it,  which  is  injurious 
or  beneficial.  The  organism  reacts  as  if  to  something 
else  than  the  change  actually  occurring.  The  change  has 
the  function  of  a  sign.  We  may  appropriately  call  stimuli 
of  this  sort  representative  stimuli."2 

The  same  general  principle  applies  to  the  higher  organ- 
ism, Socrates.  That  which  releases  his  action  is  a  represen- 
tation. His  friends  come  to  his  prison  and  urge  him  to 
escape.  Their  actions  and  words  are  a  sign  to  him  of 
law-breaking  and  as  such  he  resists  them ;  or  his  presence 
in  prison  represents  to  him  submission  to  law,  and  as  repre- 
senting that,  he  holds  to  it.  Let  us  now  refine  this  notion 
of  interest  or  purpose  by  comparing  it  with  other  notions 
which  approximate  it,  but  in  some  respect  fall  short  of  it 
or  depart  from  it. 

IV.   THE   PATHETIC   FALLACY. 

The  most  familiar  error  regarding  purpose  is  the  so- 
called  "pathetic  fallacy."  It  will  be  worth  our  while  to 
inquire  just  wherein  the  fallacy  lies.  Suppose  that  in  spite 
of  my  most  painstaking  efforts  to  execute  a  powerful  stroke, 
the  golf  ball  rolls  ingloriously  from  the  tee.  I  then  turn 
and  rend,  my  new  driver  or  call  down  maledictions  upon  it. 
I  am  angry  not  with  myself  but  with  it.  I  feel  resentment 

-  H.  S.  Jennings,  Behavior  of  the  Lower  Organisms,  p.  297. 


362  THE    MONIST. 

toward  it  precisely  as  though  it  had  meant  to  spite  me. 
I  virtually  attribute  malice  to  it.  Now  this,  as  my  less 
heated  partner  may  remind  me,  is  unreasonable,  because 
the  golf  stick  really  didn't  mean  it  or  do  it  "on  purpose." 
It  is  true  that  in  effect  the  stick  thwarts  me.  The  stick  is 
a  cause  of  my  displeasure.  But  the  error  consists  in  im- 
puting that  displeasure  to  it  as  a  motive  or  ground  for  its 
action.  In  other  words,  it  is  not  sufficient  for  purposive 
action  that  its  effect  should  occasion  displeasure;  it  is  neces- 
sary that  this  displeasure  as  a  prospective  contingency 
should  determine  the  act.  Or  take  another  example.  Bask- 
ing in  its  warmth,  I  praise  the  sun  and  feel  gratefully  dis- 
posed to  it.  If  I  knew  what  the  sun  liked  I  would  gladly  re- 
ciprocate. This  is  an  innocent  error,  a  kind  of  poetic  license, 
but  error  it  is  none  the  less.  For  I  have  responded  to  the 
sun  as  though  the  pleasure  which  its  rays  were  about  to 
give  me  had  actuated  the  sun  in  shedding  them;  whereas 
this  effect  upon  my  sensibilities  is  accidental  and  in  no  way 
needed  in  order  to  account  for  the  radiation  of  the  sun's 
light  and  heat. 

But  there  is  also  a  positive  implication  in  this  criticism. 
My  own  action  in  each  case  is  purposive.  My  addressing 
the  ball,  or  lying  in  the  sun,  is  to  be  accounted  for  by 
reference  to  the  stroke  or  the  bodily  comfort  that  is  to 
come.  My  error  lies  not  in  employing  such  a  mode  of 
explanation  but  in  misapplying  it.  There  is  a  human  weak- 
ness, doubtless  one  of  the  major  motives  in  religion,  which 
prompts  one  to  extend  to  all  the  agencies  involved  in  any 
event  that  purposive  type  of  determination  which  really 
holds  only  of  one's  own  participation  in  it.  In  the  case  of 
one's  own  agency  the  prospective  sequel  does  account  for 
the  act,  but  in  the  case  of  the  other  contributory  agencies 
this  explanation  is  out  of  place ;  or,  some  but  not  all  antece- 
dent agencies  are  determined  by  the  sequel.  Not  to  dis- 
criminate is  to  commit  the  inverse  of  a  common  fallacy. 


PURPOSE  AS  SYSTEMATIC  UNITY.  363 

It  would  not  be  inappropriate  to  term  this  characteristic 
teleological  error  the  fallacy  of  "ante  hoc  ergo  propter  hoc." 
There  is  a  further  point  which  this  error  brings  to  light. 
In  so  far  as  I  like  it  the  sun's  warming  my  body  is  good. 
The  effect  of  the  sun's  action  is  therefore  good;  and  it 
might  even  be  that  the  sun  "tended"  to  warm  my  body  and 
so  to  do  good.  But  that  is  evidently  not  sufficient  to  make 
the  sun's  action  purposive.  Action  resulting  in,  or  tending 
to,  good  is  not  ipso  facto  purposive  action.  It  would  be 
purposive  only  provided  that  result  were  somehow  ac- 
countable for  the  action.  In  other  words  we  are  forced 
to  recognize  the  essentially  dynamic  character  of  purpose. 
It  is  not  the  quality  of  the  results,  whether  good,  bad  or 
indifferent,  that  implies  the  purposiveness  of  its  antece- 
dents, but  the  function  of  that  result  as  somehow  partici- 
pating in  the  determination  of  the  process. 

V.  PURPOSE  AND  SYSTEMATIC  UNITY. 

i.  Among  the  widely  accepted  notions  of  purpose  or 
interest  which  we  shall  find  it  profitable  to  examine,  the 
next  is  that  which  identifies  purpose  with  systematic  unity. 
This  notion  is  distinguished  by  the  fact  that  it  disregards 
the  time  factor,  or  regards  it  as  accidental.  Purpose  of  this 
sort  may  characterize  the  world  sub  specie  eternitatis.  It 
may  qualify  a  static  whole,  and  appear  in  its  mere  structure 
or  arrangement,  regardless  of  its  origin  or  history.  It  fol- 
lows that  the  purposiveness  of  any  given  reality  may  be 
judged  by  internal  evidence,  even  when  it  is  supposed  that 
the  reality  in  question  was  produced  by  conscious  design. 
A  purposive  object  is  believed,  like  Paley's  watch,  to  ex- 
hibit its  "designedness"  in  its  very  form.  This  formal, 
static  purposiveness  is  identified  with  order,  system, — the 
interrelation  of  parts  in  a  whole.  Let  us  first  consider 
examples,  beginning  with  an  example  in  which  the  time 
factor  is  clearlv  eliminated. 


364  THE    MONIST. 

An  ellipse  is  more  than  a  mere  collection  of  individual 
points;  it  is  a  curve  having  a  distinctive  character  as  a 
whole,  which  may  be  expressed  by  the  equation  x  -\-  y  =  c. 
Every  individual  point  in  the  curve  is  a  value  of  the  vari- 
ables in  this  equation,  and  its  position  is  determined  ac- 
cording to  the  law  by  the  position  of  the  other  points.  Al- 
though the  position  of  each  point  differs  from  that  of  every 
other  point,  there  is  at  the  same  time  a  certain  identical 
character  among  them  all,  namely  the  "x  -\-  y"  character, 
or  the  sum  of  the  distances  from  two  fixed  points  called 
the  "foci."  To  call  this  a  unified  whole  means  that  there 
is  a  definite  whole-character  in  terms  of  which  all  of  the 
constituents  can  be  described.  This  whole-character  is 
the  law  of  the  parts,  prescribes  their  positions,  or,  as  it  is 
sometimes  expressed,  "generates"  them.  In  the  case  of  a 
broken  line  or  a  curve  having  no  equation,  there  is  no 
whole-character  except  the  merely  collective  aspect  of  the 
several  points.  In  that  case  the  parts  are  prior  to  the 
whole,  and  to  speak  of  them  as  parts  of  the  whole  is  there- 
fore circular  or  redundant.  But  in  the  case  of  the  ellipse 
the  whole  is  prior  to  the  parts,  or  comes  first  in  the  order 
of  explanation.  The  parts,  therefore,  are  said  to  be  gov- 
erned by  something  ulterior  to  them.  The  ellipse  does  not 
exist  except  in  so  far  as  all  the  points  are  in  their  proper  po- 
sitions, and  yet  their  being  so  disposed  is  determined  by  the 
nature  of  the  ellipse.  The  ellipse  is  then  said  to  be  the  pur- 
pose which  regulates  the  several  points.  Each  point  is  deter- 
mined by  what  is  necessary  in  order  that  there  shall  be  an 
ellipse. 

Let  us  now  turn  to  examples  in  which  time  figures  as 
one  of  the  internal  factors  of  a  unified  whole.  The  whole 
is  not  in  time,  but  time  is  in  the  whole.  First  let  us  take  an 
example  of  what  is  commonly  regarded  as  mechanism. 
Suppose  a  body  to  be  moving  in  a  straight  line  at  a  uniform 
velocity,  governed  by  the  law  of  inertia.  Although  each 


PURPOSE  AS  SYSTEMATIC  UNITY.  365 

successive  position  of  the  body  is  new,  a  certain  ratio  of  its 
distance-interval  and  its  time-interval  measured  from  any 
previous  position  is  always  the  same.  Its  kinematic  his- 
tory as  a  whole  exhibits  a  definite  character  which  pre- 
scribes what  its  position  must  be  at  each  particular  mo- 
ment. It  may  in  its  actual  behavior  be  construed  as  a  reali- 
zation of  the  principle  of  uniform  velocity.  This  principle 
in  itself  is  a  universal  or  ideal  entity.  It  does  not  exist 
except  in  and  through  the  successive  positions  of  a  moving 
body  which  obeys  it.  And  yet  these  positions  are  them- 
selves somehow  determined  by  it. 

Let  us  take  one  more  example,  one  that  is  less  precise 
but  is  drawn  from  the  context  of  life.  Modern  civilization 
may  be  said  to  possess  a  characteristic  flavor,  which  dis- 
tinguishes it  as  a  form  of  life.  It  is  conditioned  by  the 
co-presence  and  cooperation  of  a  thousand  factors,  such 
as  the  present  phase  of  geological  evolution,  temperate 
climate,  fertility  of  soil,  racial  blend,  cultural  tradition  etc. 
But  these  many  factors  compose  something.  There  is  a 
unique  and  simple  quality  which  somehow  supervenes  when 
all  these  factors  are  aggregated, — a  quality  which  is  iden- 
tical with  none  of  them  and  yet  somehow  takes  them  all 
up  into  itself.  In  terms  of  this  one  quality  we  can  construe 
all  the  various  conditions  as  contributing  this  or  that  to  it. 
Through  it  they  become,  not  so  many  miscellaneous  par- 
ticulars, but  various  aspects  or  phases  of  one  thing.  This 
resultant  quality,  or  Gcstaltsqualitat,  is  their  purpose  by 
reference  to  which  they  are  now  seen  to  be  for  something. 
They  may  now  be  understood  not  merely  severally  but  col- 
lectively. There  is  a  reason  why  they  should  be  together ; 
or,  over  and  above  that  determination  which  accounts  for 
each  by  itself,  there  is  a  determination  which  accounts  for 
each  in  its  relation  to  the  others.  But  this  determination 
springs  somehow  from  a  character  which  does  not  come 
into  existence  until  after  they  are  all  in  place. 


366  THE    MONIST. 

These  examples  serve  to  give  plausibility  to  the  notion 
that  is  now  before  us.  Let  us  analyze  them  more  carefully. 
It  will  be  found,  I  believe,  that  the  notion  of  unity  which 
they  illustrate  is  divisible  into  two  types,  which  I  shall  call 
"ideal"  and  "existential"  unity.  The  first  is  based  on  the 
conception  of  a  universal,  A  universal  unifies  its  instances. 
Furthermore  it  has  this  peculiar  relation  to  any  instance 
of  itself:  it  explains  the  instance,  or  serves  as  a  description 
of  it,  and  in  that  sense  appears  to  be  prior  to  it ;  but  on  the 
other  hand  it  exists,  or  is  exemplified  only  through  the  in- 
stance, and  in  that  sense  appears  to  be  posterior  to  it.  So 
that  a  case  of  a  universal  seems  to  be  something  that  is 
only  through  itself.  Interrelation  is  an  example  of  ideal 
unity.  When  a  number  of  terms  possess  a  mutual  relation 
exclusively,  that  is,  when  they  are  related  among  them- 
selves as  none  of  them  is  related  to  any  term  without,  they 
compose  a  whole.  Or  they  may  all  sustain  a  common  rela- 
tion to  a  term  outside  the  group.  Or  they  may  be  instances 
of  the  same  set  of  universals  where  the  universals  are  them- 
selves interrelated. 

The  second,  or  existential,  type  of  unity  consists  of  the 
convergence  or  fusion  of  many  existences  into  one.  The 
unity  lies  not  in  any  universal  or  set  of  universals  under 
which  many  particulars  may  be  subsumed,  but  in  an  ul- 
terior particular.  Whereas  unity  of  the  first  type  is  intelli- 
gible or  apprehended  by  reason,  unity  of  this  second  type 
is  sensible  or  is  a  matter  of  empirical  fact.  The  several 
particulars  work  together  to  produce  a  singular  result,  or 
blend  into  an  individuality  that  is  directly  felt.  Let  us  in- 
quire, then,  whether  either  of  these  conceptions  of  unity, 
that  of  universality  or  that  of  individuality,  will  serve  as  a 
definition  of  purpose. 

2.  It  is  to  be  noted  at  the  outset  that  purpose  would  be 
an  all-pervasive  feature  of  the  world  we  live  in.  Instead 
of  its  being  the  exception  it  would  be  the  rule.  Instead  of 


PURPOSE  AS  SYSTEMATIC  UNITY.  367 

its  being  a  residual  aspect  of  the  world,  complementary  to 
that  aspect  with  which  the  physical  sciences  have  to  do,  it 
would  coincide  with  that  orderly  and  law-abiding  aspect 
of  nature  of  which  physical  science  has  been  the  principal 
exponent.  Instead  of  its  being  the  antithesis  to  mechanism, 
mechanism  would  itself  supply  the  most  perfect  instances 
of  it.  This  will  doubtless  serve  to  recommend  it  in  the 
judgment  of  those  who  have  a  predilection  for  teleological 
monism.  But  such  philosophers  cannot  escape  the  price  of 
their  easy  speculative  victory.  For  in  so  far  as  a  conception 
is  universal  it  is  relatively  colorless.  To  characterize  the 
world  as  purposive  in  this  general  formal  sense  is  to  say 
nothing  more  than  every  scientist  or  materialist  asserts.  It 
does  not  differ  from  saying  that  it  is  determined  and  in- 
telligible in  terms  of  laws.  Democritus  and  Spinoza  would 
then  be  as  good  teleologists  as  Plato  or  Leibniz.  And  quite 
apart  from  its  philosophical  barrenness  such  a  view  would 
be  wholly  inept  for  the  purpose  of  a  theory  of  value.  It 
would  wholly  disregard  the  peculiar  or  differential  feature 
of  those  phenomena  which  biology,  economics,  ethics  and 
esthetics  study,  and  would  be  of  no  service  whatever  in 
distinguishing  and  coordinating  these  sciences.  Although 
this  pragmatic  objection  might  be  thought  to  justify  our 
dismissing  it,  it  will  be  instructive  to  discover  if  possible 
just  wherein  this  view  fails  to  agree  with  our  provisional 
conception. 

3.  Unity  may  be  thought  to  constitute  purpose,  or  to 
imply  a  purposive  origin.  In  other  words  the  purpose  in 
question  may  be  thought  of  as  internal  to  the  system,  or  as 
external.  When  intelligible  or  ideal  unity  is  thought  of  as 
itself  constituting  purposiveness  it  is  evident  that  the  com- 
mon view  from  which  the  teleological  terms  get  their  initial 
meaning,  is  virtually  abandoned.  Consider  first  the  simple 
relation  of  a  universal  to  its  instance.  A  certain  given 
curve  is,  let  us  say,  an  ellipse.  The  universal  ellipse  gives 


368  THE   MONIST. 

the  curve  its  character,  or  serves  as  a  description  of  it; 
while  on  the  other  hand  the  curve  gives  existence  or  em- 
bodiment to  the  general  nature  ellipse.  There  is  no  para- 
dox here  provided  we  distinguish  the  sort  of  status  which 
a  universal  enjoys  from  the  status  of  existence.  There  is 
a  peculiar  relation  between  a  universal  and  its  instance 
whereby  the  first  qualifies  the  second  and  the  second  real- 
izes the  first.  Now  it  means  nothing  to  say  that  the  curve 
exists  in  order  to  realize  the  ellipse.  It  simply  does  realize 
the  ellipse.  The  ideal  nature  of  the  ellipse  explains  what 
the  curve  is;  but  it  does  not  explain  the  fact  that  the  curve 
exists.  Compare  the  case  of  Socrates  cited  above.  The 
purposiveness  of  Socrates'-s  act  lay  not  in  the  fact  that  it 
was  an  instance  of  submission  to  law,  but  in  the  fact  that  its 
being  such  in  some  sense  accounted  for  its  occurrence.  We 
express  this  by  saying  that  Socrates  performed  the  act  be- 
cause he  deemed  it  such.  In  other  words,  the  particular 
case  of  being  submissive  to  law  which  in  fact  ensued  was  a 
condition  of  its  own  occurrence,  through  being  referred  to 
as  a  hypothetical  possibility  by  the  mind  of  Socrates.  To 
construe  the  curve  similarly  it  would  be  necessary  to  impute 
to  the  curve  as  determining  its  existence  some  reference 
to  the  possibility  of  its  being  an  ellipse ;  which  would  imply 
a  complexity  of  determination  for  which  there  is  here  no 
justification. 

In  the  case  of  existential  unity  or  individuality,  it  is 
admitted  that  a  variety  does  possess  a  unitary  aspect,  but  it 
cannot  be  said  that  any  term  of  the  manifold  exists  for  the 
sake  of  that  unity.  The  peculiar  flavor  which  supervenes 
upon  an  assemblage  of  historical  conditions  is  not  neces- 
sarily accountable  for  any  of  them.  It  is  not  necessary  to 
suppose  that  the  conditions  were  in  any  sense  determined 
by  their  composing  a  unity.  This  would  be  the  case  only 
provided  among  the  determining  factors  of  each  condition 
there  were  one  which  referred  to  the  composite  sequel; 


PURPOSE  AS  SYSTEMATIC  UNITY.  369 

which  might,  of  coursej  be  the  case,  but  could  not  be  argued 
merely  from  the  fact  of  the  supervening  unity. 

The  situation  is  not  altered  if  we  suppose  any  degree 
or  any  combination  of  these  types  of  systematic  unity.  If 
nature  throughout  observes  the  law  of  gravitation,  or  that 
of  the  conservation  of  energy,  so  that  every  bodily  event  is 
an  instance  of  the  same  set  of  interrelated  universals — if 
it  be  possible  to  describe  everything  in  nature  by  one  form- 
ula, this  would  not  in  the  least  imply  that  nature  exists  for 
the  sake  of  realizing  the  formula.  If  the  world  as  a  whole 
should  possess  a  simple  flavor  or  quality  to  which  every 
existence  and  every  event  contributed  an  indispensable 
condition,  this  would  not  in  the  least  imply  that  such  a 
cosmic  quale  determined  its  conditions.  In  short  mere  unity 
as  such,  whether  it  be  a  conceptual  unity  or  a  perceptual 
unity,  does  not  constitute  purpose.  This  does  not  prove 
that  purpose  does  not  involve  unity,  but  only  that  its  dif- 
ferentia must  lie  in  something  else. 

4.  But  it  may  still  be  supposed  that  unity  argues  an  ex- 
ternal agency  of  a  purposive  sort,  that  unity  is  a  product 
of  purpose.  In  the  first  place,  it  is  to  be  observed  that  unity 
furnishes  an  almost  irresistible  opportunity  for  the  pathetic 
fallacy.  There  is  a  strong  human  interest  in  unity,  an  in- 
tellectual and  practical  interest  in  ideal  unity,  and  an  es- 
thetic interest  in  existential  unity.  When  nature  is  found 
to  obey  relatively  simple  laws,  and  so  to  be  predictable  and 
workable,  the  mind  rejoices  and  praises  God.  When  sky 
and  sea  and  land  compose  a  pleasing  landscape,  or  when  a 
thousand  different  conditions  conspire  to  bring  about  the 
existence  of  fuel  or  food,  one  feels  instinctively  grateful. 
And  so  strong  is  the  instinct  that  it  creates  its  own  object. 
But  we  may  dismiss  this  impulse  as  an  amiable  weakness. 
We  have  already  seen  that  the  fact  that  a  state  of  things 
is  an  object  of  interest,  is  no  proof  that  that  state  of  things 
is  owing  to  interest. 


37O  THE   MONIST. 

A  second  argument  for  the  purposive  origin  of  unity 
is  the  argument  from  analogy,  the  argument  that  Paley 
employed  in  the  case  of  the  watch.  A  thing  of  the  type 
which  man  makes  on  purpose  is  presumably  made  on  pur- 
pose, if  not  by  man  then  by  God.  There  is  a  curious  para- 
dox connected  with  this  argument.  Man  is  peculiarly  ad- 
dicted to  making  machines,  or  things  which  work  uni- 
formly and  automatically.  That  being  the  case  those  parts 
of  nature  which  argue  a  purposive  creation  ought  to  be 
those  parts  which  are  most  mechanical,  such  as  the  periodic 
motions  of  the  stars,  or  the  conservation  of  energy.  A 
living  organism  differs  from  the  typical  human  artefact 
just  in  so  far  as  it  is  spontaneous  and  unpredictable;  and 
ought  therefore  to  be  the  last  thing  to  be  attributed  to  a  cre- 
ative will.  As  a  matter  of  fact,  however,  the  mechanical  parts 
of  nature  are  the  originals  of  which  human  artefacts  are 
adaptations  and  imitations.  Machines  are  made  after  the 
analogy  of  nature,  and  t^heir  machine-like  character  is  due 
to  what  they  borrow  from  its  independent  and  self-sufficient 
forces.  Invention  does,  it  is  true,  correlate  these  forces  in 
new  ways ;  but  there  is  nothing  in  the  principle  of  correla- 
tion that  is  new.  One  could  not  look  for  a  prettier  correla- 
tion of  forces  than  that  between  the  centrifugal  and  centri- 
petal forces  of  a  planet  moving  in  an  elliptical  orbit.  The 
fact  is  that  man  can  contrive  for  his  own  ends  physical 
systems  which  resemble  those  which  he  finds  in  nature. 
The  remarkable  or  unaccountable  thing  is  not  that  system- 
atic unity  should  appear  in  the  absence  of  purpose,  but  that 
purpose  should  have  anything  to  do  with  it  at  all.  The  orig- 
inal mechanisms  of  nature  are  relatively  intelligible,  and 
human  artefacts  relatively  doubtful  and  obscure.  Purposive 
origination  is  not  to  be  invoked  as  a  helpful  hypothesis  to 
account  for  a  mystery;  it  is  itself  the  mystery  which  the 
mechanical  laws  of  nature  will  presumably  help  to  solve. 


PURPOSE  AS  SYSTEMATIC  UNITY.  371 

If  the  argument  from  analogy  is  to  be  employed  at  all, 
there  is  more  justification  for  arguing  from  the  case  of 
nature  to  that  of  human  conduct  than  for  arguing  in  the 
reverse  direction.  If  the  hypothesis  of  purpose  is  needed 
at  all,  it  is  needed  to  explain  not  the  existence  of  systematic 
unity  in  the  world,  but  the  peculiar  case  of  human  conduct 
or  animal  behavior. 

Nor  is  the  case  for  the  argument  from  analogy 
strengthened  if  the  emphasis  is  put  on  the  aspect  of  utility. 
A  systematic  unity  which  serves  human  needs  does  not 
require  an  explanation  which  refers  to  these  needs.  The 
periodic  motions  of  the  earth  evidently  provide  the  heat 
and  light  and  intervals  of  rest  without  which  human  life 
would  be  impossible.  Their  utility  exceeds  that  of  any 
man-made  agency.  But  to  suppose  that  they  have  come 
about  for  the  sake  of  this,  is  simply  to  lapse  into  that 
pathetic  fallacy  which  we  have  already  dismissed. 

5.  There  is  one  further  argument  from  unity  which 
deserves  consideration,  the  argument,  namely  which  em- 
ploys the  notion  of  probability.  It  is  argued  that  in  pro- 
portion as  a  coincidence  is  remarkable  it  must  have  been 
designed.  Thus,  for  example,  Professor  Henderson  has 
shown  that  the  physico-chemical  constitution  of  the  natural 
world  is  uniquely  favorable  to  life.  It  constitutes  a  maxi- 
mum of  fitness. 

"The  fitness  of  the  environment  results  from  character- 
istics which  constitute  a  series  of  maxima  —  unique  or 
nearly  unique  properties  of  water,  carbonic  acid,  the  com- 
pounds of  carbon,  hydrogen,  and  oxygen  and  the  ocean — 
so  numerous,  so  varied,  so  nearly  complete  among  all 
things  which  are  concerned  in  the  problem  that  together 
they  form  certainly  the  greatest  possible  fitness.  No  other 
environment  consisting  of  primary  constituents  made  up 
of  other  known  elements,  or  lacking  water  and  carbonic 


372  THE   MONIST. 

acid,  could  possess  a  like  number  of  fit  characteristics  or 
such  highly  fit  characteristics,  or  in  any  manner  such  great 
fitness  to  promote  complexity,  durability,  and  active  metab- 
olism in  the  organic  mechanism  which  we  call  life."3 

The  author  then  goes  on  to  argue  that  "there  is  not  one 
chance  in  millions  of  millions"  that  all  these  properties 
should  simultaneously  occur,  and  that  they  should  be  thus 
uniquely  favorable  to  life,  unless  we  assume  some  general 
law  that  determines  them  so  to  be. 

Now,  in  the  first  place,  this  appears  to  be  a  misuse  of 
the  principle  of  probability.  It  is  not  proper  to  infer  a  law 
from  a  single  simultaneity,  but  only  from  a  succession  of 
simultaneities.  If  the  first  throw  of  a  pair  of  dice  happens 
to  be  a  double-six,  that  does  not  prove  that  the  dice  are 
loaded,  in  spite  of  the  fact  that  the  chances  were  thirty-six 
to  one  against  that  particular  combination.  There  would 
be  ground  for  suspecting  a  partiality  for  double-sixes  only 
provided  in  the  long  run  this  combination  turned  up  more 
frequently  than  once  in  thirty-six  times.  The  general  or 
original  physico-chemical  composition  of  the  cosmos  is 
like  a  single  throw  of  dice ;  the  chances  are  heavily  against 
it,  but  this  proves  nothing  as  to  any  determining  principle 
over  and  above  chance.  It  would  be  possible  to  make  such 
an  inference  only  provided  it  were  possible  to  gather  in  the 
cosmic  elements  and  throw  them  again.  It  makes  no  dif- 
ference whatever  how  heavy  the  odds  are  against  any  par- 
ticular combination,  provided  there  is  only  one  instance  of 
the  combination;  for  it  is  entirely  in  keeping  with  a  com- 
bination's unusual  or  remarkable  character  that  it  should 
occur  first.  In  other  words,  the  principle  of  chance  has  to 
do  with  the  frequency  of  a  combination  and  not  with  its 
place  in  the  series.  Where  the  range  of  alternatives  is 
large  the  first  combination  will  always  be  highly  improb- 

3  I..  J.  Henderson,  The  Fitness  of  the  Environment,  p.  272. 


PURPOSE  AS  SYSTEMATIC  UNITY.  373 

able;  but  this  fact  follows  from  the  principle  of  chance, 
and  cannot  create  a  presumption  against  chance.4 

The  same  reasoning  holds  of  the  "fitness"  of  the  en- 
vironment for  life.  Let  us  suppose  life  to  be  a  constant. 
It  will  then  be  comparable  to  a  die  having  the  same  num- 
ber on  all  of  its  faces.  The  environment,  on  the  other 
hand,  has  millions  of  faces  only  one  of  which  matches  the 
first  die.  That  the  two  should  match  in  any  single  instance 
is  highly  improbable;  the  chances  are  millions  to  one 
against  it.  But  if  it  should  happen  that  there  was  only 
one  trial,  its  happening  to  be  successful  would  prove  noth- 
ing as  to  there  beiag  anything  more  than  chance  at  work. 
Professor  Henderson  insists  that  the  relation  of  fitness 
between  life  and  its  environment  is  reciprocal;  but  he  ap- 
pears to  ignore  this  essential  fact,  that  it  is  the  environment 
which  is  given  once  and  for  all,  while  the  die  of  life  is 
thrown  again  and  again.  It  may  be  argued  that  life  agrees 
with  its  environment  too  often  to  permit  one  to  suppose 
that  on  the  part  of  life  it  is  a  matter  of  chance.  But  nothing 
of  the  sort  can  be  inferred  on  the  part  of  the  cosmic  en- 
vironment because  that  lies  unchanged  upon  the  board. 
The  relation  of  matching  where  one  term  is  cast  once  and 
the  other  repeatedly  is  not  a  reciprocal  relation.  If  the 
matching  is  uniformly  successful,  it  may  prove  that  the 
matcher  is  not  trusting  to  chance,  but  it  proves  nothing  as 
to  the  matched. 

Suppose  that  we  vary  the  illustration.  It  is  a  remark- 
able fact  that  a  given  individual  likes  the  world  just  as 
he  finds  it.  The  world  agrees  with  his  taste.  In  view  of 
the  vast  range  of  possibilities,  the  countless  worlds  that 
would  offend  him,  this  is  prodigiously  improbable.  But  it 
does  not  follow  that  the  world  is  determined  to  please  him. 

4  Bosanquet  makes  this  clear  when  he  says :  "We  have  very  small  ground 
for  being  surprised  at  the  actual  occurrence  of  that  alternative  which  had 
fewest  chances  in  its  favor;  and  absolutely  none  for  being  surprised  at  the 
occurrence  of  a  marked  or  interesting  alternative  which  has  against  it  enormous 
odds."  {Logic,  second  edition,  Vol.  I,  p.  342.) 


374  THE  MONIST. 

That  would  follow  only  provided  the  world  came  up  again 
and  again  according  to  his  taste.  But,  unfortunately  for 
the  argument,  the  world  does  not  come  up  again  and  again, 
but  only  once.  Suppose,  on  the  other  hand,  that  sentient 
beings  come  up  again  and  again  always  liking  the  given 
world.  This,  then,  would  argue  that  the  taste  of  sentient 
creatures  was  somehow  determined  with  reference  to  their 
environment,  and  did  not  originate  independently  of  it. 

Even  this  would  not  prove  purpose.  Suppose  all  the 
impressions  on  a  given  area  of  sand  to  correspond  exactly 
and  uniquely  to  the  feet  of  a  certain  child  that  is  at  play 
in  the  neighborhood.  This  would  presumably  not  be  an 
accident ;  but  would  be  accepted  as  evidence  that  one  of  the 
terms  of  the  fitness  relation,  namely  the  feet  of  the  child, 
was  the  cause  of  the  other,  namely  the  impressions  on  the 
sand.  It  would  be  necessary,  however,  to  distinguish  this 
case  from  the.  relation  between  the  same  child's  feet  and 
the  shoes  in  his  closet.  There  is  fitness  in  both  cases;  and 
in  both  cases  the  fitness  is  determined,  not  accidental.  But 
in  the  latter  case  alone  would  one  say  that  the  fitness  was 
due  to  purpose.  One  would  not  argue  the  purposiveness 
from  the  bare  relation  of  fitness,  or  from  the  non-accidental 
character  of  the  fitness,  but  from  the  peculiar  way  in  which 
the  fitness  was  in  this  case  determined.  The  shoes  in  the 
closet  are  of  a  certain  shape  because  of  being  judged  or 
expected  to  fit  their  owner.  And  this  might  still  be  the 
case  even  though  they  should  as  a  matter  of  fact  fit  very 
poorly. 

6.  We  conclude,  then,  that  purpose  in  the  provisional 
sense  adopted  at  the  outset,  cannot  be  said  to  consist  in  the 
structural  unity  of  any  system  taken  as  a  whole;  nor  can 
it  be  inferred  from  such  a  unity,  as  necessary  to  account 
for  its  uniqueness,  maximal  character,  aptness  or  any  other 
peculiarity.  The  same  condition  of  unity  might  or  might 
not  have  been  due  to  purpose.  It  is  necessary  in  each  case 


PURPOSE  AS  SYSTEMATIC  UNITY.  375 

to  observe  the  actual  course  of  its  coming  into  existence. 
In  other  words,  purpose  is  not  to  be  defined  in  general 
formal  terms,  any  more  than  chemical  reaction.  It  is  not 
the  same  thing  as  determinateness  or  law  in  general.  If 
there  be  such  a  thing,  it  consists  in  a  particular  sort  of 
agency  that  appears  in  some  cases  of  determination  and 
not  in  others.  We  dissent,  then,  from  the  view  that  pur- 
pose is  exhibited  in  all  cases  of  system  and  unity ;  being  ex- 
hibited most  unmistakably  in  those  realms  of  nature  that 
science  has  already  set  in  order,  and  more  doubtfully, 
therefore,  in  the  phenomena  of  life.5  We  agree  with  those 
who  find  purpose  to  be  a  peculiarity  attaching  to  some  parts 
of  the  existent  world,  most  unmistakably  to  the  behavior 
of  man;  purpose  in  the  inorganic  world  being  a  doubtful 
extension  of  a  conception  derived  from  the  datum  of  life. 

RALPH  BARTON  PERRY. 
HARVARD  UNIVERSITY. 

5  I  understand  that  this  latter  is  the  view  to  which  "objective"  idealists 
incline,  as  illustrated  by  the  case  of  Bosanquet.  Cf.  his  "Meaning  of  Teleol- 
ogy," Proceedings  of  the  British  Academy,  Vol.  II :  "The  foundations  of 
teleology  in  the  universe  are  far  too  deeply  laid  to  be  accounted  for  by,  still 
less  restricted  to,  the  intervention  of  finite  consciousness.  Everything  goes  to 
show  that  such  consciousness  should  not  be  regarded  as  the  source  of  teleol- 
ogy, but  as  itself  a  manifestation,  falling  within  wider  manifestations,  of  the 
immanent  individuality  of  the  real.  It  is  not  teleological  because,  as  a  finite 
subject  of  desire  and  volition,  it  is  'purposive.'  It  is  what  we  call  'purposive' 
because  reality  is  individual  and  teleological,  and  manifests  this  character  partly 
in  finite  intelligence,  partly  in  appearances  of  a  far  greater  range  and  scope" 
(pp.  8-9).  This  "individuality  of  the  real"  which  manifests  itself  in  the  larger 
cosmic  and  historical  processes,  where  we  cannot  suppose  it  to  be  designed 
or  commanded  by  any  finite  mind,  would  appear  to  consist  in  systematic  unity 
of  the  sorts  which  we  have  defined. 


THE  ORIGIN  OF  TAOISM.* 

THE  western  world  is  apt  to  regard  Chinese  reflection 
as  predominantly  ethical.  This  is  due  largely  to  the 
fact  that  the  system  of  Confucius  is  taken  as  typical.1  But 
this  view  is  misleading  and  requires  to  be  supplemented. 
In  reality  the  Chinese  mind  is  fundamentally  concerned  for 
the  health  of  the  inner  man,  and  accordingly  it  is  more 
properly  described  as  ethico-spiritual.  This  appears  to  the 
careful  reader  in  the  teachings  of  Confucius  himself,  and 
it  is  notably  true  of  the  mystical  doctrine  of  Lao-tze  and 
his  more  immediate  followers. 

Taoism  is  well  named  after  the  central  principle  (Tao) 
which  pervades  this  system  of  thought.  The  original  mean- 
ing of  the  term  was  "way"  (path),  which  in  the  realm  of 

*  Partial  publication  (Part  I,  revised  and  abridged)  of  thesis  entitled: 
"The  Thought  of  Lao-tze;  its  origin,  content  and  development,"  presented  to 
Northwestern  University  in  partial  fulfilment  of  the  requirements  for  the 
attainment  of  the  degree  of  Doctor  of  Philosophy.  The  whole  will  appear  in 
book  form  in  the  publications  of  the  Open  Court  Publishing  Company. 

1  The  common  view  is  well  seen  in  Grube  when  he  says  ("Die  chinesische 
Philosophic,"  in  Kultur  der  Gegemvart,  I,  v,  p.  66,  2d  ed.,  Berlin  1913)  that 
"iiberhaupt  das  Chinesentum  in  Konfuzius  seine  vollendetste  und  ausgeprag- 
teste  Verkorperung  gefunden  hat.... Will  man  die  chinesische  Kultur  mit 
einem  kurzen  ScMagwort  charakterisieren,  so  wird  man  sie  als  konfuzianisch 
bezeichnen."  This  is  very  misleading.  Confucianism  came  to  be  dominant 
over  Taoism  in  China  partly  because  of  the  royal  edict  of  Wu-Ti  (139-85 
B.C.),  which  exalted  this  thought  at  the  expense  of  all  other,  and  partly 
because  of  the  universal  difficulty  of  popularizing  mysticism  or  adapting  it  to 
institutional  life.  But  while  Confucius  has  had  more  visible  effect  in  China 
the  effect  of  Lao-tze  has  been  more  profound.  "It  is  not  Confucianism  so 
much  as  Taoism  which  has  most  profoundly  influenced  the  Chinese  mind." 
This  statement  by  Chang-Tai-Yen,  a  noted  scholar  and  my  former  revered 
teacher,  I  believe  gives  the  real  truth  of  the  matter,  and  it  should  be  carried 
in  mind  always  in  studying  Chinese  thought. 


THE  ORIGIN  OF  TAOISM.  3/7 

moral  inquiry  came  to  mean  "norm  of  conduct" ;  in  time  it 
was  narrowed  to  mean  "the  rational  principle  in  man,"  and 
then  later  it  was  extended  to  signify  "reason  in  man  and 
reality."  This  transformation  was  brought  to  definite  ac- 
complishment by  the  real  founder  of  the  system,  as  I  be- 
lieve, Lao-tze  (sixth  century  B.  C.),  who  was  concerned 
to  find  a  metaphysical  basis  for  his  ethico-spiritual  convic- 
tions and  to  that  end  hypostatized  the  principle  of  Tao. 
Thus  a  convenient  analogue  in  western  thought  is  Reason 
or  Logos  mystically  conceived.2 

Concerning  the  life  of  the  founder  we  know  very  little 
in  detail,  and  of  his  work  we  have  only  the  Tao-Teh-king 
which  tradition  attributed  to  him.3  Both  the  historicity 
of  Lao-tze  and  the  authenticity  of  his  work  have  been  ques- 
tioned. But  it  is  my  belief  that,  in  the  existing  state  of 
our  data,  these  doubts  have  been  disposed  of  definitively  by 
Carus.4  Certainly  the  proper  procedure  here  is  first  to  at- 

-  The  term  "Tao"  of  course  long  antedates  the  time  of  Lao-tze.  As  early 
as  the  Shu-King  its  ambiguity  is  already  evident,  where  it  means  "way"  (wan- 
tao,  or  "royal  way,"  as  the  norm  to  which  all  should  conform)  and  also 
"rational  part  of  man"  (tao-sin,  or  rational  heart,  as  distinguished  from  jhren- 
sin,  or  human  heart).  Herein  lay  the  germ  for  the  development  from  the 
moral  to  the  definitely  metaphysical.  The  transition  was  therefore  from  "way" 
to  "right  way  of  life,"  to  "life  according  to  reason,"  to  life  in  accordance  with 
the  rational  principle  of  all  reality,  including  man."  It  was  this  last  idea  which 
was  elaborated  by  Lao-tze  in  a  world-view. 

3  The  Tao-Teh-King  is  accessible  to  the  English  reader  in  the  excellent 
translation  by  Carus   (Chicago,  1898;   [rev.  ed. 1913])   where   (pp.  95, 96)   the 
brief  account  of  Lao-tze's  life,  by  Sze-Ma-Chien,  may  also  be  found  in  English 
translation.     This  account  gives  his  place  of  birth,  family,  official  connection 
(custodian  of  the  royal  archives  and  state  historian)  and  relates  an  encounter 
with  Confucius.     "He  practised  reason  and  virtue"  we  are  told,  and  that  his 
teaching  was  directed  to  "self-concealment  and  namelessness."    When  he  fore- 
saw the  decline  of  his  state  he  left  for  the  frontier,  where  the  custom-house 
officer  urged  him  to  write  a  book  before  leaving  his  country.     "Thereupon," 
concludes  the  account,  "he  wrote  a  book  of  two  parts  consisting  of  five  thou- 
sand and  odd  words,  in  which  he  discussed  the  concepts  of  reason  and  virtue. 
Then  he  departed.     No  one  knows  where  he  died."     The  term  Tao-Teh-King 
was  not  employed  before  the  second  century  A.  D.,  but  the  sayings  which  con- 
stitute this  work  were  uniformly  referred  to  Lao-tze  as  their  author.     It  had 
been  customary  to  name  books  after  the  writer. 

4  See  his  admirably  judicious  article,  "The  Authenticity  of  the  Tao-Teh- 
King,"  in  The  Monist,  Vol.  XI,  1901,  pp.  574-601.     It  is  my  belief  that  the 
western  reader  of  Chinese  literature  is  in  danger  of  hasty  conclusions  from  the 
difficulty  of  understanding  the  Chinese  way  of  thinking.     The  Chinese  mind 


378  THE   MONIST. 

tempt  to  justify  the  tradition  before  rejecting  it  because  of 
difficulties  in  the  Tao-Teh-King.  The  determining  factor 
in  this  connection  must  be  a  real  understanding  of  that 
work.  If  it  can  be  viewed  as  a  unitary  whole  produced  by 
a  single  mind,  the  tradition  may  be  taken  as  confirmed 
beyond  question.  In  considering  its  content  systematically 
I  will  hope  to  show  that  this  can  be  done.  For  the  present 
my  concern  is  to  indicate  how  the  thought  of  Lao-tze  can  be 
considered  in  the  historical  continuity  of  Chinese  reflection, 
after  the  manner  of  the  western  treatment  of  the  history 
of  philosophy.  To  that  end  we  must  deal  with  it  as  a 
product  of  preceding  thought  and  immediate  environment 
and  the  genius  of  our  author. 

The  rise  of  a  new  viewpoint  in  the  development  of 
thought  cannot  be  an  entirely  isolated  affair,  however  novel 
the  addition  may  be.  This  may  be  safely  assumed  for  the 
progress  of  Chinese  thought  as  it  is  for  that  of  the  west. 
Hence  one  may  properly  expect  that  a  system  such  as  that 
of  Lao-tze's  in  the  Tao-Teh-King  could  not  have  appeared 
without  a  preceding  development  and  that  accordingly  it 
should  be  studied  in  its  historical  setting. 

The  earliest  Chinese  reflection  centered  in  the  conduct 
of  man  and  is  embodied  in  the  Hong-Fan,  which  dates  back 
to  2205-2198  B.  C.  and  forms  a  part  of  the  Shu-King  (the 
oldest  book  of  China).  Therein  we  find  rules  laid  down  for 

does  not  move  normally  in  the  channels  of  discursive  reasoning  because  it  is 
essentially  intuitionistic.  Insight  rather  than  dialectic  engages  their  attention. 
Hence  the  westerner  may  too  readily  suspect  forgery  in  what  appears  to  be 
nonsense  (cf.  La  Couperie,  Western  Origin  of  Chinese  Thought,  p.  124,  where 
he  shrewdly  observes  this).  The  cautious  reader  will  bear  in  mind  the  con- 
ciseness of  diction  in  the  Tao-Teh-King  as  standing  for  thought  far  deeper 
than  appears,  and  also  that  the  circumstances  of  writing  precluded  fuller 
elaboration,  as  well  as  the  inevitable  errors  of  copyists  where  the  thought  of 
the  text  is  obscure  in  itself.  In  particular  it  is  important  to  pay  due  regard 
to  the  purity  of  style  and  soberness  of  thought  which  signalize  the  Tao-Teh- 
King  in  contrast  with  the  later  works  of  the  school.  A  stream  cannot  rise 
higher  than  its  source,  and  a  forgery  would  necessarily  have  revealed  those 
fantasies  and  vagaries  which  are  so  conspicuous  in  the  writings  of  the  later 
Taoists.  The  unsympathetic  reader  is  apt  to  be  robbed  of  insight  both  for 
seeing  this  obvious  fact  and  also  for  getting  the  real  meaning  at  the  heart  of 
the  perversions  and  aberrations. 


THE  ORIGIN  OF  TAOISM.  379 

the  ordering  of  one's  inner  life,  the  securing  of  proper  bal- 
ance between  conflicting  tendencies  in  one's  nature,  the 
relation  that  subsists  between  man  and  the  natural  world- 
order  as  well  as  that  between  man  and  his  fellows.  In  it 
we  find  too  the  conception  of  the  king  as  the  embodiment  of 
the  eternal  moral  principles,  the  "royal  way"  (wan-tao) 
which  was  conceived  of  as  the  objective  criterion  to  which 
men  should  conform  their  personal  preferences.  And  in 
it  we  find  the  notion  of  Tao  also  as  "rational  part  of  man," 
as  above  indicated.  The  idea  of  Tao  therefore  goes  very 
far  back  in  Chinese  thought. 

In  addition  to  the  Shu-King  there  is  also  the  Yih-King, 
or  "Book  of  Changes."5  Therein  is  outlined  the  first 
Chinese  cosmological  scheme,  as  well  as  an  ethical  doctrine 
based  on  this  cosmology.  It  posits  an  original  principle 
called  Tai-Chi,  the  "Great  Origin,"  and  two  primary  forces 
called  Yin  and  Yang.  It  was  thought  that  the  world  was 
formed  through  the  action  and  reaction  between  these  two 
principles.  A  cosmos  was  regarded  as  possible  only  when 
there  was  a  perfect  balance  between  these  two  basic  ele- 
ments, otherwise  chaos  would  ensue.  The  attendant  eth- 
ical doctrine  centered  in  the  notion  of  moderation.  As  in 
the  objective  order  so  in  man  an  equilibrium  of  opposite 
forces  was  the  aim.  Going  to  extremes  was  regarded  as 
disastrous,  because  contrary  to  the  course  of  nature.  The 
cosmology  and  the  ethics  of  the  Yih-King  were  therefore 
constituent  elements  in  Chinese  reflection  long  before  they 
appeared  in  the  Tao-Teh-King. 

In  addition  to  these  two  sources  there  were  probably 
other  documents  which  were  later  lost,  as  the  quotations 
in  the  Tao-Teh-King  would  indicate.  Moreover,  the  ac- 
counts  of  the  lives  of  ascetics  make  plain  that  from  early 
times  there  had  been  men  who  lived  in  seclusion,  insulated 

6  The  rudiments  of  this  work  were  in  existence  prior  to  the  date  of  the 
Shu-King,  but  were  not  elaborated  until  about  1200  B.  C. 


380  THE   MONIST. 

from  the  currents  of  social  and  political  life.  With  the 
advent  of  the  period  of  storm  and  stress,  at  Lao-tze's  time, 
this  ascetic  spirit  became  much  intensified.  It  took  deep 
hold  on  the  thoughtful  and  serious-minded  men  of  that 
age,  some  of  whom  betook  themselves  to  rural  pursuits 
while  others  moved  about  apparently  without  profession, 
eccentric  and  mysterious  in  behavior. 

In  the  Tao-Teh-King  the  connection  with  the  past  is 
evidenced  by  certain  expressions8  which  indicate  clearly 
a  consciousness  of  debt  to  preceding  thought.  This  has 
long  been  recognized  by  Chinese  scholars  and  has  been 
largely  responsible  for  the  impulse  to  find  the  origin  of 
Taoism  in  reflection  antecedent  to  Lao-tze.  Thus  Hwang- 
ti,  the  legendary  emperor  of  the  Chinese,  has  been  regarded 
as  the  founder  of  Taoism,  though  on  very  meagre  evi- 
dence.7 Again  it  has  been  suggested  that  Lao-tze  was 
simply  the  transmitter  of  wise  sayings  and  proverbs  out 
of  the  past.8  Another  account  makes  Lao-tze  to  have  sat 
under  a  master,  Shan  Yung,  who  was  already  advanced  in 
years.9  Still  another  view  finds  the  origin  of  Taoism  in  the 
Yih-King,  whose  cosmology  and  ethics  bear  so  striking 
a  resemblance  to  those  of  the  Tao-Teh-King.10  In  short, 
Chinese  scholars  have  been  amply  aware  of  a  continuity 
between  preceding  reflection  and  that  of  Lao-tze,  and  the 
connection  is  so  obvious  that  there  is  danger  of  thereby 
overlooking  his  originality.11 

6  Such,  for  example,  as  "The  Ancients  say,"  "The  Poet  says,"  "The  Sage 
says"  and  the  like. 

7  Based  on  the  fact  that  a  passage  of  the  Tao-Teh-King  is  quoted  from  a 
book  attributed  to  Hwang-ti  no  longer  extant.     The  same  passage  is  found 
at  the  beginning  of  the  work  of  Lieh-tze.    The  existence  of  such  a  book  was 
denied  by  Hwai-Nan-tze. 

8  By  Chu-Hsi  (1130-1200  A.D.) 

9  See  Hwai-Nan-tze  (ch.  10)  Lao-tze  "learned  the  lesson  of  tenderness  by 
watching  the  tongue."    The  allusion  is  to  old  age  when  the  teeth  have  fallen 
out. 

10  See  Yih-King,  especially  Books  III,  VI  and  XI,  Engl.  transl.  by  Legge 
(Sacred  Books  of  the  East,  Vol.  XVI). 

11  Cf.  Carus,  op.  cit.,  p.  31,  and  Strauss,  Lao-tze's  Tao-Teh-King,  pp.  Ixiii  ff. 


THE  ORIGIN   OF  TAOISM.  381 

That  Lao-tze  had  free  and  full  access  to  the  literature 
of  his  day  is  sufficiently  attested  by  the  tradition  which 
made  him  custodian  of  the  royal  archives  and  state  his- 
torian. This  included  the  classical  literature  which  has 
survived  and  probably  much  that  has  since  been  lost.12  It 
is  inconceivable  that  a  contact  of  this  kind  should  have 
failed  to  influence  the  development  of  his  thought.  In 
addition  there  were  certain  records  of  the  hermits  or  re- 
cluses who  preceded  him  and  to  whose  general  circle  he  is 
supposed  to  have  belonged.  The  contempt  for  temporal 
goods,  the  effort  to  create  a  world  of  their  own  beyond 
that  of  ordinary  values,  the  spirit  of  thoroughgoing  re- 
nunciation which  characterized  this  group  are  essentially 
the  marks  of  the  thought  of  Lao-tze.  Such  influence  of 
his  predecessors  and  contemporaries  in  thought  must  there- 
fore be  assumed  if  we  are  to  avoid  the  impossible  idea  that 
the  construction  of  Lao-tze  was  wholly  de  novo.13 

Thus  it  is  clear  that  Lao-tze  enjoyed  the  intellectual 
heritage  of  his  age.  But  we  must  recall  that  this  heritage 
reveals  no  such  systematic  character  as  may  be  found  in 
the  Tao-Teh-King.  This  work  is  so  characterized  by  sim- 
plicity and  unity,  it  so  bears  the  impress  of  a  single  indi- 
vidual, that  it  suggests  inevitably  to  the  reader  who  has 
entered  into  its  spirit  a  seamless  fabric  woven  from  the 
deeply  experienced  convictions  of  a  distinct  personality.  One 
must  therefore  assume  some  genius  operative  in  revital- 
izing and  bringing  en  rapport  with  his  age  the  inherited 

12  The  Shi-King,  Yih-King  and  Lih-King  would  have  been  accessible  to 
Lao-tze  in  their  ancient  form  and  not  as  revised  by  Confucius. 

13  The  possibility  of  foreign  influence  in  the  shaping  of  Lao-tze's  thought, 
either  direct  or  indirect,  I  do  not  consider  here.     Where  the  effort  is  made 
(e.  g.,  by  Harlez,  Douglas,  La  Couperie,  Strauss,  Remusat,  in  varying  degrees) 
the  proof  rests  upon  mere  resemblance  in  mystical  or  mythological  or  re- 
ligious conceptions.     Such  procedure  is  too  open  to  the  charge  of  precipitate 
generalization  on  the  basis  of  fancied  resemblance  and  too  hazardous  in  the 
absence  of  supporting  external  evidence  to  win  more  than  doubtful  assent. 
It  may  be  true  that  such  foreign  influence  did  exist  in  fact.     But  the  state  of 
historical  knowledge  is  at  present  entirely  inadequate  to  furnish  satisfactory 
conclusions.     It  seems  therefore  to  me  more  desirable  to  seek  to  account  for 
Lao-tze  by  reference  to  indigenous  conditions. 


382  THE   MONIST. 

thought  of  the  past.  As  Eucken  well  says,  with  the  western 
philosophy  in  mind,  "It  is  not  so  much  the  past  which 
decides  as  to  the  present  as  the  present  which  decides  as 
to  the  past,  and  that  in  accordance  with  this,  our  picture 
of  the  past  continually  changes,  depending  upon  the  spirit- 
ual nature  of  the  present.14  So  in  the  Chinese  constructive 
activity  of  the  sixth  century  B.  C,  for  which  the  historical 
evidence  is  ample,  the  living  present  served  to  stimulate 
and  illuminate  the  obscure  potentialities  of  the  past.  Cer- 
tainly the  writer  of  the  Tao-Teh-King  was  possessed  of  a 
genius  for  illuminating  even  the  homeliest  wisdom  in  the 
literature  and  tradition  at  hand,  and  by  new  insight  into 
the  significance  of  Tao  he  was  enabled  to  unfold  the  possi- 
bilities lying  inherent  in  this  supreme  principle.15 

But  with  all  his  genius  Lao-tze  was  a  part  of  his  age, 
and  hence  he  must  be  considered  in  relation  to  the  con- 
ditions then  prevailing.  What  has  been  so  distinctly  true 
in  the  progress  of  western  philosophic  thought  again  must 
be  taken  to  maintain  in  its  degree  for  the  development  of 
Chinese  thought.  "Philosophy,"  says  Windelband,  "receives 
both  its  problems  and  the  materials  for  their  solutions  from 
the  ideas  of  the  general  consciousness  of  the  time  and  from 
the  needs  of  society.  The  general  conquests  and  the  newly 
emerging  questions  of  the  special  sciences,  the  movements 
of  the  religious  consciousness,  the  intuitions  of  art,  the 
revolutions  in  social  and  political  life — all  these  give  phi- 
losophy new  impulses  at  irregular  intervals,  and  condition 
the  directions  of  the  interest  which  forces,  now  these,  now 
those,  problems  into  the  foreground,  and  crowds  others 
aside  for  the  time  being."1  Here  we  have  the  course  indi- 

14  Main  Problems  of  Modern  Philosophy,  1912,  p.  319. 

15  The  unfolding  of  the  past  by  synthesis  of  the  various  elements  therein 
is  perfectly  familiar  to  the  student  of  western  philosophy  in  its  development. 
It  is  so  much  a  condition  of  progress  in  that  thought  that  its  history  is  replete 
with  illustrations.    I  believe  the  same  may  be  safely  assumed  for  the  develop- 
ment of  Chinese  thought,  however  more  measured  its  progress  is. 

I"  History  of  Philosophy,  1893,  p.  13. 


THE  ORIGIN  OF  TAOISM.  383 

cated  which  must  be  followed  in  our  inquiry  concerning 
the  origin  of  Taoism.  The  rise  of  this  system  of  thought 
must  remain  an  obscure  mystery  unless  we  regard  the  en- 
vironment of  Lao-tze,  in  connection  with  his  heritage  and 
his  genius,  and  seek  to  understand  the  Tao-Teh-King  at- 
tributed to  him  in  relation  to  the  cultural  milieu  in  which  it 
arose.  To  this  we  turn  now. 

The  first  form  of  government  that  Chinese  history  dis- 
closes to  us  may  be  designated  an  elective  monarchy,  in 
the  sense  that  the  successor  to  the  throne  was  chosen  by 
the  nobles  and  ministers.  In,  this  way  Yao  (2357-2255 
B.  C.)  and  Shun  (2255-2205  B.  C.)  came  to  hold  the  im- 
perial scepter.  A  change  came  with  Yu  (2205-2197  B.  C.) 
who  chose  his  own  son  to  succeed  him  and  so  departed 
from  the  established  mode  of  procedure,  and  who  laid  the 
basis  for  the  feudal  system  by  assigning  portions  of  the 
empire  to  members  of  the  imperial  family.  The  exact 
course  of  the  ensuing  development  it  is  impossible  to  fol- 
low. But  with  the  Cheo  dynasty  (1122-249  B.  C)  feudal- 
ism had  become  established  as  a  well-defined  political  in- 
stitution. As  elsewhere  in  political  history  it  consisted  in 
dividing  the  empire  into  fiefs  or  estates  to  be  distributed 
among  the  various  nobles  for  the  purpose  of  consolidating 
the  empire. 

This  feudal  system  worked  well  at  first,  largely  because 
strong  emperors  held  the  scepter  of  state  and  the  fief- 
holders  served  as  a  bulwark  to  the  throne.  But  as  time 
went  on  the  emperors  forgot  the  labors  of  their  forefathers 
and  turned  more  and  more  away  from  the  responsibilities 
of  government  to  the  gratification  of  personal  desires.  As 
a  result  of  this  there  came  about  gradually  a  decline  of  the 
central  power.  The  various  nobles  and  princes,  who  had 
theretofore  been  kept  within  control,  began  to  show  signs 
of  recalcitrancy  and  to  assert  their  own  powers.  This 
process  of  encroaching  upon  the  royal  prerogatives  in- 


384  THE    MONIST. 

creased  more  and  more  until  the  emperor  became  a  mere 
figurehead,  a  negligible  factor,  and  the  real  power  passed 
into  the  hands  of  the  vassals.  With  this  came  a  contest 
among  the  various  states  for  supremacy,  and  so  the  nation 
was  precipitated  into  a  tumultuous  maelstrom  of  strife. 
The  balance  between  the  forces  which  make  law  and  order 
possible  had  become  violently  disturbed.  Factional  strife 
and  internecine  feuds  became  the  order  of  the  day.  There 
ensued  a  reckless  rush  for  self-aggrancjizement  and  an 
unscrupulous  disregard  of  rights,  and  brute  power  replaced 
reason.  To  supplement  the  military  force  the  resources 
of  craft  and  cunning  were  pressed  into  service  and  the 
Machiavellian  attitude  became  dominant. 

Along  with  the  political  decline  went  hand  in  hand  a 
cultural  deterioration.  In  place  of  the  earlier  devotion  to 
peaceful  pursuits,  with  its  cultivation  of  arts  and  literature, 
there  arose  an  exaggerated  emphasis  upon  material  values, 
and  the  earlier  simplicity  was  supplanted  by  sophistication 
both  in  thought  and  in  action.  In  this  rule  of  unreason  the 
complex  social  organization,  which  the  first  few  rulers  had 
succeeded  in  building  up,  had  completely  collapsed.  At  the 
beginning  of  the  dynasty,  especially  in  the  reign  of  Chen- 
Wang  (1115-1079  B.  C),  there  had  been  worked  out  an 
elaborate  system  of  etiquette,  which  in  point  of  complexity 
has  no  parallel  in  history.17  But  in  these  troublous  times 
this  fell  to  pieces.  Neither  the  weaklings  on  the  throne 
nor  the  contending  vassals  were  inclined  to  maintain  this 
elaborate  system.  And  where  all  forces  were  working  for 
disintegration  naturally  all  phases  of  the  social  life  were 
affected.  The  established  ethical  standards  also  broke 
down  to  be  superseded  by  personal  whim  and  caprice.  No- 

17  In  its  ramifications  it  extended  to  every  phase  of  social  and  political 
life.  Regulations  were  prescribed  even  for  such  details  as  mode  of  dress, 
eating,  toilet,  form  of  address,  etc.,  etc.  Its  apparently  immutable  and  fixed 
character  testifies  to  the  genius  for  organization  of  its  author,  Cheo-King,  and 
also  accounts  for  the  fascination  which  it  exercised  over  the  mind  of  Con- 
fucius later  who  felt  impelled  to  refer  to  that  period  as  the  great  age  of  culture. 


THE  ORIGIN   OF  TAOISM.  385 

where  could  universal  rules  of  conduct  be  found,  as  in  the 
ancient  days.  Unjust  laws  were  enacted  in  place  of  the 
old  regulations,  which  had  been  so  nicely  calculated  to  pro- 
mote orderly  life.  The  life  of  the  people  was  made  mis- 
erable by  all  sorts  of  oppressive  measures,  and  their  very 
life-blood  was  drained  that  the  craving  of  the  rulers  for 
military  glory  and  the  excitement  of  the  chase  might  be 
satisfied.  In  short,  a  condition  of  affairs  existed  which 
was  strikingly  similar  to  that  which  prevailed  in  France 
prior  to  the  Revolution.  Wherever  one  looks  he  is  con- 
fronted with  unreason  and  disorder  resulting  from  the 
chase  after  worldly  gain  and  the  abuse  of  power. 

Such  were  the  conditions  prevailing  in  the  world  into 
which  both  Lao-tze  and  Confucius  were  born.  The  in- 
tensity of  the  crisis  may  be  measured  by  the  fact  that 
China's  two  greatest  creative  thinkers  arose  at  this  time, 
after  whom  really  significant  thought  in  that  country  con- 
tinued to  develop.  The  system  of  each  was  adapted  to 
solve  from  its  angle  the  problem  set  by  the  aggravated 
situation.  Confucius  was  conservative  and  sought  to  re- 
construct in  harmony  with  the  past,  while  Lao-tze  was 
radical  and  could  be  satisfied  with  nothing  short  of  com- 
plete breach.  Each  may  be  conceived  as  crystallizing  the 
spirit  and  thought  of  the  type  which  he  represented.  The 
temperament  of  the  one  was  essentially  institutional  and 
accordingly  gave  itself  to  reconstructing  the  social  -fabric 
as  existing,  as  is  abundantly  clear  out  of  all  his  writings. 
The  temperament  of  the  other  was  wholly  impatient  with 
all  temporal  expedients  and  would  not  stop  short  of  per- 
manent peace  in  some  eternal  principle;  this  he  found  by 
reconstructing  the  ancient  Tao  as  supreme  principle  of  men 
and  reality,  as  also  amply  appears  in  his  work,  the  Tao- 
Teh-King. 

The  contrast  between  the  two  men  was  really  antip- 


386  THE   MONIST. 

odal18  and  by  reference  to  it  the  signficance  of  the  genius 
of  our  author  stands  out  at  its  highest.  Confucius  was 
characterized  by  moderation  and  sanity  as  the  world  of 
common  sense  measures  these  qualities.  In  his  efforts  at 
reform  he  confined  himself  wholly  to  the  attainable,  in 
conformity  with  the  sagacity  of  the  plain  man.  His  keen 
sense  for  concrete  reality  forbade  him  to  step  forth  with 
anything  like  a  Utopian  program.  He  clung  to  the  solid 
ground,  with  never  a  desire  to  soar  in  the  empyrean  realms. 
He  was  no  doctrinaire,  no  mere  theorist  in  any  sense,  but 
a  practical  reformer.  To  mend  the  situation  as  he  saw  it 
he  set  about  to  abolish  the  feudal  system,  as  the  source  of 
disintegration,  and  to  reestablish  the  monarchy  with  its 
stabilizing  force  of  imperial  power.  To  counteract  the 
forces  that  were  making  against  law  and  order  he  set  out 
to  revive  the  doctrines  of  the  ancient  sages,  the  system  of 
Cheo-li,  whose  exact  and  rigid  orderliness  very  naturally 
fascinated  his  type  of  mind.  Hence  his  supreme  emphasis 
on  ritual  and  his  belief  that  the  golden  age  lay  in  the  past. 
But  the  spirit  of  Lao-tze  was  radically  different  and 
permitted  no  such  direction  as  that  of  Confucius  in  his  solu- 
tion of  the  problem.  His  genius  impelled  him  to  make  a 
clean  sweep  and  led  him  to  a  very  different  reconstruction. 
He  felt  that  the  world  had  gone  so  far  astray  that  it  could 
not  be  reformed  by  mere  revival  of  ancient  traditions  or 
by  any  other  patching-up  process.  He  demanded  some 
radical  procedure,  a  complete  reversal  of  the  existing  order. 
He  felt  deeply  the  insecurity,  nay,  the  utter  collapse  of 
the  foundations  of  life  in  his  age,  and  he  sought  a  basis  so 

18  This  contrast  is  revealed  in  beautiful  simplicity  in  the  report  by  Sze- 
Ma-Chien  concerning  the  interview  between  the  two  men  (Carus,  Tao-Teh- 
King,  pp.  95,  96).  The  difference  between 'these  men  is  vividly  portrayed  by 
Grube,  who  writes :  "Auf  der  einen  Seite  ein  Mann,  der  mit  beiden  Fiissen 
auf  dem  Boden  der  Wirklichkeit  steht  und....nur  nach  dem  Erreichbaren 
strebt.  Auf  der  anderen  Seite  das  Wolkenkuckucksheim  eines  einsamen,  welt- 
fremden  Denkers.  Dort  zielbewusstes  Streben  nach  staatlicher  Reform  auf 
sittlicher  Grundlage,  hier  asketische  Weltflucht  und  mystisches  Versenken  ins 
ewige  Tao." 


THE  ORIGIN   OF  TAOISM.  387 

secure  that  it  might  not  be  shaken.  Like  Plato,  so  much  in 
this  his  fellow-spirit  of  the  Occident  a  century  and  a  half 
later,  he  regarded  the  present  order  as  wholly  bad  and 
not  to  be  compromised  with.  And  like  Plato  he  turned 
away  from  the  immediate  world  of  strife  to  the  life  of 
reflection  and  contemplation,  to  find  a  world  that  was  char- 
acterized by  the  eternal  as  opposed  to  the  temporal.  But 
more  mystical  than  Plato  he  found  his  solution  by  way  of 
the  inner  life  and  communing  with  nature.  In  revolting 
against  the  existing  order  he  was  driven  to  withdraw  from 
externals  like  the  true .  mystic  that  he  was.  And  in  so 
withdrawing  he  found  within  his  inner  self  the  supreme 
principle  of  his  own  and  of  all  being.  Thus  he  was  enabled 
to  give  new  life  and  meaning  to  the  doctrine  of  Tao,  as  a 
simple  and  unitary  principle  of  all  reality. 

To  this  abiding  principle  he  called  his  wayward  people 
to  return.  In  opposition  to  the  spirit  of  self-assertion  that 
pervaded  the  age,  he  called  for  complete  renunciation,  for 
the  surrender  of  the  petty  ambitions  of  the  ego  which  only 
in  this  way  could  realize  Tao.  Instead  of  the  feverish  and 
scattered  haste  so  common  in  his  day,  he  enjoined  quiet 
confidence  in  the  fundamental  reason  of  the  universal  order. 
Against  over-regulation  and  the  multiplication  of  laws  and 
statutes  he  therefore  went  the  full  length  of  a  doctrine  of 
laissez-faire.  He  would  have  none  of  the  ceremonies  and 
rules  of  etiquette  on  which  the  conciliating  Confucius  later 
laid  such  stress ;  they  were  for  him  the  most  prolific  source 
of  the  great  evil  of  hypocrisy,  being  merely  external  show. 
All  parading  of  virtue  or  even  conscious  well-doing  was 
for  him  an  evil.  He  would  eliminate  all  virtue  except 
that  of  acting  according  to  Tao  and  all  knowledge  save  that 
of  Tao.  This  was  the  sum  and  substance  of  his  thought. 
And  the  solution  which  he  disclosed  to  his  age  as  the  way 
of  salvation  was  an  unfolding  of  this. 

But  Lao-tze  did  not  stand  alone  in  this  negative  atti- 


388  THE    MONIST. 

tude  toward  the  existing  order  of  things.  He  was  a  true 
spokesman  for  those  fellow  spirits  of  his  race  and  day  who 
had  also  turned  unreservedly  to  the  inner  life  for  refuge 
from  the  storm  of  the  external  world.  Like  all  great  lead- 
ers of  thought,  our  philosopher  gave  form  and  body  to 
the  longings  and  aspirations  in  the  minds  of  the  many  less 
gifted.  He  is  clearly  the  concentrated  embodiment  of  the 
quietistic  and  mystical  spirit  of  the  recluses  already  re- 
ferred to.  They  were  in  need  of  a  spokesman  to  make 
clearly  articulate  what  they  felt  and  experienced,  and  this 
was  supplied  by  Lao-tze.  As  the  genius  of  Confucius 
enabled  him  to  serve  as  a  constructive  guide  for  the  type 
he  represented,  so  the  genius  of  Lao-tze  enabled  him  to 
create  for  and  direct  the  less  numerous  but  relatively  wide- 
spread number  of  the  opposite  type.19 

Such  then  was  the  place  of  Lao-tze  in  the  origin  of 
Taoism.  He  was  its  real  founder  because  it  was  his  genius 
that  established  it.  What  had  grown  up  during  long  cen- 
turies and  undergone  gradual  transformation  was  brought 
by  him  to  articulate  formulation  under  the  impulse  of  an 
environment  which  pressed  to  a  mystical  solution.  His 
fundamental  doctrine  was  the  long  familiar  Tao,  but  its 
central  position  and  multiple  unfolding  in  man  and  in 
reality  required  the  labor  of  genius  for  establishment.  Lao- 
tze  was  that  genius,  and  so  Chinese  history  has  recorded 

10  In  the  Confucian  Analects  alone  reference  is  made  to  fourteen  such 
recluses  who  ridiculed  the  effort  to  reform  a  decadent  society.  The  fortuitous 
character  of  these  meetings  and  the  fact  that  they  are  recorded  by  Confucius 
and  his  disciples  attest  how  widespread  the  movement  was.  Strauss  (op.  cit., 
pp.  xliii  ff )  has  suggested  the  ingenious  theory  that  there  was  already  in  ex- 
istence a  Taoist  sect  (Tao-Gemeinde),  whose  teachings  were  reduced  to  writ- 
ing by  Lao-tze.  There  is  no  basis  in  fact  for  this  conjecture,  and  it  overlooks 
the  real  ability  of  Lao-tze.  But  this  is  undoubtedly  a  more  correct  direction 
for  interpretation  than  that  which  disregards  the  widespread  nature  of  the 
movement. 

In  this  connection  it  is  of  great  importance  to  bear  in  mind,  contrary  to 
a  too  prevalent  misconception,  that  even  Confucius  had  to  give  up  his  efforts 
at  reform  in  despair  in  his  later  years,  and  that  he  was  forced  to  content  him- 
self with  the  more  quiet  work  of  teaching  and  of  editing  books.  The  real  sig- 
nificance of  his  work  lay  in  this  preparation  for  posterity  rather  than  in  his 
actual  effect  on  his  own  age. 


THE  ORIGIN  OF  TAOISM.  389 

him  as  one  of  its  two  great  creative  thinkers.  Accordingly 
his  doctrine,  as  set  down  in  the  Tao-Teh-King,  is  found 
to  exhibit  the  unity  and  simplicity  which  signalize  that 
work.  It  is  essentially  the  reaction  to. a  most  difficult  situa- 
tion of  a  born  mystic  who  was  able  to  give  full  expression 
to  the  mysticism  of  his  people.  And  what  has  been  said 
of  the  mystic  in  general  maintains  for  Lao-tze  in  an  emi- 
nent degree.  "What  the  world,  which  truly  knows  nothing, 
calls  'mysticism,'  is  the  science  of  ultimates,.  . .  .the  science 
of  self-evident  reality,  which  cannot  be  'reasoned  about,' 
because  it  is  the  object  of  pure  reason  or  perception."20 
Herein  is  contained  the  key  to  the  true  understanding  of 
Lao-tze's  work. 

KING  SHU  Liu. 
NANKING  UNIVERSITY. 

20  Quoted  from  Patmore  by  Underbill  (Mysticism,  4th  ed.,  1912,  p.  29). 


THE  CONTRIBUTIONS  OF  PARACELSUS  TO 
MEDICAL  SCIENCE  AND  PRACTICE. 

THERE  appears  to  be  little  doubt  as  to  the  real  value 
of  many  specific  contributions  of  Paracelsus  to  med- 
ical knowledge  and  practice,  although  competent  author- 
ities differ  widely  as  to  the  extent  and  character  of  his 
influence  upon  medical  progress.  It  may  be  admitted  that 
his  vigorous  assaults  upon  the  degenerate  Galenism  of 
his  day  were  effective  in  arousing  an  attitude  of  criticism 
and  questioning  which  assisted  greatly  the  influence  of 
other  workers  whose  labors  were  laying  less  sensationally 
but  more  soundly  the  foundation  stones  of  scientific  medi- 
cine. 

Vesalius,  often  called  the  founder  of  the  modern  science 
of  anatomy,  and  Pare,  the  "father  of  surgery,"  were  both 
contemporaries  of  Paracelsus,  though  their  great  works 
appeared  only  after  the  death  of  Paracelsus.  The  "Greater 
Surgery"  of  Paracelsus  had  appeared  nearly  thirty  years 
before  Pare's  classical  work  and  had  passed  through  sev- 
eral editions,  and  it  is  said  that  Pare  acknowledged  his  in- 
debtedness to  Paracelsus  in  the  preface  to  the  first  edition 
of  his  work.1 

Admitting  that  none  of  the  medical  treatises  of  Para- 
celsus has  the  scientific  value  of  the  works  o^his  great 
contemporaries,  it  should  nevertheless  not  be  forgotten 

1  Cf.  Stoddart,  The  Life  of  Paracelsus.    London,  1911,  p.  65. 


THE  CONTRIBUTIONS  OF  PARACELSUS. 

that  his  work  may  have  had  an  influence  for  progress  in 
his  own  time  much  greater  than  its  present  value  in  the 
light  of  later  knowledge.  Dr.  Sudhoff  records  some  nine- 
teen editions  of  the  "Greater  Surgery"  by  the  close  of  the 
sixteenth  century,  in  German,  French,  Latin  and  Dutch 
languages,  and  other  works  of  his  shared  in  somewhat  less 
degree  in  this  popularity. 

The  disapproval  and  hostility  of  the  universities  and  the 
profession  toward  Paracelsus  should  not  be  permitted  to 
mislead  us  into  underrating  his  influence,  as  it  may  be  re- 
called that  both  Vesalius  and  Pare  also  suffered  from  this 
hostility.  Vesalius  was  denounced  by  his  former  teacher 
Sylvius  as  an  insane  heretic  and  his  great  work  on  anatomy 
was  denounced  to  the  Inquisition.  Though  he  was  not 
condemned  by  that  body  his  professorship  at  Padua  be- 
came untenable,  and  he  was  forced  to  return  to  his  native 
city  Brussels  and  is  said  to  have  become  a  hypochondriac 
as  the  result  of  his  persecutions. 

Pare  was  more  successful  in  maintaining  his  profes- 
sional position  through  official  support  though  the  faculty 
of  the  University  of  Paris  protested  his  tenure  of  office. 

The  history  of  medical  science  and  discovery  has  been 
the  subject  of  more  thorough  study  than  most  of  the  nat- 
ural sciences,  and  a  number  of  competent  critics  of  early 
medical  history  have  estimated  the  place  of  Paracelsus  in 
the  development  of  various  departments  of  that  science. 
From  such  sources  may  be  best  summarized  the  contribu- 
tions of  Paracelsus. 

Thus  with  respect  to  surgery,  Dr.  Edmund  Owen  in 
the  Encyclopaedia  Britannica  (eleventh  edition,  article 
"Surgery")  says: 

"The  fourteenth  and  fifteenth  centuries  are  almost  en- 
tirely without  interest  for  surgical  history.  The  dead  level 
of  tradition  is  broken  first  by  two  men  of  originality  and 
genius,  Paracelsus  (1493-1541)  and  Pare,  and  by  the  re- 


392  THE    MONIST. 

vival  of  anatomy  at  the  hands  of  Andreas  Vesalius  (1514- 
64)  and  Gabriel  Fallopius  (1523-1562),  professors  at 
Padua.  Apart  from  the  mystical  form  in  which  much  of 
his  teaching  was  cast  Paracelsus  has  great  merits  as  a 
reformer  of  surgical  practice.  It  is  not,  however,  as  an 
innovator  in  operative  surgery,  but  rather  as  a  direct  ob- 
server of  natural  processes  that  Paracelsus  is  distinguished. 
His  description  of  hospital  gangrene,  for  example,  is  per- 
fectly true  to  nature ;  his  numerous  observations  on  syphilis 
are  also  sound  and  sensible;  and  he  was  the  first  to  point 
out  the  connection  between  cretinism  of  the  offspring  and 
goitre  of  the  parents." 

So  also  Proksch,2  the  historian  of  syphilitic  diseases, 
credits  Paracelsus  with  the  recognition  of  the  inherited 
character  of  this  disease  and  states  that  there  are  indeed 
but  few  and  subordinate  regulations  in  modern  syphilis- 
therapy  which  Paracelsus  has  not  enunciated.  Iwan  Bloch 
also  attributes  the  first  observation  of  the  hereditary  char- 
acter of  that  disease  to  Paracelsus.3  That  Paracelsus  de- 
voted so  much  attention  to  the  consideration  of  these  dis- 
eases was  evidently  made  a  subject  of  contemptuous  criti- 
cism by  his  opponents  as  may  be  inferred  from  his  replies 
to  them  in  the-Paragranum.* 

"Why  then  do  you  clowns  (Gugelfritzen}  abuse  my 
writings,  which  you  can  in  no  way  refute  other  than  by 
saying  that  I  know  nothing  to  write  about  but  of  luxus 
and  venere?  Is  that  a  trifling  thing?  or  in  your  opinion 
to  be  despised?  Because  I  have  understood  that  all  open 
wounds  may  be  converted  into  the  French  disease  (i.  e., 
syphilis),  which  is  the  worst  disease  in  the  whole  world,— 
no  worse  has  ever  been  known, — which  spares  nobody  and 
attacks  the  highest  personages  the  most  severely — shall  I 

2  Quoted  by   Baas,   Geschichtliche  Entunckelung   des   arztUchen   Standcs, 
p.  210. 

3  Neuburger  und  Pagel.    Handbuch  der  Geschichte  der  Medisin,  III,  403. 

4  Paracelsus,  Opera,  Strassburg  Folio,  1616.     I,  201-2. 


THE  CONTRIBUTIONS  OF  PARACELSUS.  393 

therefore  be  despised  ?  Because  I  bring  help  to  princes,  lords 
and  peasants  and  relate  the  errors  that  I  have  found,  and 
because  this  has  resulted  in  good  and  high  reputation  for 
me,  you  would  throw  me  .down  into  the  mire  and  not  spare 
the  sick.  For  it  is  they  and  not  I  whom  you  would  cast 
into  the  gutter." 

Dr.  Bauer"  calls  attention  to  the  rational  protest  of 
Paracelsus  against  the  excessive  blood-letting  in  vogue  at 
the  time,  his  objections  being  based  on  the  hypothesis  that 
the  process  disturbed  the  harmony  of  the  system,  and  upon 
the  argument  that  the  blood  could  not  be  purified  by  merely 
lessening  its  quantity. 

''For  the  healing  art  and  for  pharmacology  in  connec- 
tion therewith,"  says  Dr.  E.  Schaer  in  his  monograph  on 
the  history  of  pharmacology,6  reform  is  in  the  first  instance 
attached  to  the  name  of  Theophrastus  Paracelsus  whose 
much  contested  importance  for  the  rebirth  of  medicine  in 
the  period  of  the  Reformation  has  been  in  recent  times 
finally  established  in  a  favorable  direction  by  a  master 
work  of  critical  investigation  of  sources ....  But  however 
much  overzealous  adherents  of  the  brilliant  physician  may 
have  misunderstood  him  and  have  gone  at  times  beyond 
the  goal  he  established,  nevertheless  the  historical  con- 
sideration of  pharmacology  will  not  hesitate  to  yield  to 
Paracelsus  the  merit  of  the  effective  repression  of  the  me- 
dieval polypharmacy  often  as  meaningless  as  it  was  super- 
stitious and  to  credit  him  with  having  effectively  called 
attention  to  the  pharmacological  value  of  many  metallic 
preparations  and  analogous  chemical  remedies." 

Dr.  Max  Neuburger7  thus  summarizes  the  claims  of 
Paracelsus  to  a  place  in  the  history  of  the  useful  advances 
in  medicine: 

r>  G esc hie lite  der  Aderlasse,  1870,  p.  147. 

6  Neuburger  and  Pagel,  II,  565-6. 

7  Neuburger  and  Pagel,  II,  36ff. 


394  THE  MONIST. 

"Under  the  banner  of  utilitarianism  Paracelsus  ren- 
dered the  practical  art  of  healing  so  many  services  that  in 
this  respect  his  preeminent  historical  importance  cannot 
he  doubted.  In  bringing  chemistry  to  a  higher  plane  and 
in  making  the  new  accessory  branch  useful  to  medicine,  in 
comprehending  the  value  of  dietetics,  in  teaching  the  use 
of  a  great  number  of  mineral  substances  (iron,  lead,  cop- 
per, antimony,  mercury),  and  on  the  other  hand  in  teaching 
the  knowledge  of  their  injurious  actions;  in  paving  the 
way  to  the  scientific  investigation  of  mineral  waters  (de- 
termination of  the  iron  contents  by  nut  galls),  in  essen- 
tially improving  pharmacy  (with  his  disciples  Oswald  Croll 
and  Valerius  Cordus)  by  the  preparation  of  tinctures  and 
alcoholic  extracts.  . .  .he  has  achieved  really  fundamental 
merit  for  all  time." 

It  was  also  no  unimportant  service  that  Paracelsus 
rendered  to  medical  science  in  attributing  to  natural  rather 
than  to  the  mystical  influence  of  devils  or  spirits  such 
nervous  maladies  as  St.  Vitus'  dance.  It  is  doubtful  per- 
haps if  his  influence  in  this  direction  was  very  immediate 
upon  contemporary  thought,  at  least  if  we  may  judge  from 
the  sad  history  of  the  trials,  tortures  and  executions  of 
witches  during  a  century  after  the  activity  of  Paracelsus. 

Doubtless  also  the  fantastic  character  of  the  philosophy 
of  Paracelsus  itself  served  to  diminish  the  effect  of  his 
sounder  and  saner  thought. 

A  distinguished  student  of  the  history  of  science,  An- 
drew D.  White,  thus  characterizes  the  services  of  Para- 
celsus in  this  direction.8 

"Yet  in  the  beginning  of  the  sixteenth  century  cases  of 
'possession'  on  a  large  scale  began  to  be  brought  within  the 
scope  of  medical  science,  and  the  man  who  led  in  this  evo- 
lution of  medical  science  was  Paracelsus.  He  it  was  who 
first  bade  modern  Europe  think  for  a  moment  upon  the 

8  History  of  Warfare  of  Science  and  Theology,  II,  139. 


THE  CONTRIBUTIONS  OF  PARACELSUS.  3Q5 

idea  that  these  diseases  are  inflicted  neither  by  saints  nor 
demons,  and  that  the  'dancing  possession'  is  simply  a  form 
of  disease  of  which  the  cure  may  be  effected  by  proper 
remedies  and  regimen.  Paracelsus  appears  to  have  escaped 
any  serious  interference;  it  took  some  time,  perhaps,  for 
the  theological  leaders  to  understand  that  he  had  'let  a  new 
idea  loose  upon  the  planet/  but  they  soon  understood  it 
and  their  course  was  simple.  For  about  fifty  years  the  new 
idea  was  well  kept  under,  but  in  1563  another  physician, 
John  Wier  of  Cleves,  revived  it  at  much  risk  to  his  position 
and  reputation." 

An  interesting  thesis  maintained  by  Paracelsus  was  the 
doctrine  that  every  disease  must  have  its  remedy.  The 
scholastic  authorities  had  pronounced  certain  diseases  as 
incurable,  and  they  were  accordingly  so  considered  by  the 
profession.  Rejecting  as  he  did  the  ancient  authorities, 
Paracelsus  naturally  enough  rejected  this  dogma  as  neces- 
sarily true.  Manifestly  also  he  believed  that  he  himself 
had  with  his  new  remedies  effected  cures  of  certain  of  these 
diseases,  though  he  makes  no  pretension  to  be  able  to  cure 
all  diseases.  The  history  of  medical  thought  and  discus- 
sion shows  that  this  thesis  of  Paracelsus  was  a  frequent 
subject  of  partizan  debate  during  the  century  after  Para- 
celsus. 

Paracelsus  sustains  his  thesis,  however,  not  by  the 
method  of  modern  science — upon  evidence  of  experiment 
and  observation — but  by  the  philosophical  or  rather  meta- 
physical argument  of  its  a  priori  reasonableness  in  the 
divine  purpose,  and  by  his  interpretation  of  the  doctrines  of 
Christ. 

"Know  therefore  that  medicine  is  so  to  be  trusted  in 
relation  to  health — that  it  is  possible  for  it  to  heal  every 
natural  disease,  for  whenever  God  has  entertained  anger 
and  not  mercy,  there  is  always  provided  for  every  disease 
a  medicine  for  its  cure.  For  God  does  not  desire  us  to  die 


396  THE   MONIST. 

but  to  live,  and  to  live  long,  that  in  this  life  we  may  bear 
sorrow  and  remorse  for  our  sins  so  that  we  may  repent  of 
them."8 

"There  is  yet  another  great  error  which  has  strongly 
influenced  me  to  write  this  book, — namely,  because  they 
say  that  diseases  which  I  include  in  this  book  are  incurable. 
Behold,  now,  their  great  folly:  How  can  a  physician  say 
that  a  disease  is  incurable  when  death  is  not  present ;  those 
only  are  incurable  in  which  death  is  present.  Thus  they 
assert  of  gout,  of  epilepsy.  O  you  foolish  heads,  who  has 
authorized  you  to  speak,  because  you  know  nothing  and 
can  accomplish  nothing?  Why  do  you  not  consider  the 
saying  of  Christ,  where  he  says  that  the  sick  have  need  of 
a  physician?  Are  those  not  sick  whom  you  abandon"  T 
think  so.  If  then  they  are  sick  as  proven,  then  they  need 
the  physician.  If  then  they  need  the  physician,  why  do 
you  say  they  cannot  be  helped?  They  need  the  physician 
that  they  may  be  helped  by  him.  Why  then  do  you  say 
that  they  are  not  to  be  helped?  You  say  it  because  you 
are  born  from  the  labyrinth  [of  errors]  of  medicine,  and 
Ignorance  is  your  mother.  Every  disease  has  its  medicine. 
For,  it  is  God's  will  that  he  be  manifested  in  marvelous 
ways  to  the  sick."1 

This  is  obviously  setting  dogma  against  dogma,  and 
opposing  to  scholasticism  the  methods  of  scholasticism. 
Yet  that  this  dictum  of  Paracelsus  was  not  without  in- 
fluence upon  contemporary  thought  is  evidenced  by  a  pas- 
sage in  the  writings  of  Robert  Boyle  in  the  century  follow- 
ing.11 

"Though  we  cannot  but  disapprove  the  vainglorious 
boasts  of  Paracelsus  himself  and  some  of  his  followers, 
who  for  all  that  lived  no  longer  than  other  men,  yet  I  think 

9  Paracelsus,  Liber  de  religione  perpetua.     Sudhoff,  Versuch  eincr  Kritik, 
etc.,  II,  415. 

10  Par.,  Op.  I,  253.    "Die  erste  Defension." 

11  Boyle's  Works,  Birch's  ed.,  I,  481. 


THE  CONTRIBUTIONS  OF  PARACELSUS.  397 

mankind  owes  something  to  the  chymists  for  having  put 
some  men  in  hope  of  doing  greater  cures  than  have  been 
formerly  aspired  to  or  even  thought  possible  and  thereby 
engage  them  to  make  trials  and  attempts  in  order  thereto. 
For  not  only  before  men  were  awakened  and  excited  by 
the  many  promises  and  some  great  cures  of  Arnaldus  de 
Villanova,  Paracelsus,  Rulandus,  Severinus,  and  Helmont, 
many  physicians  were  wont  to  be  too  forward  to  pronounce 
men  troubled  with  such  and  such  diseases  as  incurable  and 
rather  detract  from  nature  and  art  than  confess  that  these 
two  could  do  what  ordinary  physick  could  not,  but  even 
now,  I  fear,  there  are  but  too  many  who  though  they  will 
not  openly  affirm  that  such  and  such  diseases  are  absolutely- 
incurable,  yet  if  a  particular  patient  troubled  with  them  is 
presented,  they  will  be  very  apt  to  undervalue  (at  least) 
if  not  deride  those  who  shall  attempt  to  cure  them." 

His  rational  consideration  and  treatment  of  wounds 
and  open  sores  is  worthy  of  note.  Instead  of  the  customary 
treatment  of  closing  up  by  sewing  or  plastering,  or  cov- 
ering them  with  poultices  and  applications,  he  advocated 
cleanliness,  protection  from  dirt  and  "external  enemies." 
and  regulation  of  diet,  trusting  to  nature  to  effect  the  cure. 
"Every  wound  heals  itself  if  it  is  only  kept  clean."12 

There  is  no  doubt  that  Paracelsus  enjoyed  a  consider- 
able reputation  as  a  skilful  and  successful  practitioner,  and 
there  is  contemporary  testimony,  as  well  as  his  own  state- 
ments, to  show  that  he  was  frequently  sent  for  even  from 
long  distances  to  treat  wealthy  and  prominent  patients 
whose  maladies  had  baffled  the  skill  of  the  Galenic  phy- 
sicians. 

It  is  of  course  true  that  popular  reputations  of  phy- 
sicians are  not  always  the  true  measure  of  ability  even  in 
our  day.  Nevertheless  there  seems  little  reason  to  doubt 
in  spite  of  the  assertions  of  hostile  critics  of  his  time,  that 

12  Cf.  Helfreich  in  Neuburger  and  Pagel,  III,  p.  15. 


398  THE    MONIST. 

with  his  new  remedies,  his  keen  observation,  and  his  un- 
usually open  mind,  he  was  indeed  able  to  afford  relief  or 
to  effect  cures  where  the  orthodox  physicians  trammeled 
by  their  infallible  dogmas  were  unsuccessful.  That  his 
new  methods  sometimes  did  harm  rather  than  good  is  quite 
possible.  That  would  naturally  be  the  result  of  breaking 
radically  new  paths.  And  an  independent  empiricism — a 
practice  founded  upon  experiment  and  personal  observation 
seems  to  have  been  his  practicce  and  his  teaching,  "Expe- 
rentia  ist  Sciential  It  seems  probable  that  in  his  dealings 
with  the  sick,  his  fantastic  natural  philosophy  was  rather 
subordinated  to  a  native  common  sense  and  practical  logic. 
As  stated  by  Professor  Neuburger  (op.  cit.,  II,  35),  "We 
see  in  Paracelsus.  ..  .the  most  prominent  incorporation 
of  that  enigmatic,  intuitive,  anticipative  intelligence  of  the 
people,  which  drawing  upon  the  unfathomable  sources  of 
a  rather  intuitive  than  consciously  recognized  experience, 
not  infrequently  puts  to  shame  the  dialectically  involved 
reasoning  of  scholasticism." 

Paracelsus  has  indeed  clearly  expressed  his  opinion 
that  theories  should  not  be  permitted  to  dominate  the  prac- 
tice of  the  physician. 

"For  in  experiments  neither  theories  nor  other  argu- 
ments are  applicable,  but  they  are  to  be  considered  as  their 
own  expressions.  Therefore  we  admonish  every  one  who 
reads  these,  not  to  oppose  the  methods  of  experiment  but 
according  as  its  own  power  permits  to  follow  it  out  without 
prejudice.  For  every  experiment  is  like  a  weapon  which 
must  be  used  according  to  its  peculiar  power,  as  a  spear  to 
thrust,  a  club  to  strike, — so  also  is  it  with  experiments.  . .  . 
To  use  experiments  requires  an  experienced  man  who  is 
sure  of  his  thrust  and  stroke  that  he  may  use  and  direct 
it  according  to  its  fashion."1 

That  he  endeavored  to  keep  an  open  mind  toward  the 

»  Chir.  Bucher,  Fol.  1618,  pp.  300-301. 


THE  CONTRIBUTIONS  OF  PARACELSUS.  399 

symptoms  of  his  patients,  not  too  much  governed  by  pre- 
conceived dogmas,  is  also  indicated  in  his  defense  against 
certain  attacks  of  his  opponents  in  which  they  accuse  him 
of  not  at  qnce  recognizing  symptoms  and  treatment: 

"They  complain  of  me  that  when  I  come  to  a  patient, 
I  do  not  know  instantly  what  the  matter  is  with  him,  but 
that  I  need  time  to  find  out.  It  is  indeed  true  that  they 
pronounce  judgment  immediately;  their  folly  is  to  blame 
for  that,  for  in  the  end  their  first  judgment  is  false,  and 
from  day  to  day  as  time  passes  they  know  less  what  the 
trouble  is  and  hence  betake  themselves  to  lying,  while  I 
from  day  to  day  endeavor  to  arrive  at  the  truth.  For  ob- 
scure diseases  cannot  be  at  once  recognized  as  colors  are. 
With  colors  we  can  see  what  is  black,  green,  blue  etc.  If 
however  there  were  a  curtain  in  front  of  them  we  could 
not  recognize  them.  . .  .  What  the  eyes  can  see  can  be 
judged  quickly,  but  what  is  hidden  from  the  eyes — it  is 
vain  to  grasp  as  if  it  were  visible.  Take,  for  instance  the 
miner;  be  he  as  able,  experienced  and  skilful  as  may  be, 
when  he  sees  for  the  first  time  an  ore,  he  cannot  know  what 
it  contains,  what  it  will  yield,  nor  how  it  is  to  be  treated, 
roasted,  fused,  ignited  or  burned.  He  must  first  run  tests 
and  trials  and  see  whither  these  lead.  . .  .Thus  it  is  with 
obscure  and  serious  diseases,  that  so  hasty  judgments  can- 
not be  made  though  the  humoral  physicians  do  this."14 

Admitting  the  value  of  the  positive  contributions  of 
Paracelsus  to  medical  knowledge  and  practice,  the  net 
value  of  the  reform  campaign  which  he  instituted  is  vari- 
ously estimated  by  historians  of  medicine.  For  it  must  be 
remembered  that  Paracelsus  fought  against  dogmas  in- 
trenched in  tradition,  by  dogmas  of  his  own.  To  the  fan- 
tastic theories  of  the  Greek-Arabian  authorities  he  opposed 
many  equally  fantastic  theories.  That  by  his  assault  upon 
the  absurdities  and  weaknesses  of  the  Galenic  medicine  of 

14  Of.  foi,  I,  262.    (Die  siebente  Defension.) 


4OO  THE    MONIST. 

his  time  he  paved  the  way  for  greater  hospitality  to  new 
and  progressive  ideas  is  unquestionable,  but  that  by  this 
assault  he  also  did  much  to  discredit  the  valuable  elements 
as  well  as  the  corruptions  of  ancient  medical  achievements 
is  also  true.  It  is  very  difficult  to  balance  justly  the  pro- 
gressive and  the  reactionary  influences  he  exerted  upon 
the  progress  of  medicine,  and  naturally,  therefore,  author- 
ities differ  upon  this  question.  Thus  Neuburger  (op.  cit.) 
appreciates  the  value  of  the  accomplishments  of  Paracelsus, 
yet  doubts  that  he  is  to  be  considered  as  a  reformer  of 
medicine  in  the  sense  that  was  Vesalius  or  Pare,  that  is, 
he  laid  no  foundation  stones  of  importance  and  the  real 
value  of  much  of  his  thought  required  the  later  develop- 
ments of  modern  scientific  thought  for  its  interpretation. 
His  aim  was  to  found  medicine  upon  physiological  and 
biological  foundation  but  the  method  he  chose  was  not  the 
right  method,  and  his  analogical  reasons  and  fantastic  phi- 
losophy of  macrocosm  and  microcosm  were  not  convincing 
and  led  nowhere.  The  disaffection  and  discontent  with 
conditions  in  medicine  produced  by  his  campaign,  can, 
thinks  Neuburger,  hardly  be  called  a  revolution.  That 
was  to  come  later  through  the  constructive  work  of  more 
scientific  methods. 

In  a  similar  vein  Haeser  (op.  cit.)  remarks  "Scarcely 
ever  has  a  physician  seized  the  problem  of  his  life  with 
purer  enthusiasm,  served  it  with  truer  heart,  or  with 
greater  earnestness  kept  in  view  the  honor  of  his  calling 
than  the  reformer  of  Einsiedeln.  But  the  aim  of  his  scien- 
tific endeavors  was  a  mistaken  one  and  no  less  mistaken 
was  the  method  by  which  he  sought  to  attain  it." 

A  recent  writer,  Professor  Hugo  Magnus,15  presents 
a  more  critical  point  of  view : 

"We  must  then  summarize  our  judgment  to  this  effect, 
that  Paracelsus  keenly  felt  the  frightful  corruption  which 

15  Hugo  Magnus,  Paracelsus  der  Ucberarzt.    Breslau,  1906. 


THE  CONTRIBUTIONS  OF  PARACELSUS.  4OI 

medicine  and  the  investigation  of  nature  suffered  from  the 
hands  of  the  Scholastics,  but  that  he  did  not  understand 
how  to  penetrate  to  the  causes  of  this  condition  of  his 
science.  Instead  of  seeking  in  the  scholastic  system  the 
root  of  this  medical  degeneration,  he  believed  that  it  must 
.be  found  exclusively  in  the  healing  art  of  the  ancients.  And 
thus  he  sought  to  shatter  in  blind  hatred  all  that  existed, 
without  being  in  position  to  be  able  to  replace  the  old  theory 
he  maligned  by  a  new  and  better  concept  of  nature  and 
medicine.  So  Paracelsus  wore  away  in  unclear  struggling, 
his  bodily  and  mental  energy,  and  lived  indeed  as  a  re- 
former,— a  medical  superman,  in  his  own  imagination,  in 
his  own  valuation,  but  not  in  the  recognition  of  his  own 
times,  nor  in  the  judgment  of  posterity." 

"If  therefore  I  can  find  no  relationship  between  the 
general  methods  of  medicine  to-day  and  the  Theophrastic 
concept  of  nature,  nevertheless  our  supercolleague  must 
be  considered  in  an  essentially  limited  respect,  to  be  sure, 
as  the  pioneer  in  certain  modern  points  of  view.  He  was 
the  first  to  attempt  the  consideration  of  the  phenomena  of 
organic  life  in  a  chemical  sense,  and  I  do  not  need  to  em- 
phasize that  he  thereby  paved  the  way  to  a  very  powerful 
advance  in  our  science.  In  this  respect  was  Paracelsus  a 
reformer,  here  he  has  pointed  new  paths  in  the  valuation 
of  pathologic  phenomena  as  well  as  in  therapy,  even  if  here 
also  he  has  theorized  enough  and  allowed  his  neo-Platon- 
ism  to  play  him  many  a  trick." 

By  discarding  and  condemning  all  the  ancient  author- 
ities, thinks  Magnus,  Paracelsus  assailed  not  only  the  cor- 
rupted Galenism  of  his  time  but  did  much  to  discredit  the 
positive  achievements  of  the  Greeks,  and  although  the  orig- 
inal Greek  authorities  were  not  the  then  prevailing  texts, 
they  were  at  least  accessible  in  newly  translated  versions, 
and  the  good  in  them  might  have  been  incorporated  and 
built  upon  by  Paracelsus  if  he  had  possessed  the  scientific 


4O2  THE   MONIST. 

point  of  view.  To  the  extent  of  his  influence  in  this  direc- 
tion Paracelsus  was  therefore  an  opponent  rather  than  a 
promoter  of  the  progress  of  medical  science.  "Through 
his  irrational  theories  he  gave  impulse  to  all  sorts  of  mis- 
taken notions  among  his  followers,  so  that  the  wildest 
vagaries  existed  among  the  Paracelsists  of  the  succeeding 
century." 

The  above  will  serve  to  illustrate  the  trend  of  modern 
critical  judgment  of  Paracelsus  as  a  reformer  of  medicine. 

However  estimates  may  vary  as  to  the  extent  of  the 
influence  of  Paracelsus  as  a  reformer  of  medicine,  credit 
must  certainly  be  given  him  as  a  forceful  agent  in  the 
downfall  of  the  scholastic  medical  science  of  his  time.  The 
real  reform  in  medical  science,  its  establishment  upon  a 
basis  of  modern  scientific  method,  was  not  the  work  of  his 
century  nor  of  the  century  to  follow.  Indeed  it  may  not 
be  too  much  to  say  that  that  great  reform  was  mainly  the 
work  of  the  nineteenth  century,  and  was  made  possible  only 
through  the  patient  labors  of  many  investigators  in  the 
domains  of  physics,  chemistry,  anatomy,  and  biology. 

If,  however,  we  cannot  claim  for  Paracelsus  the  un- 
challenged place  of  the  reformer  of  medicine,  we  may  at 
least  recognize  in  him  an  earnest,  powerful,  and  prophetic 
voice  crying  in  the  wilderness. 

J.  M.  STILLMAN. 

LELAND  STANFORD  JUNIOR  UNIVERSITY. 


THE  ORIGIN  OF  THE  MUTATION  THEORY. 

AP  the  time  when  Darwin  published  his  book  on  the 
Origin  of  Species  biological  science  was  in  a  very 
different  condition  from  what  it  is  now.  Hardly  ten  years 
had  elapsed  since  Schleiden  and  Schwann  discovered  the 
fundamental  law  that  all  living  organisms  are  built  up  of 
one  or  more  ordinarily  almost  innumerable  cells. 

Mohl's  contention  that  protoplasm  is  the  essential  and 
in  fact  the  only  living  part  of  the  cell  is  almost  contempo- 
raneous with  Darwin's  book  (1849  and  1851).  The  pres- 
ence of  a  nucleus  within  the  cells  began  to  be  recognized. 
Hereditary  problems  were  almost  only  discussed  by  breed- 
ers. 

The  Textbook  of  Botany  by  Julius  Sachs  appeared  in 
1868;  it  was  the  first  to  introduce  into  botany  really  scien- 
tific methods.  When  I  was  a  student  at  the  University  of 
Leiden  (1866-1870)  systematic  and  descriptive  morpho- 
logical studies  prevailed.  Microscopical  study  of  tissues 
was  new  and  cytology  had  hardly  reached  us.  Under  these* 
conditions  a  student  interested  in  the  causal  relations  of 
the  phenomena  of  life  naturally  turned  his  mind  to  physics 
and  chemistry.  The  prominent  question  of  those  days 
was  the  validity  of  physical  and  chemical  laws  in  the  living 
body.  The  idea  dawned  upon  us  that  this  question  chiefly 
related  to  the  protoplasm  but  hardly  needed  a  proof  for  the 
cell  walls  and  the  tissues  built  up  of  them. 

Once  convinced  that  the  phenomena  of  life  are  regu- 


404  THE   MONIST. 

/ 

lated  by  the  protoplasm  we  naturally  looked  for  methods  of 
studying  this  relation.  Many  different  ways  presented 
themselves,  and  among  these  four  seemed  to  me  the  most 
promising.  They  were  the  study  of  respiration,  of  galls, 
of  osmosis  and  of  variability.  I  tried  all  of  them  and  at 
the  end  chose  the  last.  Respiration  was  the  source  of 
energy;  it  was  a  phenomenon  common  to  animals  and 
plants,  and  one  of  the  main  links  which  connected  both 
kingdoms  in  our  knowledge  at  that  time.  I  devoted  many 
years  to  its  study,  chiefly  in  a  comparative  way,  and  chose 
it  for  the  subject  of  my  inaugural  address  when  I  was 
called  to  the  chair  of  plant  physiology  in  the  University  of 
Amsterdam  (1878). 

But  galls  seemed  to  promise  far  more.  They  are  built 
up  of  the  ordinary  qualities  of  the  plants  combined  in  a 
new  way  to  fit  the  requirements  of  their  insects,  and 
this  combination  is  brought  about  under  the  influence  of 
some  stimulus  given  off  by  the  insect.  To  discover  the 
nature  of  these  stimuli  and  the  laws  by  which  they  so  effec- 
tively change  the  growth  of  the  tissues,  seemed  to  me  a 
scope  worth  the  devotion  of  a  whole  life.  I  made  a  large 
collection  of  galls,  in  search  of  the  species  which  \vould 
be  the  most  appropriate  to  attack  this  line  of  research. 
I  concluded  for  those  of  the  willows,  belonging  to  the 
genus  Nematus.  But  at  that  period  I  met  with  Mr.  M.  W. 
Beyerinck  who  was  far  beyond  me  in  the  study  of  the  life 
history  of  the  galls,  and  so  I  left  this  pathway.  I  have, 
however,  read  a  course  upon  galls  and  their  bearing  on  the 
broad  problems  of  biology  about  every  third  year  from  that 
time  on. 

The  study  of  osmosis  and  of  the  turgidity  of  the  cells 
led  to  the  discovery  of  the  semi-permeable  membranes  of 
the  protoplasm  and  their  significance  for  growth  and  move- 
ments as  well  as  for  the  study  of  isotonic  coefficients  and 
the  determination  of  atomic  weights,  as,  e.  g.,  in  the  case 


THE  ORIGIN  OF  THE  MUTATION  THEORY.  405 

of  the  sugar  raffinose.     But  its  promise  of    elucidating 
hereditary  questions  diminished  with  every  new  discovery. 

In  1880  I  started  a  course  on  variability.  I  had  been 
interested  in  this  question  chiefly  by  making  a  herbarium 
of  monstrosities,  and  monstrosities  were  at  that  time  almost 
all  we  knew  of  variability.  Moreover  I  had  visited  the 
celebrated  agriculturist  W.  A.  Rimpau  at  Schlanstedt  in 
Saxony  and  stayed  repeatedly  for  some  weeks  on  his  estate 
in  order  to  study  his  selection  of  cereals  and  sugarbeets. 
This  induced  me  to  take  up  a  thorough  study  of  agricul- 
tural and  horticultural  selection  and  I  soon  found  that 
Darwin's  books  were  the  best  guides  for  this  literature. 
Especially  from  the  pamphlets  of  Vilmorin,  Verlot  and 
Carriere  I  took  a  large  part  of  the  facts  for  elaboration  of 
my  lessons. 

I  read  this  course  every  second  year  from  1880  to  1900, 
and  each  time  introduced  into  it  the  principles  and  methods 
which  I  found  in  the  literature.  This  consisted  partly  in- 
rare  pamphlets  which  I  succeeded  in  collecting  only  grad- 
ually, partly  in  articles  scattered  in  agricultural  and  horti- 
cultural journals.  In  the  meantime  I  increased  my  collec- 
tion of  monstrosities  but  soon  perceived  that  collecting  is 
not  the  right  way  to  gain  an  insight  into  them.  Therefore 
I  preferred  revisiting  the  same  spots  in  nature  for  succes- 
sive years  and  found  the  monstrosities  regularly  repeated. 
This  induced  the  idea  of  their  being  heritable  phenomena, 
a  conception  wholly  new  at  that  time,  although  the  in- 
heritance of  the  cockscomb  or  Celosia  was,  of  course,  known 
to  every  horticulturist.  Then  I  turned  to  cultivation,  made 
races  of  fasciated  and  twisted  forms  and  studied  the  in- 
heritance of  pitchers  and  analogous  deviations. 

Parallel  to  these  experimental  studies  I  tried  to  pene- 
trate into  the  theoretical  side  of  the  question,  and  this  led 
to  the  publication  of  my  book  on  Intracellular  Pangenesis 
in  1889,  of  which  the  Open  Court  Publishing  Company 


406  THE   MONIST. 

published  an  English  translation  by  Prof.  C.  Stuart  Gager 
in  1910.  Freed  from  the  hypothesis  of  the  transportation 
of  germs  through  the  tissues,  Darwin's  pangenesis  coin- 
cided with  my  own  conception  of  the  material  basis  of 
protoplasmic  life  and  of  the  hereditary  qualities.  This 
study  brought  about  the  conviction  that  variability  must 
at  least  consist  in  two  essentially  different  principles.  One 
of  them  is  the  origin  of  new  qualities  and  their  accumula- 
tion through  geological  times,  producing  the  continuous 
development  of  higher  forms  from  lower.  This  form  is 
what  we  now  call  mutability.  The  other  is  our  present 
fluctuating  variability.  It  determines  the  degree  in  which 
the  single  qualities  will  show  in  different  individuals.  I 
proposed  this  difference  between  mutability  and  fluctu- 
ating variability  at  the  conclusion  of  my  book,  but  said  to 
myself:  It  is  all  right  to  deduce  the  theoretical  necessity 
of  this  conclusion,  but  it  would  be  of  far  higher  importance 
to  prove  the  actual  existence  of  these  two  types  of  variation. 

I  set  at  work  at  once,  first  in  the  field  but  soon  in  the 
garden.  I  cultivated  over  a  hundred  wild  species,  and 
some  of  them  through  many  years.  Fluctuating  variabil- 
ity was  everywhere  present.  Then  I  chanced  to  meet  with 
Quetelet's  Anthropometrie,  which  had  appeared  in  1870, 
applied  his  methods  to  plants  and  saw  that  here  the  same 
general  laws  prevail.  Different  forms  of  curves  of  varia- 
tion were  determined  in  the  corn  marigold  (Chrysanthe- 
mum segetum)  and  other  plants  (1894-1899),  and  it  be- 
came clear  that  they  changed  the  properties  only  in  the 
directions  of  more  or  less  development,  but  gave  no  indi- 
cation whatever  of  an  origin  of  new  qualities.  Fluctua- 
tion and  mutability  must  therefore  be  principally  distinct. 

Mutations  must  of  course  be  rare,  but  some  few  of  them 
occurred  in  my  garden  in  well-guarded  breeds.  They  were 
sudden,  without  visible  preparation  or  transitions.  The 
peloric  toadflax  appeared  in  1894,  the  double  corn  marigold 


THE  ORIGIN  OF  THE  MUTATION  THEORY.  407 

in  1896;  they  sufficed  to  prove  the  reality  of  mutations 
and  gave  an  experimental  basis  for  the  appreciation  and 
the  study  of  the  sudden  appearance  of  new  varieties  in 
horticulture. 

Besides  them,  one  species  proved  to  be  rich  in  such 
sudden  changes.  It  was  Lamarck's  evening  primrose,  a 
species  originally  wild  in  the  eastern  United  States  and 
collected  there  by  Michaux,  but  which  has  since  disap- 
peared in  America.  It  has,  however,  won  an  extensive  dis- 
tribution in  England,  Holland,  Belgium  and  France,  pre- 
ferring the  sand  dunes  along  the  coast.  I  observed  its  muta- 
tions for  the  first  time  in  1888  and  since  then  it  has  never 
ceased  to  produce  them.  The  number  of  mutants  amounts 
to  more  than  a  dozen,  some  of  them  being  progressive,  as 
for  instance  the  giant  type  or  Oenothera  Lamarckiana 
gigas,  published  in  1900,  others  retrogressive  like  the  dwarfs 
and  a  brittle  race  called  O.  rubrineri'is.  Ordinarily  they 
are  constant  from  seed,  but  some  show  a  splitting  and  are 
therefore  considered  to  be  half-mutants  only,  as  O.  lata 
and  allied  forms.  The  changes  are  always  sudden  and 
without  transitions  and  occur  so  regularly  in  about  i%  of 
the  individuals  that  they  constitute  an  unexpected  but  ex- 
cellent material  for  experimental  researches. 

In  my  course  on  variability  I  laid  especial  stress  on  the 
pedigrees  of.  definite  systematic  groups.  The  families  of 
the  euphorbiaceous  and  the  umbelliferous  plants  afforded 
a  very  demonstrative  material,  and  the  hypothesis  of  the 
descent  of  the  Monocotyls  from  the  Dicotyls  through  types 
allied  with  the  common  buttercups,  proposed  at  that  time 
by  Delpino,  proved  to  be  very  convincing  and  instructive. 
Systematic  atavisms,  as  shown  in  the  leaf-bearing  seedlings 
of  the  leafless  species  of  Acacia  and  analogous  instances 
were  added  to  these  discussions.  They  showed  that  evo- 
lution in  nature  is  partly  progressive  and  partly  retro- 
gressive. Progression  means  differentiation  and  speciali- 


408  THE   MONIST. 

zation,  it  governs  the  main  lines  of  the  pedigree  of  the 
animal  and  vegetable  kingdoms.  But  retrogression,  con- 
sisting in  the  loss  of  previously  developed  qualities,  must 
be  responsible  for  a  large  part  of  the  diversity  of  forms  in 
nature.  And  since  it  is  easier  to  lose  a  thing  than  to  acquire 
a  new  quality,  the  cases  of  retrogression  must  be  far  more 
numerous  in  nature  than  those  of  actual  progression. 

Therefore  there  must  be  two  kinds  of  mutations  and 
even  in  our  experimental  cultures  progressive  ones  must 
be  rare,  and  retrogressive  ones  comparatively  more  fre- 
quent. This  is  exactly  what  we  see  in  the  mutations  of  the 
evening  primrose. 

Alongside  of  these  studies  I  tried  hybridization.  Opium 
poppies  afforded  a  useful  material  and  led  to  the  rediscov- 
ery of  Mendel's  law.  At  that  time  this  conception  was  be- 
lieved in  by  nobody,  it  was  rather  considered  as  an  ideal- 
istic fiction.  But  the  splitting  of  the  poppies  confirmed  that 
of  Mendel's  peas,  and  numerous  garden  varieties  behaved 
in  the  same  way.  I  was  fortunate  enough  to  be  the  first 
to  publish  this  result  (1900)  and  pointed  out  that  it  is 
especially  retrogressive  variations  which  follow  this  law, 
whereas  progressive  ones  produce  constant  hybrids,  at 
least  in  many  instances. 

Paleontological  studies  strengthened  the  idea  of  the 
origin  of  species  by  means  of  sudden  variations  instead  of 
a  slow  and  gradual  development.  This  side  of  the  question 
has  since  been  taken  up  by  Charles  A.  White  and  other 
paleontologists.  From  my  own  studies  I  deduced  the  con- 
tention, that  life  on  this  earth  has  not  lasted  long  enough 
for  such  a  slow  development  as  Darwin's  theory  of  selection 
supposed.  Darwin  calculated  some  thousands  of  millions 
of  years  as  required  for  his  theory,  but  geologists  and 
physicists  only  allow  about  forty  or  at  most  a  hundred 
millions  of  years  for  the  development  of  all  animals  and 
plants.  The  hypothesis  of  sudden  mutations  delivers  us 


THE  ORIGIN  OF  THE  MUTATION  THEORY.  409 

from  this  difficulty.  And  so  it  does  for  many  other  objec- 
tions which  were  still  being  used  as  weapons  against  the 
whole  principle  of  evolution  in  the  form  proposed  by  Dar- 
win. 

It  has  always  been  my  conviction  that  the  improvement 
of  industrial  practice  is  the  main  aim  of  all  science.  Bio- 
logical science  has  to  be  a  basis  for  agriculture  and  horti- 
culture. The  discipline  of  heredity  should  be  crowned  by 
the  advance  in  our  knowledge  concerning  the  breeding  of 
animals  and  plants.  With  Dr.  Wakker  I  studied  the  dis- 
eases of  the  flower  bulbs  cultivated  all  around  Haarlem 
(1883-1885),  and  since  then  I  regularly  sent  contributions 
to  the  journal  of  our  agricultural  society.  From  1892  to 
1894  I  was  editor  of  the  journal  of  the  Dutch  Horticultural 
Society  in  order  to  have  an  easy  access  to  horticultural 
establishments  in  the  Netherlands  as  well  as  abroad,  and 
collected  all  the  evidence  I  could  find  concerning  practical 
plant-breeding.  As  a  matter  of  fact  this  was  very  scanty 
but  it  led  me  to  a  connection  with  the  Director  of  the  Swed- 
ish agricultural  station  at  Svalof,  Dr.  Hjalmar  Nilsson, 
whose  celebrated  method  of  plant  improvement  rested  on 
the  same  scientific  basis  as  my  own  experiments. 

My  book  on  the  mutation  theory  is  the  combination  of 
all  these  preliminary  studies  into  a  regular  discussion  of 
the  main  principle.  I  had  the  great  advantage  of  my 
steadily  repeated  courses  on  heredity,  which  constituted, 
if  I  may  say  so,  a  first  unpublished  edition,  with  all  the 
many  faults  inherent  to  first  trials  on  a  new  field.  The 
book  appeared  in  190x3,  and  an  English  edition,1  prepared 
by  Prof.  J.  B.  Farmer  and  A.  D.  Darbishire,  was  published 
by  the  Open  Court  Publishing  Company  in  1909.  It  tries 
to  show  that  the  origin  of  species  is  a  natural  phenomenon 
and  that  it  is  possible  to  subject  it  to  experimental  study. 
In  nature  the  mutations  have  produced  the  whole  evolution 

1  The  Mutation  Theory.    2  vols. 


4IO  THE   MONIST. 

of  all  living  beings;  in  the  garden  we  can,  of  course,  only 
expect  to  see  their  very  smallest  steps.  The  identity  of 
retrogressive  mutations  in  nature,  in  horticulture  and  agri- 
culture and  in  the  experimental  garden  seems  now  to  be 
beyond  doubt.  But  progressive  changes,  which  are  the 
most  important,  are  at  the  same  time  the  rarest,  in  nature 
as  well  as  in  cultivation.  In  regard  to  these  the  theory 
relies  on  its  broad  arguments  and  the  question  whether 
the  directly  observed  progressive  mutations  afford  a  mate- 
rial for  the  interpretation  of  the  ways  of  nature  is  still 
under  discussion. 

The  theory  is  based  upon  arguments  taken  from  widely 
different  branches  of  nearly  all  natural  sciences.  It  con- 
duces of  necessity  to  experimental  research,  but  this,  of 
course,  is  still  in  its  first  infancy.  It  promises,  however, 
to  become  some  day  of  important  service  to  science  at  large 
as  well  as  to  the  practice  of  breeders. 

HUGO  DE  VRIES. 

LUNTEREN,  HOLLAND. 


THE  MANUSCRIPTS  OF  LEIBNIZ  ON  HIS  DIS- 
COVERY OF  THE  DIFFERENTIAL  CALCULUS. 

PART  II  (CONTINUED). 

§§  XI— XV. 

Between  the  date  of  the  manuscript  last  considered 
and  the  one  which  follows  there  is  a  gap  of  seven  months, 
for  which  Gerhardt  does  not  appear  to  have  found  any- 
thing. This  is  very  unfortunate ;  for  in  this  interval  Leib- 
niz has  attained  to  the  important  conclusion  that  the  true 
general  method  of  tangents  is  by  means  of  differences. 
We  saw  that  in  November  1675  he  had  started  to  investi- 
gate more  thoroughly  the  direct  method  of  tangents;  but 
the  method  is  that  of  the  auxiliary  curve,  and  there  is  no 
indication  whatever  of  the  characteristic  triangle.  Does 
this  interval  correspond  with  the  time  taken  by  Leibniz 
for  his  final  reading  of  Barrow  from  Lect.  VI  to  Lect.  X, 
comparing  all  the  geometrical  theorems  with  his  own  nota- 
tion? Or  is  it  only  a  strange  coincidence  that  Leibniz's 
order  is  the  same  as  that  of  Barrow,  first  the  auxiliary 
curve,  and  lastly  the  method  of  differences?  One  could 
form  a  more  definite  opinion,  if  Leibniz  had  given  a  dia- 
gram for  the  first  problem  he  considers,  the  one  in  the  next 
following  manuscript,  which  amounts  to  the  differentiation 
of  an  inverse  sine.  Such  a  diagram  he  must  have  had 
beside  him  as  he  wrote;  for  I  think  the  reader  will  find 
that  he  wants  one  to  follow  the  argument;  with  the  idea 


412  THE   MONIST. 

of  verifying  this  argument,  I  have  not  endeavored  to  supply 
the  omission. 

The  consideration  of  the  direct  method  of  tangents  is 
apparently,  however,  only  as  a  means  and  not  as  an  end; 
for  Leibniz  harks  back  to  the  inverse  method,  and  to  the 
catalogue  of  quadrible  curves,  which  he  seems  to  say  he 
has  in  hand.  It  is  not  until  November  1676  that  he  seems 
to  be  coming  into  his  own;  and  it  is  not  until  July  1677  that 
he  has  a  really  definite  statement  of  his  rules.  On  the  other 
hand,  in  July  1676,  he  is  consistently  using  the  differential 
factor  with  all  his  integrals,  and  before  the  end  of  that 
year  he  has  the  differential  of  a  product,  whether  obtained 
as  the  inverse  of  his  theorem  fy  dx  =  xy  —  $x  dy,  or  by 
the  use  of  the  substitution  x  +  dx,  y  -f-  dy,  is  not  certain ; 
but  this  substitution  appears  in  the  manuscript  for  No- 
vember 1676.  Finally,  in  July  1677,  appears  the  general 
idea  of  the  substitution  of  other  letters,  in  order  to  eliminate 
the  difficulty  caused  by  the  appearance  of  the  variable 
under  a  root  sign  or  in  the  denominator  of  a  fraction ;  and 
with  this  the  whole  thing  is  now  fairly  complete  for  all 
algebraical  functions.  There  is  as  yet  no  equally  clear 
method  for  the  treatment  of  exponentials,  logarithms,  or 
trigonometrical  functions;  for  the  latter  he  refers  to  a 
geometrical  diagram,  strongly  reminiscent  of  Barrow. 

§xi. 
26  June,  1676. 

Nova  methodus  Tangentium. 
(New  Method  of  Tangents.) 

I  have  many  beautiful  theorems  with  regard  to  the  method  of 
tangents  both  direct  as  well  as  inverse.  D^scartes's  method  of 
tangents  depends  on  finding  two  equal  roots,  and  it  cannot  be  em- 
ployed, except  in  the  case  when  all  the  undetermined  quantities 
occurring  in  the  work  are  expressible  in  terms  of  one,  for  instance, 
in  terms  of  the  abscissa. 

But  the  true  general  method  of  tangents  is  by  means  of  dif- 


THE  MANUSCRIPTS  OF  LEIBNIZ.  413 

ferences.  That  is  to  say,  the  difference  of  the  ordinates,  whether 
direct  or  converging,  is  required.  It  follows  that  quantities  that 
are  not  amenable  to  any  other  kind  of  calculus  are  amenable  to 
the  calculus  of  tangents,  so  long  as  their  differences  are  known. 
Thus  if  we  are  given  an  equation  in  three  unknowns,  in  which  x 
is  an  abscissa,  3;  an  ordinate,  and  z  the  arc  of  a  circle  of  which  x 
is  the  sine  of  the  complement,  e.  g.,  the  equation  bzy  =  cx*  +  fz-.  To 
find  the  next  consecutive  y,  in  place  of  x  take  x  +  f$,  and  in  place  of 

,  Pr  Pr    (51) 

z  take  2  -as.  or,  since  dz=  -7    —  —  ,  we  may  take  z-  —~=- 

/*  9  •  /     9  9      ' 


hence  we  have 


Hence  the  difference  between  y  and  (y)  is  given  by 

=  t>*  dy  ; 

dy      =?2cxT}~r*^x**2/a:r       t          ttf 
Therefore          a=~    — 2   r« — ~g —    -  = -  = — 2 — -^2 . 

From  this  the  flexure  or  sinuosity  of  the  curve  can  be  found, 
according  as  now  2cz\Jrz  -x*,  now  2fzr  predominates ;  for  when 
they  are  equal,  the  ordinate  on  that  side  on  which  it  was  previously 
the  greater  then  becomes  the  less.  It  is  just  the  same,  if  several 
other  undetermined  quantities,  such  as  logarithms  and  other  things 
occur,  no  matter  how  they  are  affected,  as  for  instance  in  the  equa- 
tion b2y  =  cx2  +  fz"+xzl,  where  s  is  supposed  to  be  an  arc,  and  /  a 
logarithm,  .v  the  sine  of  the  complement  of  the  arc,  and  y  the  num- 
ber of  the  logarithm,  b  being  the  radius  and  unity,  equal  to  r.  Also 
it  is  just  the  same,  whenever  an  undetermined  transcendental  has 
been  derived  from  some  dimension  or  quadrature  that  has  not  been 
investigated.32 

For  the  rest,  many  noteworthy  and  useful  theorems  now  arise 
from  the  foregoing  by  the  inverse  method  of  tangents.  Thus  gen- 
eral equations,  or  equations  of  any  indefinite  degree  may  be  formed, 
at  first  indeed  in  two  unknowns,  x  and  y,  only.  But  if  in  this  way 
the  matter  does  not  work  out  satisfactorily,  it  will  easily  do  so  when 

51  In  this  and  the  following  line  I  have  corrected  two  obvious  misprints; 
they  are  evidently  not  the  fault  of  Leibniz,  for  the  lines  that  follow  from  them 
are  correct. 

82  There  is  some  doubt  here  as  to  whether  Leibniz  could  have  given  an 
example ;  but  it  must  be  remembered  that  these  are  practically  only  notes, 
mostly  for  future  consideration. 


414  THE   MONIST. 

the  tables  which  I  am  investigating  are  finished ;  then  it  will  be 
possible  to  take  one  or  more  other  letters,  and  to  take  the  difference 
as  an  arbitrary  known  formula,  and  when  this  is  done  it  is  certain 
that  finally  in  any  case  a  formula  will  be  found  such  as  is  re- 
quired, and  in  this  way  also  a  curve  which  will  satisfy  the  conditions 
given ;  but  in  truth  the  description  of  the  curve  will  need  diagrams 
for  these  symbols,  representing  the  sums  of  the  arbitrarily  chosen 
differences. 

Now  once  a  curve  is  found  having  the  tangent  property  that 
we  want,  it  will  be  more  easy  afterwards  to  find  simpler  construc- 
tions for  it.  We  have  this  also  as  a  convenient  means  enabling 
us  to  use  many  quantities  that  are  transcendent,  yet  depending  the 
one  on  the  other,  such  for  example  as  are  all  those  that  depend 
on  the  quadrature  of  the  circle  or  the  hyperbola.  From  these 
investigations  it  will  also  appear  whether  or  no  other  quadratures 
can  be  reduced  to  the  quadrature  of  the  circle  or  the  hyperbola. 
Lastly,  since  the  finding  of  maxima  and  minima  is  useful  for  the 
inscription  and  circumscription  of  polygons,  hence  also,  by  employ- 
ing these  transcendent  magnitudes,  convergent  series  can  be  found, 
and  in  the  same  way  their  terminations  ;  or  of  any  quantities  formed 
in  the  same  way.  However  in  that  case  it  may  not  be  so  easy  to 
argue  about  impossibility ;  at  least  indeed  by  the  same  method. 
Only  I  do  not  see  how  we  can  find  whether  from  the  quadrature  of 
the  circle,  say,  any  sum  can  be  found,  when  no  quantity  depending 
on  the  dimensions  of  the  circle  enters  into  the  calculation. 

§XII. 
July,  1676. 

Methodus  tangentium  inversa. 
[Inverse  method  of  tangents.] 

In  the  third  volume  of  the  correspondence  of  Descartes,  I  see 
that  he  believed  that  Fermat's  method  of  Maxima  and  Minima  is 
not  universal;  for  he  thinks  (page  362,  letter  63)  that  it  will  not 
serve  to  find  the  tangent  to  a  curve,  of  which  the  property  is  that 
the  lines  drawn  from  any  point  on  it  to  four  given  points  are  to- 
gether equal  to  a  given  straight  line. 

[Thus  far  in  Latin;  Leibniz  then  proceeds  in  French.] 

Mons.  des  Cartes  (letter  73,  part  3,  p.  409)  to  Mons.  de  Beaune. 

"I  do  not  believe  that  it  is  in  general  possible  to  find  the  con- 
verse to  my  rule  of  tangents,  nor  of  that  which  Mons.  Fermat  uses, 


THE  MANUSCRIPTS  OF  LEIBNIZ.  415 

although  in  many  cases  the  application  of  his  is  more  easy  than 
mine ;  but  one  may  deduce  from  it  a  posteriori  theorems  that  apply 
to  all  curved  lines  that  are  expressed  by  an  equation,  in  which  one 
of  the  quantities,  x  or  y,  has  no  more  than  two  dimensions,  even 
if  the  other  had  a  thousand.  There  is  indeed  another  method  that 
is  more  general  and  a  priori,  namely,  by  the  intersection  of  two 
tangents,  which  should  always  intersect  between  the  two  points  at 
which  they  touch  the  curve,  as  near  one  another  as  you  can  im- 
agine ;  for  in  considering  what  the  curve  ought  to  be,  in  order  that 
this  intersection  may  occur  between  the  two  points,  and  not  on  this 
side  or  on  that,  the  construction  for  it  may  be  found.  But  there 
are  so  many  different  ways,  and  I  have  practised  them  so  little,  that 
I  should  not  know  how  to  give  a  fair  account  of  them." 

Mons.  des  Cartes  speaks  with  a  little  too  much  presumption 
about  posterity ;  he  says  (page  449,  letter  77)  that  his  rule  for  re- 
solving in  general  all  problems  on  solids  has  been  without  compari- 
son the  most  difficult  to  find  of  all  things  which  have  been  discovered 
in  geometry  up  to  the  present,  and  one  which  will  possibly  remain 
so  after  centuries,  "unless  I  take  upon  myself  the  trouble  of  finding 
others"  (as  if  several  centuries  would  not  be  capable  of  producing 
a  man  able  to  do  something  that  would  be  of  greater  moment). 

(Page  459.)  The  question  of  the  four  spheres  is  one  that  is 
easy  to  investigate  for  a  man  who  knows  the  calculus.  It  is  due 
to  Descartes,  but  as  it  is  given  in  the  book,  it  appears  to  be  very 
prolix. 

The  problem  on  the  inverse  method  of  tangents,  which  Mons. 
des  Cartes  says  he  has  solved  (Vol.  3,  letter  79,  p.  460) 

[Leibniz  then  continues  in  Latin.] 

EAD  is  an  angle  of  45  degrees.     ABO  is  a  curve,  BL  a  tan- 
gent to  it ;  and  BC,  the  ordinate,  is  to  CL  as  N  is  to  BJ.    Then 
c          BC  =  *y 

BJ  =y  —  x 
ny       n       y—x  x 

hence,  ,__,  _.  _  ml__t 

hence,  £  =  '-«.  bu,  <  J±; 

y        t  y    dy 

therefore  fe^-JL.    or  dx  y—x  dx  =  dy  «; 

dy    y—x 

hence  §dxy-  $xdx=-n$dy. 


416 


THE   MONIST. 


Now,  fdy  =  y,  and  $.vdx  =  x*/2,  and  fdxy  is  equal  to  the 
area  ACBA,  and  the  curve  is  sought  in  which  the  area  ACBA  is 
equal  to  (x*/2)  +ny=  (AC2/2)  +«BC  B3 


Let  this  .r2/2,  i.  e.,  the  triangle  ACJ  be  cut  off  from  the  area, 
then  the  remainder  AJBA  should  be  equal  to  the  rectangle  ny. 

The  line  that  de  Beaune  proposed  to  Descartes  for  investigation 
reduces  to  this,  that  if  BC  is  an  asymptote  to  the  curve,  BA  the 
axis,  A  the  vertex,  AB,  BC,  fixed  lines,  for  BAG  is  at  right  angles. 

B  T    A 


Let  RX  be  an  ordinate,  XN  a  tangent,  then  RN  is  always  to 
be  constant  and  equal  to  BC ;  required  the  nature  of  the  curve. 

This  is  how  I  think  it  should  be  done. 

Let  PV  be  another  ordinate,  differing  from  the  other  one  RX 
by  a  straight  line  VS,  found'  by  drawing  XS  parallel  to  RN ;  then 

53  Leibniz  has  a  footnote  to  this  manuscript :  "I  solved  in  one  day  two 
problems  on  the  inverse  methods  of  tangents,  one  of  which  Descartes  alone 
solved,  and  the  other  even  he  owned  that  he  was  unable  to  do." 

This  problem  is  one  of  them,  the  first  mentioned  in  the  footnote  given  by 
Leibniz.  But  it  requires  a  stretch  of  imagination  to  consider  Leibniz's  result 
as  a  solution.  For  he  ends  up  with  a  geometrical  construction,  that  is  at 
least  as  hard  as  the  construction  that  can  be  made  by  the  use  of  the  original 
data.  There  is  of  course  the  usual  misprint  that  one  is  becoming  accustomed 
to;  but  there  is  also  the  unusual,  for  Leibniz,  mistake  of  using  his  data  in- 
correctly. Starting  with  the  hypothesis  that  BC :  CL  =  N  :  BJ,  he  writes  CL  = 
N.BC/BJ  (correcting  the  omission  of  the  factor  N),  instead  of  CL  == 
BC.BJ/N. 

The  solution  of  the  problem  is  y-\-n\og(y  —  x+n)—Q,  as  originally 
stated,  or  .r  =  «log(tt  —  y-\-x),  if  we  continue  from  Leibniz's  erroneous  re- 
sult dx/dy  =  n/(y —  x). 

The  point  to  be  noted,  however,  is  that  Leibniz  does  not  remark  that  "this 
curve  appertains  to  a  logarithm." 


THE  MANUSCRIPTS  OF  LEIBNIZ.  417 

the  triangles  SVX,  RXN  are  similar,  RN  -t-c,  a  constant,  RX  =  y, 
SY  =  dy,  and  therefore 

-  -  =  —  -  ;  hence  cy  —    I  y  dx    or  c  dy  =  y  dx.       " 

Ct  X  t  —  C        ,  J 


If  AQ  or  TR  =  r,  and  AC  =  /,  while 

AC       /       TR        z  az 


If  dx  is  constant,  then  dz  is  also  constant.    Hence 
c  dy=  jy  dz,  or  cy=  -,  \  y  dz  ,  and  ry  dy^jy1  dz,  therefore 

>'2       a    (*      — 

~o  —j  \  y1  dz-    Hence  we  have  both  the  area  of  the  figure  and  the 


'2 
c 


moment   to   a   certain   extent    (for   something   must   be  added   on 
account  of  the  obliquity)  ;  also 


cz  dy=-,yz  dz ,  and  therefore  c  I  z  dy—  -7  I  yz  dz. 

Also  £&.  —  -  dz,  and  hence,  c  I  -y-  =  -z.     Now,  unless  I  am 
y       f  J    y      f 

greatly  mistaken,    J  ^  is  in  our  power.55    The  whole  matter  reduces 

J    y 

to  this,  we  must  find  the  curve5"  in  which  the  ordinate  is  such  that 


54  Leibniz  does  not  see  that  this  result  immediately  gives  him  the  equation 
that  he  requires.     Thus  jr  =  cLogy,  as  he  would  have  written  it;  the  usual 
omission  of  the  arbitrary  constant  does  not  matter  in  this  case,  so  long  as  BA 
is  taken  as  unity,  which  is  possible  with  Leibniz's  data. 

55  Here  he  seems  to  recognize  that  he  has  the  solution.    The  next  sentence 
is,  however,  very  strange.    As  long  ago  as  Nov.  1675  he  has  written  fa-/y  as 
Logy,  and  recognized  the  connection  between  the  integral  and  the  quadrature 
of  the  hyperbola ;  and  yet  he  says  "unless  I  am  mistaken,  fdy/y  is  always  in 
our  power."    Now  notice  that  in  the  date  there  is  no  day  of  the  month  given, 
contrary  to  the  usual  custom  with  these  manuscripts  so  far;  can  it  be  possible 
that  this  date  was  afterward  added  from  memory,  and  that  the  manuscript 
should  bear  an  earlier  date?     If  not  we  must  conclude  that  Leibniz  has  not 
yet  attained  to  a  correct  idea  of  the  meaning  of  his  integral  sign,  and  is  still 
worried  by  the  necessity  (as  it  appears  to  him)  of  taking  the  y's  in  arithmet- 
ical progression. 

56  The  passage  in  the  original  Latin  is  very  ambiguous,  and  it  may  be  that 
it  is  not  quite  correctly  given  ;  I  think,  however,  that  I  have  given  the  correct 
idea  of  what  Leibniz  intended.    One  has  to  draw  an  auxiliary  curve,  in  which 
y  =  dy/dx,  and  then  find  its  area ;  in  that  case  it  should  be  "divided  by  the 
differences  of  the  abscissae"  instead  of  "divided  by  the  abscissae." 


41 8  THE    MONIST. 

it  is  equal  to  the  differences  of  the  ordinates  divided  by  the  ab- 
scissae, and  then  find  the  quadrature  of  that  figure. 

1  (57) 

d^ay  =    , 

Figures  of  this  kind,  in  which  the  ordinates  are  dy/y,  dy/y2 
dy/y3,  are  to  be  sought  in  the  same  way  as  I  have  obtained  those 
whose  ordinates  are  y  dy,  y~dy,  etc.  Now  w/a  --  dy/y,  and  since  dy 
may  be  taken  to  be  constant  and  equal  to  ft,56  therefore  the  curve, 
in  which  •w/a  =  dy/y,  will  give  wy  =  aj3,  which  would  be  a  hyper- 
bola.53 Hence  the  figure,  in  which  dy/y  =  2,  is  a  hyperbola,  no  mat- 
ter how  you  express  y,  and  if  y  is  expressed  by  </r  we  have  dy  =  2<f>, 

,  2<t>        2  Cdy       a  fcC-l 

and  -TO-  =  T  .       Now,  c  I    -  =  -f  z  ,    and   therefore   -       •    -  =  z, 
•r  •      •*  J   y        f  a  J    y 

which  thus  appertains  to  a  logarithm.00 

Thus  we  have  solved  all  the  problems  on  the  inverse  method 
of  tangents,61  which  occur  in  Vol.  3  of  the  Correspondence  of  Des- 
cartes, of  which  he  solved  one  himself,  as  he  says  on  page  460, 
letter  79,  Vol.  3 ;  but  the  solution  is  not  given ;  the  other  he  tried 
to  solve  but  could  not,  stating  that  it  was  an  irregular  line,  which 
in  any  case  was  not  in  human  power,  nay  not  within  the  power  of 
the  angels  unless  the  art  of  describing  it  is  determined  by  some  other 
means. 

§XIII. 

This  manuscript  bears  no  date:  however,  it  was  prob- 
ably written  very  shortly  after  his  call  on  Hudde  at  Am- 
sterdam, on  his  way  home  from  England  (the  second  visit) 

57  An  interpolated  note,  marking  a  sudden  thought  or  guess ;  for  the  next 
sentence  carries  on  the  train  of  thought  that  has  gone  before.     Query,  some 
interval  of  time,  either  short  (such  as  for  a  meal)  or  long  (continued  the  next 
day),  may  have  occurred  here. 

58  This  cannot  be  referred  back  to  the  present  problem,  since  Leibniz  has 
already  assumed  in  it  that  dz  and  dx  are  constant.    This  may  account  for  the 
fact  that  he  has  hesitated  to  say  that  the  integral  represents  a  logarithm. 

59  This  working  is  intended  to  apply  to  the  auxiliary  curve  mentioned 
above,  w  standing  for  dx,  and  /3  for  dy ;  hence  the  curve  is  not  a  hyperbola ; 
Leibniz  seems  to  have  been  misled  by  the  appearance  of  the  equation  suggest- 
ing xy  =  constant. 

60  Here  apparently  he  leaves  the  muddle,  in  which  he  has  entangled  him- 
self, and  returns  to  his  original  equation ;  he  then  remembers  that  he  has  found 
before  that  the  integral  in  question  leads  to  a  logarithm. 

61  He  has  not  solved  either  of  them ;  nor  can  it  be  said  from  this  that 
"Leibniz  in  1676  sought  and  found  the  curve  whose  subtangent  is  constant." 
Of  all  the  work  that  Leibniz  has  done  hitherto,  there  is  none  that  is  so  incon- 
clusive as  this  in  comparison. 


THE  MANUSCRIPTS  OF  LEIBNIZ.  419 

to  Hanover.  Leibniz  stayed  in  Holland  from  October  1676 
to  December  of  that  year;  hence  the  date  may  be  fairly 
accurately  assigned. 

Hudde  showed  me  that  in  the  year  1662  he  already  had  the 
quadrature  of  the  hyperbola,  which  I  found  was  the  very  same  as 
Mercator  also  had  discovered  independently,  and  published.  He 
showed  me  a  letter  written  to  a  certain  van  Duck,  of  Leyden  I 
think,  on  this  subject.  His  method  of  tangents  is  more  complete 
than  that  of  Sluse,  in  that  he  is  able  to  use  any  arithmetical  pro- 
gression, as  in  a  simple  equation,  whereas  Sluse  and  others  can 
use  only  one.  Hence  constructions  can  be  made  simple,  while  terms 
can  be  eliminated  at  will.  This  also  can  be  made  use  of  for  elim- 
inating any  letter  with  greater  facility,  for  numerous-equations  of 
all  sort  are  thereby  rendered  fit  for  elimination. 

x3    +  px*  +  qx       =0  x*+  xy  +   y1    +  x  +  y+  *=  Q 

y'2  y,  2xdx  +  xdy  +  2ydy+dx+dy      =  0 

y3  ydx 


* 


2yx*  +  yx  y         dy          y~+2x+l 


yzx 

What  I  had  observed  with  regard  to  triangular  numbers  for 
three  equal  roots,  and  pyramidal  numbers  for  four,  was  already 
known  to  him,  and  indeed  even  more  generally, 

-10123456 
-3-1      0    0     1     3    6     10  15 
-4-1      0      0    0     1     4     10  20 

Here  it  must  be  observed  that  the  number  of  zeros  increases,  as 
this  is  of  the  greatest  service  in  separating  roots. 

He  has  also  rules  for  multiplying  equations,  so  that  they  are 
not  only  determined  for  equal  roots,  but  also  for  roots  increasing 
arithmetically,  or  geometrically,  or  according  to  any  progression. 

Hudde  has  a  most  elegant  construction  for  describing  two 
curves,  one  outside  and  the  other  inside  a  circle,  which  are  capable 
of  quadrature,  and  by  means  of  these  curves  he  finds  the  true  area 
of  a  circle  so  nearly,  that  with  the  help  of  the  dodecagon,  in 
a  number  of  six  figures,  there  is  an  error  of  only  three  units,  or 
3/100000. 


42O  THE    MONIST. 

He  has  a  method  for  finding  the  real  roots  of  equations,  having 
some  roots  real  and  the  rest  impossible,  by  the  help  of  another 
equation  having  all  its  roots  real,  and  as  many  in  number  as  he 
previously  had  of  real  and  impossible  together. 

He  had  an  example  of  a  beautiful  method  of  finding  sums  of 
series  by  the  continuous  subtractions  of  geometrical  progressions. 
He  subtracts  geometrical  progressions  whose  sums  are  also  geo- 
metrical progressions,  and  thus  he  can  find  the  sums  of  the  sums, 
and  so  he  obtains  the  sum  of  the  series.  This  method  is  excellent  for 
a  series  whose  numerators  are  arithmetical,  and  denominators  geo- 
metrical, such  as, 

1  2  3  ± 

2  4  8  16 

He  has  three  series,  like  those  of  Wallis,  for  interpolations  for  the 
circle.    He  says  that  there  are  no  more  by  that  method,  I  think. 

Also  he  can  very  often  write  down  the  quadratures  of  irra- 
tionals, as  also  their  tangents,  without  eliminating  irrationals,  or 
fractions,  etc. 

8  XIV. 

November,  1676. 

Calculus  Tangentium  differentialis. 
[Differential  calculus  of  tangents.] 

dx=l,    dx-  =  2x,    dx3  =  3;r2,    etc. 
U  l';;      Jl          2         ,1     _  3 

a       —   --  9  ,      a     o  =  --  n  ,      «     •»  —        o  ,      etc. 

x  xl        x          x          x6      x1 

1 


d  V*=    /-  ,    etc. 

\x 

From  these  the  following  general  rules  may  be  derived  for  the 
differences  and  sums  of  the  simple  powers: 

_  s*  —       x'+i 

dx'  —  e,x*-1  ,  and  conversely    I  x'  =  --  . 

J  e+l 

~T  2 

Hence,  d-^-dx'"1  will  be  —  2*~3  or  —  —., 
x3  Xs 

and  d  Jx  or  dx*  will  be—  \x~'A  or  —4/1. 

\* 

Let  y  =  X*,  then  dy  =  2*  dx  or  ^  =  2  x  . 

dx 


THE  MANUSCRIPTS  OF  LEIBNIZ.  42! 

This  reasoning  is  general,  and  it  does  not  depend  on  what  the  pro- 
gression for  the  .r's  may  be.0"  By  the  same  method,  the  general 
rule  is  established  as: 

d* 

-  =  ^  x     ,  am 
dx 

Suppose  that  we  have  any  equation  whatever,  say, 

ay2  +  byx  +  cs2  +  f2x  +  g-y  +  h3  =  0, 

and  suppose  that  we  write  y  +  dy  for  y,  and  x  +  dx  for  x,  we  have, 
by  omitting  those  things  which  should  be  omitted,  another  equation 

ayz  +  byx  +  ex2  +  f2x  +  g2y  +  h3  =  0    ~* 


a2dyy  +  by  dx  +  2cxdx  +  f2d.v  +  g-dy 
bxdy 


>  =  0 


ady2  -f  bdxdy  +  cdx2  =  0 

This  is  the  origin  of  the  rule  published  by  Sluse.  It  can  be  extended 
indefinitely:  Let  there  be  any  number  of  letters,  and  any  formula 
composed  from  them ;  for  example,  let  there  be  the  formula  made 
up  of  three  letters, 

ay2    bx2    cs2    fyx    gyx    hxs    ly    mx    nz.    p  =  0. 
From  this  we  get  another  equation 

ay2        bx2        cz~      fyx     simi-    ly     mx    simi-  p 

2adyy  2bdxx  2cdzz  fydx   larly   Idy  mdx   larly 
fxdy 

a  dy2     bdx2     cdz2  fdxdy  

It  is  plain  from  this  that  by  the  same  method  tangent  planes 

02  AT  LAST !     The  recognition  of  the  fact  that  neither  dx  nor  dy  need 
necessarily  be  constant,  and  the  use  of  another  letter  to  stand  for  the  function 
that  is  being  differentiated,  mark  the  beginning,  the  true  beginning,  of  Leib- 
niz's development  of  differentiation.     Later  in  this  manuscript  we  find  him 
using  the  third  great  idea,  probably  suggested  by  the  second  of  those  given 
above,  namely,  the  idea  of  substitution,  by  means  of  which  he  finally  attains 
to  the  differentiation  of  a  quotient,  and  a  root  of  a  function. 

It  is  very  suggestive  that  this  remarkable  advance  occurs  after  his  second 
visit  to  London,  while  he  is  staying  in  Holland.  Did  some  one  tell  then  of 
the  work  of  Newton,  or  of  Barrow's  method  (which  is  geometrically  an  exact 
equivalent  of  substitution),  pointing  out  those  things  of  which  he  had  not 
perceived  the  drift,  or  is  it  the  result  of  his  intercourse  with  Hudde?  For 
the  date  is  that  of  his  stay  at  The  Hague.  (For  the  answer  to  this  query  see 
an  article  to  follow,  entitled  "Leibniz  in  London." — ED.) 

03  This  is  Barrow  all  over;  even  to  the  words  omissis  omittendis  instead 
of  Barrow's  -rcjcdis  rcjicietidis.     Lect.  X,  Ex.  1  on  the  differential  triangle  at 
the  end  of  the  lecture. 


422  THE   MONIST. 

to  surfaces  may  be  obtained,  and  in  every  case  that  it  does  not 
matter  whether  or  no  the  letters  x,  y,  z  have  any  known  relation, 
for  this  can  be  substituted  afterward. 

Further,  the  same  method  will  serve  admirably,  even  though 
compound  fractions  or  irrationals  enter  into  the  calculation,  nor  is 
there  any  need  that  other  equations  of  a  higher  degree  should  be 
obtained  for  the  purpose  of  getting  rid  of  them  ;  for  their  differences 
are  far  better  found  separately  and  then  substituted ;  hence  the 
ordinary  method  of  tangents  will  not  only  proceed  when  the  ordi- 
nates  are  parallel,  but  it  can  also  be  applied  to  tangents  and  any- 
thing else,  aye,  even  to  those  things  that  are  related  to  them,  such 
as  proportions  of  ordinates  to  curves,  or  where  the  angle  of  the 
ordinates  changes  according  to  some  determined  law.  It  will  be 
worth  while  especially  to  apply  the  method  to  irrationals  and  com- 
pound fractions.64 

d  fa  +  t>z  +  cz* .    Let  a  +  bs  +  cz*  =  x ; 

then  dv/x  =  — ;r~7~  >  and  ~r  =  b  +  2cz  • 

2\/x  dz 

therefore  d  V  a  +  bz+ c£  =  — 


Taking  any  equation  between  two  letters  x  and  y  for  a  curve, 
and  determining  the  equation  of  the  tangent,  either  of  the  two  let- 
ters x  or  y  can  be  eliminated,  so  that  all  that  remains  is  the  other 
together  with  dx  and  dy ;  and  this  will  be  worth  while  doing  in  all 
cases  to  facilitate  the  calculation. 

If  three  letters  are  given,  say  x,  y  and  z,  and  the  value  of  dz 
is  expressed  in  terms  of  x  or  y  (or  even  of  both),  an  equation  for 
the  tangents  will  at  length  be  obtained,  in  which  again  there  will 
be  left  only  one  or  other  of  the  letters  x  or  y  together  with  the 
two,sd.r  and  dy ;  sometimes  z  itself  cannot  be  eliminated.  Also 
this  can  be  deduced  in  all  cases  of  an  assumed  value  of  dz,  and  in 
the  same  way  more  additional  letters  can  be  taken.  Thus,  bringing 
together  every  general  calculus  into  one,  we  obtain  the  most  general 
of  them  all.  Besides,  the  assumption  of  a  large  number  of  letters 
may  be  employed  to  solve  problems  on  the  inverse  method  of  tan- 
gents, with  the  assistance  of  quadratures. 

64  Here  we  have  the  idea  of  substitutions,  which  made  the  Leibnizian 
calculus  so  superior  to  anything  that  had  gone  before.  Note  that  he  still  has 
the  erroneous  sign  that  he  obtained  for  the  differentiation  of  \ I  x  at  the  be- 
ginning of  this  manuscript.  Also  that  the  ds  is  wrongly  placed  in  the  denom- 
inator of  the  result. 


THE  MANUSCRIPTS  OF  LEIBNIZ.  423 

Thus,  if  the  following  problem  is  set  for  solution :  It  is  given 
that  the  sum  of  the  straight  lines  CB,  BP  or 


we  have 

dx  +  dy=xdx 

_  -v"' 

J  ~~"     fy 

Thus  we  have  the  curve  in  which  the  sum  of  CB  +  BP  (multi- 
plied by  a  constant  r)  is  equal  to  the  rectangle  AB.BC. 

There  are  two  marginal  notes  by  Leibniz  that  must  be  referred  to,  in  this 
manuscript.  The  first  reads  : 

It  is  especially  to  be  observed  about  my  calculus  of  differences  that,  if 

b.  ydx  -f-  xdy  -\-  etc.  =  0 

then  byx  -\-  {  etc.  =  0,  and  so  on  for  the  rest.  It  is  to  be  seen  what  is  to  be 
done  about  the  A3.  For  the  purpose  of  making  these  calculations  better,  the 
equation  ay2  -\-  byx  +  ex2  -f-  etc-  can  be  changed  into  something  else  by  means 
of  another  relation  of  the  curve,  and  if  it  turns  out  all  right  it  may  be  compared 
to  another  calculation  of  the  differences,  since  it  comes  to  the  thing  as  by  the 
first.  The  two  points  to  be  noticed  are  that  Leibniz  now  for  the  first  time  rec- 
ognizes the  need  of  considering  the  arbitrary  constant  of  integration,  though 
he  hardly  grasps  how  it  arises,  and  that  even  now  he  cannot  refrain  from 
harking  back  to  his  obsession  of  the  obtaining  of  several  equations  for  com- 
parison. This  note  is  not  made  any  the  easier  to  understand  by  its  being 
starred  by  Gerhardt  for  reference  to  the  differentiation  of  x-,  whereas  it  ob- 
viously (when  you  come  later  to  the  passage)  refers  to  the  differentiation  of 
the  equation  of  the  second  degree. 

The  second  note  refers  to  the  substitution  of  x  +  d x  for  x  and  y  -\-  dy  for 
y,  and  reads : 

Either  dx  or  dy  can  be  expressed  arbitrarily,  a  new  equation  being  ob- 
tained ;  and  either  dx  or  dy  being  taken  away,  x,  or  y,  say,  can  be  otherwise 
expressed  in  terms  of  the  quantities.  It  is  not  true,  I  think,  that  this  is  so,  for 
then  a  catalogue  of  all  curves  capable  of  quadrature  would  result,  by  sup- 
posing one  or  other  of  them  to  be  constant. 

The  point  to  be  noticed  in  this  rather  ambiguous  statement  is  that  Leibniz 
is  still  thinking  of  his  catalogue,  and  is  not  himself  convinced  of  the  com- 
pleteness of  his  method  for  all  purposes. 

§XV. 

There  is  an  interval  of  nearly  seven  months  between 
the  date  of  the  manuscript  last  considered  and  the  one  that 
now  follows.  This  interval  has  been  full  of  work;  for  we 
now  find  a  clear  exposition  of  the  rules  for  the  differentia- 


424  THE    MONIST. 

tion  of  a  sum,  difference,  product,  quotient,  etc.,  though 
these  are  without  proof,  or  indication  of  the  manner  in 
which  they  have  been  obtained.  There  is  also  no  rule 
given  for  a  logarithm,  an  exponential,  or  a  trigonometrical 
ratio.  Leibniz  may  have  known  them,  but  even  then  it 
would  not  be  surprising  to  find  them  left  out;  for  Leibniz's 
great  idea  was  the  use  of  his  method  to  facilitate  calcula- 
tion. We  must  conclude  therefore  that  these  rules  are  a 
development  of  the  method  of  substitution  outlined  in  the 
preceding  manuscript. 

This  essay  has  several  peculiar  characteristics  of  its 
own,  which  distinguish  it  from  those  that  have  gone  before. 
It  is  written  throughout  in  French;  it  is  to  some  extent 
historical  and  critical,  having  the  appearance  of  being 
prepared  for  publication,  or  possibly  as  a  letter;  this  is 
corroborated  by  the  fact  that  there  is  an  original  draft  and 
a  more  fully  detailed  revision.  Could  it  be  that  this  is  the 
original  of  Leibniz's  communication  of  this  method  to  New- 
ton and  others?  If  so,  Leibniz  is  very  careful  not  to  give 
much  away.  The  figures  are  strongly  reminiscent  of  Bar- 
row, but  the  context  does  not  deal  with  subtangents,  which 
are  such  a  feature  in  all  Barrow's  work. 

The  start  from  the  work  of  Sluse  is  peculiar ;  it  seems 
to  suggest  that  Leibniz  is  pointing  out  that  his  method  is 
a  fuller  development  of  that  of  the  former.  Leibniz  has 
already  hazarded  two  different  guesses  at  the  origin  of 
the  rules  given  by  Sluse;  the  second,  namely,  by  substitu- 
tion of  x  -f-  dx  for  x,  etc.,  being  the  more  probable.  Is 
Leibniz  trying  to  draw  a  red  herring  across  the  trail,  the 
real  trail  that  leads  to  Barrow's  a  and  e? 

1 1  July  1677. 

Methode  generate  pour  mener  les  touchantes  des  Lignes  Courbes 
sans  calcnl,  et  sans  reduction  des  quantites  irrationelles  et 
rompues. 


Til  i:  MANUSCRIPTS  OF  LEIBNIZ. 


425 


[General  method  for  drawing  tangents  to  curves  without  cal- 
culation,and  without  reducing  irrational  or  fractional  quan- 
tities.] 

Slusius  has  published  his  method  of  finding  tangents  to  curves 
without  calculation,  in  which  the  equation  is  purged  of  irrational 
or  fractional  quantities. 


For  example,  a  curve  DC  being  given,  in  which  the  equation 
expresses  the  relation  between  BC  and  AS,  which  we  will  call  y, 
and  AB  or  SC,  which  we  will  call  x;  let  this  be 

a  +  bx  +  cy  +  dxy  +  ex-  +  fy-  +  gx-y  +  hxy2  +  kx3  +  ly*  +  etc.  =  d 
One  has  only  to  write 

0  -i-  t>£  +  c v  +  dxv  +  2?*£  +  2fyv  +  gx'*y  +  hy1^  f  3&*2£  +  3fy''2t> 
dy£  2sx)'£    2hxyv 

(65; 


nxy 


ry 


that  is  to  say,  if  the  equation  is  changed  to  a  proportion, 
$  _  c  +  dx  +  2fy  -H  KX*  +  2hxy  +  3()'2  +  2ntxly  +  etc. 
v  b  +  dy  +  2fx  +  2gxy  +  hy*  +  3^'2  +  etc. 


f  TR 

and,    supposing   that     -   expresses    the    ratio  - 

v  BC  =x 


or 


CS=j> 
SV    ' 

then  TB  or  SV  can  be  obtained,  if  BC  and  SC  are  supposed  to 
be  given.  When  the  given  magnitudes,  b,  c,  d,  e,  etc.,  with  their 
proper  signs,  make  the  value  of  £/v  a  negative  magnitude,  the  tan- 
gent will  not  be  CT  which  goes  toward  A,  the  start  of  the  abscissa 
AB,  but  C(T)  which  goes  away  from  it.  That  is  all  that  has  been 

05  This  line  represents  the  "etc."  of  the  original  equation,  and  is  set  down 
for  the  purpose  of  getting  the  derived  terms ;  the  complete  derived  equation 
therefore  consists  of  the  two  lines  above  and  the  two  below.  Note  the  omis- 
sion of  the  negative  sign,  when  changing  from  the  equation  to  the  proportion. 


426  THE    MONIST. 

published  up  to  the  present  time,  easy  to  understand  by  any  one  that 
is  versed  in  these  matters.  But  when  there  are  irrational  or  frac- 
tional magnitudes,  which  contain  either  x  or  y  or  both,  this  method 
cannot  be  used,  except  after  a  reduction  of  the  given  equation  to 
another  that  is  freed  from  these  magnitudes.  But  at  times  this 
increases  to  a  terrible  degree  the  calculation  and  obliges  us  to  rise 
to  very  high  dimensions,  and  leads  us  to  equations  for  which  the 
process  of  depression  is  often  very  difficult.  I  have  no  doubt  that 
the  gentlemen06  I  have  just  named  know  the  remedy  that  it  is  neces- 
sary to  apply,  but  as  it  is  not  as  yet  in  common  use,  and  is  I  believe 
known  to  but  a  few,  also  because  it  gives  the  finishing  touch  to  the 
problem  that  Descartes  said  was  the  most  difficult  to  solve  of  all  geo- 
metrical problems,  because  of  its  general  utility,  I  have  thought  it 
a  good  thing  to  publish  it. 

Suppose  we  have  any  formula  or  magnitude  or  equation  such 
as  was  given  above, 

a  <r  b.v  +  cy  +  d.vy  +  ex*  +  fy2  +  etc. ; 

for  brevity  let  us  call  it  o>;  that  which  arises  from  it  when  it  is 
treated  in  the  manner  given  above,  namely, 
b£  +  cv  +  d.vv  +  dy£  +   etc. ; 

will  be  called  </».>;  and  in  the  same  way,  if  the  formula  is  A  or  /*, 
then  the  result  above  will  be  d\  or  dp,  and  similarly  for  everything 
else.  Now  let  the  formula  or  equation  or  magnitude  w  be  equal  to 

A//*,  then  I  say  that  dw  will  be  equal  to  M     ~        .     This  will  be 

sufficient  to  deal  with  fractions. 

dw 

z 


Again,  let  w  be  equal  to    j/  <*>  ,  then  d<»  —  z.\-l/a>  ;   and  this 

V 
will  be  sufficient  for  the  proper  treatment  of  irrationals. 

Algorithm  of  the  new  analysis  for  maxima  and  minima,  and 
for  tangents. 


Let  AB  =  .r,  and  BC  =  y,  and  let  TVC  be  the  tangent  to  the 

curve  AC  ;  then  the  ratio     TB     or  SC=*  will  be  called  —  . 

EC  =  y          SV  ay 

86  Leibniz,  at  the  beginning,  first  wrote,  "Hudde,  Sluse,  and  others"  ;  but 
later  he  struck  out  all  but  Sluse.     (Gerhardt.) 


THE  MANUSCRIPTS  OF  LEIBNIZ. 


427 


Let  there  be  two  or  more  other  curves,  AF,  AH,  and  suppose 

u 


that  BF  =  z;  and  BH  =  w,  and  that  the  straight  line  FL  is  the  tangent 

T  F*         itr 

to  the  curve  AF,  and  MH  to  the  curve  AH ;  also  -  -  ,    and 

FB       dv 

— ;  then  I  say  that  d\,  or  dwv,  will  be  equal  to  vdw  +  wdv  \ 
BH        aw 

and  if  v~w  =  x,  and  y  =  vw  =  x2,  then  by  substituting  x  for  v  and 
for  w,  we  shall  have  dvw  =  2xdx. 

(This  will  also  hold  good  if  the  angle  ABC  is  either  acute  or 
obtuse ;  also  if  it  is  infinitely  obtuse,  that  is  to  say,  if  TAG  is  a 
straight  line.) 

[Of  this  rough  draft  there  is  the  following  revision,  and  this 
obviously  comes  within  the  same  period.  (Gerhardt.)] 

Fermat  was  the  first  to  find  a  method  which  could  be  made 
general  for  finding  the  straight  lines  that  touch  analytical  curves. 
Descartes  accomplished  it  in  another  way,  but  the  calculation  that 
he  prescribes  is  a  little  prolix.  Hudde  has  found  a  remarkable 
abridgment  by  multiplying  the  terms  of  the  progression  by  those 
of  the  arithmetical  progression.  He  has  only  published  it  for  equa- 
tions in  one  unknown ;  although  he  has  obtained  it  for  those  in  two 
unknowns.  Then  the  thanks  of  the  public  are  due  to  Sluse;  and 
after  that,  several  have  thought  that  this  method  was  completely 
worked  out.  But  all  these  methods  that  have  been  published  sup- 
pose that  the  equation  has  been  reduced  and  cleared  of  fractions 
and  irrationals ;  I  mean  of  those  in  which  the  variables  occur.  I 
however  have  found  means  of  obviating  these  useless  reductions, 
which  make  the  calculation  increase  to  a  terrible  degree,  and  oblige 
us  to  rise  to  very  high  dimensions,  in  which  case  we  have  to  look 


428 


THE    MONIST. 


for  a  corresponding  depression  with  much  trouble ;  instead  of  all 
this,  everything  is  accomplished  at  the  first  attack. 

This  method  has  more  advantage  over  all  the  others  that  have 
been  published,  than  that  of  Sluse  has  over  the  rest,  because  it  is 
one  thing  to  give  a  simple  abridgment  of  the  calculation,  and  quite 
another  thing  to  get  rid  of  reductions  and  depressions.  With  respect 
to  the  publication  of  it,  on  account  of  the  great  extension  of  the 
matter  which  Descartes  himself  has  stated  to  be  the  most  useful 
part  of  Geometry,  and  of  which  he  has  expressed  the  hope  that  there 
is  more  to  follow — in  order  to  explain  myself  shortly  and  clearly, 
I  must  introduce  some  fresh  characters,  and  give  to  them  a  neiv 
Algorithm,  that  is  to  say,  altogether  special  rules,  for  their  addition, 
subtraction,  multiplication,  division,  powers,  roots,  and  also  for 
equations. 

Explanation  of  the  characters. 

Suppose  that  there  are  several  curves,  as  CD,  FE,  HJ,  con- 
nected with  one  and  the  same  axis  AB  by  ordinates  drawn  through 
one  and  the  same  point  B,  to  wit,  BC,  BE,  BH.  The  tangents  CT, 
FL,  HM  to  these  curves  cut  the  axis  in  the  points  T,  L,  M  ;  the 


\     \ 

X-J      n\ 


point  A  in  the  axis  is  fixed,  and  the  point  B  changes  with  the 
ordinates.  Let  AB  =  .r,  BC  =  y,  BF  =  w,  BH  =  t';  also  let  the  ratio 
of  TB  to  BC  be  called  that  of  dx  to  dy,  and  the  ratio  of  LB  to  BE 
that  of  d.\-  to  div,  and  the  ratio  of  MB  to  BH  that  of  dx  to  dv. 
Then  if,  for  example,  y  is  equal  to  vw,  we  should  say  dvzv  instead 
of  dy,  and  so  on  for  all  other  cases.  Let  a  be  a  constant  straight 
line ;  then,  if  y  is  equal  to  a,  that  is,  if  CD  is  a  straight  line  parallel 
to  AB,  dy  or  da  will  be  equal  to  0,  or  equal  to  zero.  If  the  magni- 
tude dxfdiv  comes  out  negative,  then  FL,  instead  of  being  drawn 


THE  MANUSCRIPTS  OF  LEIBNIZ.  429 

toward  A,  above  B,  will  be  drawn  In  the  contrary  direction,  be- 
low B. 

Addition   and  Subtraction.      Let  y  =  v±w(±)a,   then   dy   will   be 
equal  to  dv±dw(±}Q. 


Multi-plication.     Let  y  be  equal  to  atw,  then  dy  or  dawv  or  a  dvw 
will  be  equal  to  avdzv  +  aivdv. 


Division.    Let  y  be  equal  to    JL    then  dy  or  d  JL 

aw  aa> 

1    .  v  w  dv — v  dw 

or  —d —  will  be  equal  to  -       — 5 • 

a     w  aw 

The  rules  for  Powers  and  Roots  are  really  the  same  thing. 

Powers.     If  v  =  w*,  (where  r  is  supposed  to  be  a  certain  number), 
then  dy  will  be  equal  to  z,  w=~l,  div. 

dw 

Roots  or  extractions.     If    y  =  */»,  then  dz=  z  *~    /  . 

1/tP 

Equations  expressed  in  rational  integral  terms. 

a  H-  fo>  +  c  v  4-  toy  +  ez/2  +  /y2  +  gv  2y  +  hvy2  +  kv3  +  ly3 

4  ntv-y2  +  nv*y  4-  pvy3  +  qv*  +  ry*  =  0, 

supposing  that  a,  b,  c,  t,  e,  etc.  are  magnitudes  that  are  known  and 
determined ;  then  we  should  have 
0  =  bdv  +  cdy  +  tvdy  4-  2cvdv  +  2fydy  +  gv2dy  +  h\2dv 
tydv  +2gvydy  +2hvydy 

4-  Zly-dy  +  2mv2ydy  4-    mPdy   +   py*dv   +  4qv*dv  +  4r\:'dy 

4-  2mvy*dv  +  Znv-ydv  4-  3py-rdy 

This  rule  can  be  proved  and  continued  without  limit  by  the  pre- 
ceding rules ;  for,  if 

a  4  bv  4  cy  4  tvy  4  evz  4  fyz  4-  gv*y  +  etc.  =  0, 

then  da  4  dbv  +  dcy  +  tdvy  4  edv2  +  /rfv2  4  ^rf?'2 v  +  etc.  will  also  be  equal 
to  0.  Now  da  =  0,  dbv  =  bdv,  dcy  -  cdy,  di'y  =  vdy  +  ydv\  also  dv2  = 
2vdv,  since  dv*  is  equal  to  s,vs~l,dv,  that  is  to  say  (by  substituting 
2  for  z]  2vdv\  and  dv3y  =  v2dy  4- 2vydv,  for,  supposing  that  7^  =  ^, 
then  dv"y  will  be  rfwv,  and  fairy =ydw+i#dy,  and  rfw  or  dv-  =  2vdv, 
hence  in  the  value  of  dwy,  substituting  for  w  and  dw  the  values  found 


• 


43O  THE   MONIST. 

for  them,  we  shall  have  dv2y  =  v"dy  +  2vydv,  as  obtained  above. 
This  can  go  on  without  limit.  If  in  the  given  equation  a  +  bv+cy 
+  etc.  =0,  the  magnitude  v  were  equal  to  x,  that  is  to  say  if  the  line 
JH  were  a  straight  line  which  when  produced  passed  through  the 
point  A,  making  an  angle  of  45  degrees  with  the  axis,  then  the 
resulting  equation,  transformed  into  a  proportion,  would  give  the 
rule  for  the  method  of  tangents,  as  published  by  Sluse ;  and,  in 
consequence,  this  is  nothing  but  a  particular  case  or  corollary  of 
the  general  method. 

Equations  complicated  in  any  manner  with  fractions  and  irra- 
tionals. These  could  be  treated  in  the  same  way  without  any  calcu- 
lation, by  supposing  that  the  denominator  of  the  fraction  or  the 
magnitude  of  which  it  is  necessary  to  take  the  root  is  equal  to  a 
magnitude  or  letter,  which  is  to  be  treated  according  to  the  pre- 
ceding rules.07 

Also,  when  there  are  magnitudes  which  have  to  be  multiplied 
by  one  another,  there  is  no  need  to  make  this  multiplication  in 
reality,  which  saves  still  more  labor.  One  example  will  be  suffi- 
cient. 

[No  example  is  given,  however;  but  the  following  seems  to 
have  been  added  later,  according  to  Gerhardt.j 

Lastly  this  method  holds  good  when  the  curves  are  not  purely 
analytical,  and  even  when  their  nature  is  not  expressed  by  such 
ordinates,  and  in  addition  it  gives  a  marvelous  facility  for  making 
geometrical  constructions.  The  true  reason  for  an  abridgment  so 
admirable,  and  one  that  enables  us  to  avoid  reductions  of  fractions 
and  irrationals,  is  that  one  can  always  make  certain,  by  means  of 
the  preceding  rules,  that  the  letters  dy,  dv,  dw,  and  the  like,  shall 
not  occur  in  the  denominator  of  the  fraction,  or  under  the  root- 
sign. 

§XVI. 

The  next  manuscript  appears  to  be  a  more  detailed 
revision  of  the  one  last  considered.  It  bears  no  date;  but 
it  is  safe  to  say  that  it  belongs  to  a  considerably  later  period 
than  that  of  July  1677.  For  in  this  are  given,  by  means 
of  the  infinitely  small  quantities  dx  and  dy,  proofs  of  the 

67  The  complete  statement  of  the  method  of  substitutions. 


THE  MANUSCRIPTS  OF  LEIBNIZ.  43! 

fundamental  rules  for  the  first  time;  the  figure  notation 
is  changed  from  the  clumsy  C,  (C),  ((C))  to  the  neat 
iC,  2C,  3C;  the  notation  for  proportion  is  now  a:  b::  c:  d ; 
and  there  are  several  other  changes  that  readers  will  notice 
as  they  go  along.  The  ideas  of  Leibniz  are  now  approach- 
ing crystallization,  as  is  evidenced  by  the  fact  that  fy  dx  is 
clearly  stated  for  the  first  time  to  be  the  sum  of  rectangles 
made  from  y  and  dx.  It  is  rather  astonishing,  however,  in 

this  connection  to  find  J>  +  y  —  v  =  $x  +  fy  —  fv, 
which  can  have  no  significance  according  to  the  above 
definition;  and  also  to  find  the  whole  thing  explained  by 
arithmetical  series,  in  which  however  it  is  to  be  observed 
that  dx  is  not  taken  to  be  constant.  But  for  this  one  might 
almost  place  this  later  than  the  publication  of  the  method 
in  the  Acta  Eruditonim  in  1684;  in  this  essay  Leibniz  gave 
a  full  account  of  his  rules  without  proofs,  and  is  evidently 
trying  to  get  away  from  the  idea  of  the  infinitely  small,  an 
effort  which  culminates  in  the  next,  and  last,  manuscript 
of  this  set. 

If  then  we  guess  the  date  to  be  about  1680,  probably 
we  shall  not  be  very  far  out. 

A  remarkable  feature  of  this  manuscript  is  the  omission 
of  really  necessary  figures,  without  which  the  text  is  very 
hard  to  follow.  Of  course  this  manuscript  was  written 
for  publication,  and  the  suggestion  may  be  made  that  the 
diagrams  were  drawn  separately,  just  as  in  books  of  that 
time  they  were  printed  separately  on  folding  plates;  but 
then,  why  has  he  given  three  diagrams?  The  only  other 
suggestion  that  can  be  made  as  far  as  I  can  see  is  that  he 
was  referring  to  texts,  in  which  the  diagrams  were  already 
drawn,  by  Gregory  St.  Vincent,  Cavalieri,  James  Gregory 
(one  of  whose  theorems  he  quotes),  Barrow  (  who  strangely 
enough  also  quotes  the  very  same  theorem),  Wallis,  and 
others.  For  he  mentions  many  of  these  authors,  but  there 


432  THE    MONIST. 

is  never  a  word  about  Barrow.  I  consider  that  he  was 
looking  up  their  theorems  to  show  how  much  superior  his 
method  was  to  any  of  theirs. 

It  is  to  be  observed  that  not  even  in  this  manuscript  is 
there  any  mention  of  logarithms,  exponentials,  or  trigono- 
metrical ratios.  We  shall  see  later  that  Leibniz  is  reduced 
to  obtaining  the  integral  of  (a2  -f-  xzy/2  by  reference  to  a 
figure  and  its  quadrature;  that  is  to  say,  he  is  apparently 
unable  to  perform  the  integration  analytically.  It  there- 
fore follows  that,  if  he  got  a  great  deal  from  Barrow,  he 
was  unable  to  understand  the  Lect.  XII,  App.  I  of  the 
Lectiones  Geometricae. 

The  final  conclusion  that  I  personally  have  come  to, 
after  completing  this  examination  of  the  manuscripts  of 
Leibniz,  as  far  as  they  are  given  by  Gerhardt  is  this: 

As  far  as  the  actual  invention  of  the  calculus  as  he 
understood  the  term  is  concerned,  Leibniz  received  no  help 
from  Newton  or  Barrow ;  but  for  the  ideas  which  underlay 
it,  he  obtained  from  Barrow  a  very  great  deal  more  than  he 
acknowledged,  and  a  very  great  deal  less  than  he  would 
like  to  have  got,  or  in  fact  would  have  got  if  only  he 
had  been  more  fond  of  the  geometry  that  he  disliked.  For, 
although  the  Leibnizian  calculus  was  at  the  time  of  this 
essay  far  superior  .to  that  of  Barrow  on  the  question  of 
useful  application,  it  was  far  inferior  in  the  matter  of 
completeness. 

(No  date.) 

Elementa  calculi  novi  pro  differentiis  et  siimmis,  tangentibus  et 
quadratures,  maximis  et  minimi  s,  dimensionibus  linearum, 
super ficierum,  solidormn,  alilsque  communem  calculwn  trans- 
cendentibus. 

[The  elements  of  the  new  calculus  for  differences  and  sums,  tan- 
gents and  quadratures,  maxima  and  minima,  dimensions  of 
lines,  surfaces,  and  solids,  and  for  other  things  that  transcend 
other  means  of  calculation.] 


THE  MANUSCRIPTS  OF  LEIBNIZ. 


433 


Let  CC  be  a  line,  of  which  the  axis  is  AB,  and  let  BC  be  ordi- 
nates  perpendicular  to  this  axis,  these  being  called  y,  and  let  AB 
be  the  abscissae  cut  off  along  the  axis,  these  being  called  x. 


Then  CD,  the  differences  of  the  abscissae,  will  be  called  dx; 
such  are  tC  XD,  2C,D,  3C3D,  etc.  Also  the  straight  lines  jD2C, 
2D3C,  3D4C,  the  differences  of  the  ordinates,  will  be  called  dy. 
If  now  these  dx  and  dy  are  taken  to  be  infinitely  small,  or  the 
two  points  on  the  curve  are  understood  to  be  at  a  distance  apart 
that  is  less  than  any  given  length,  i.  e.,  if  iD2C,  2D3C,  etc.  are  con- 
sidered as  the  momentaneous  increments68  of  the  line  BC,  increas- 
ing continuously  as  it  descends  along  AB,  then  it  is  plain  that  the 
straight  line  joining  these  two  points,  2C  XC  say,  (which  is  an  element 
of  the  curve  or  a  side  of  the  infinite-angled  polygon  that  stands 
for  the  curve),  when  produced  to  meet  the  axis  in  /T,  will  be  the 
tangent  to  the  curve,  and  iTjB  (the  interval  between  the  ordinate 
and  the  tangent,  taken  along  the  axis)  will  be  to  the  ordinate  jB  jC  as 
,CjD  is  to  xD-jC;  or,  if  jTjB  or  L.T.,B,  etc.  are  in  general  called  /, 
then  t:y  : :  dx :  dy.  Thus  to  find  the  differences  of  series  is  to  find 
tangents. 

For  example,  it  is  required  to  find  the  tangent  to  the  hyperbola. 

a  a 
Here,  since  y=  — ,  supposing  that  in  the  diagram,  x  stands  for 

X 

AB  the  abscissa  along  an  asymptote,  and  a  for  the  side  of  the 
power,  or  of  the  area  of  the  rectangle  AB.BC;  then 

aa 

dy  —  —  — ax. 
xx 

68  Leibniz  has  evidently  seen  Newton's  work  at  the  time  of  this  composi- 
tion ;  also  the  use  of  the  word  "descends"  in  the  next  line  again  suggests 
Barrow,  while  the  figure  is  exactly  like  the  top  half  of  the  diagram  given  by 
Barrow  for  Lect.  XI,  10,  which  is  the  theorem  of  Gregory  that  is  quoted  by 
Leibniz  also.  For  this  figure,  see  the  note  to  that  passage. 


434 


THE   MONIST. 


as  will  be  soon  seen  when  we  set  forth  the  method  of  this  calculus ; 
hence  dx :  dy  or  t :  y  : :  -  xx  :aa  : :  -  x  :  —  : :  -x\y;  therefore  t  =  -y, 


that  is,  in  the  hyperbola  BT  will  be  equal  to  AB,  but  on  account  of 
the  sign  -x,  BT  must  be  taken  not  toward  A  but  in  the  opposite 
direction. 

Moreover,  differences  are  the  opposite  to  sums;  thus  4B4C  is 
the  sum  of  all  the  differences  such  as  3D  4C,  2D  3C,  etc.  as  far  as  A, 
even  if  they  are  infinite  in  number.  This  fact  I  represent  thus, 
fdy  =  y.  Also  I  represent  the  area  of  a  figure  by  the  sum  of  all 
the  rectangles  contained  by  the  ordinates  and  the  differences  of  the 
abscissae,  i.  e.,  by  the  sum  tB  jD  +  2B  2D  +  3B  3D  +  etc.  For  the  nar- 
row triangles  jC  ^  2C,  2C  2D  3C,  etc.,  since  they  are  infinitely  small 
compared  with  the  said  rectangles,  may  be  omitted  without  risk ; 
and  thus  I  represent  in  my  calculus  the  area  of  the  figure  by  fy  dx, 
or  the  sum  of  the  rectangles  contained  by  each  y  and  the  dx  that 
corresponds  to  it ;  here,  if  the  dx's  are  taken  equal  to  one  another, 
the  method  of  Cavalieri  is  obtained. 

But  we,  now  mounting  to  greater  heights,  obtain  the  area  of 
a  figure  by  finding  the  figure  of  its  summatrix  or  quadratrix ;  and 
of  this  indeed  the  ordinates  are  to  the  ordinates  of  the  given 
figure  in  the  ratio  of  sums  to  differences ;  for  instance,  let  the  curve 
of  the  figure  required  to  be  squared  be  EE,  and  let  the  ordinates 
to  it,  EB,  which  we  will  call  e,  be  proportional  to  the  differences 
of  the  ordinates  BC,  or  to  dy;  that  is  let  aB  jE :  2B  2E  : :  jD  2C :  2D  3C, 
and  so  on;  or  again,  let  AjBijBjC,  1C1D:1D2C,  etc.,  or  dx:dy 
be  in  the  ratio  of  a  constant  or  never-varying  straight  line  a  to  tB  JL 
or  e;  then  we  have 

d.v  :dy  ::  a:e,  or  e  dx  =  a  dy ; 
•'•  §e  dx  =  fady. 

But  c  dx  is  the  same  as  e  multiplied  by  its  corresponding  dx, 
such  as  the  rectangle  3B  4E,  which  is  formed  from  3B  3E  and  3B  4B ; 
hence,  fed*  is  the  sum  of  all  such  rectangles,  3B  4E  +  2B  !E  +  3B  2E 
+  etc.,  and  this  sum  is  the  figure  A  4B  4EA,  if  it  is  supposed  that  the 


THE  MANUSCRIPTS  OF  LEIBNIZ. 


435 


d.r's,  or  the  intervals  between  the  ordinates  e,  or  BC,  are  infinitely 
small.  Again,  ady  is  the  rectangle  contained  by  a  and  dy,  such  as 
is  contained  by  3D4C  and  the  constant  length  a,  and  the  sum  of 


^x 

"v     f 

<t^ 

.1 

\    .a 

}\ 

"\    B 

1     \£ 

•c,\* 

\,c 

these  rectangles,  namely  fady,  or  3D4C.a  +  2D3C.a  +  1D2C. 
is  the  same  as  gD^  +  aDgC  +  ^oC  +  etc.  into  a,  that  is,  the  same 
as  4B4C.a;  therefore  we  have  fady  =  afdy  =  ay.  Therefore  §edx 
=  ay,  that  is,  the  area  A4B4EA  will  be  equal  to  the  rectangle  con- 
tained by  4B  4C  and  the  constant  line  a,  and  generally  ABEA  is 
equal  to  the  rectangle  contained  by  BC  and  a.09 

Thus,  for  quadratures  it  is  only  necessary,  being  given  the  line 
EE,  to  find  the  summatrix  line  CC,  and  this  indeed  can  always  be 
found  by  calculus,  whether  such  a  line  is  treated  in  ordinary  geom- 
etry or  whether  it  is  transcendent  and  cannot  be  expressed  by  alge- 
braical calculation ;  of  this  matter  in  another  place. 

Now  the  triangle  for  the  line  I  call  the  characteristic  of  the 
line,  because  by  its  most  powerful  aid  there  can  be  found  theorems 
about  the  line  which  are  seen  to  be  admirable,  such  as  its  length, 
the  surface  and  solid  produced  by  its  rotation,  and  its  center  of 
gravity ;  for  jC 2C  is  equal  to  -\/d.r.d.v+  dy.dy.  From  this  we  have 

69  Leibniz  does  not  give  a  diagram,  but  it  is  not  difficult  to  construct  his 
figure  from  the  enunciation  that  he  gives  for  it.  The  whole  of  this  paragraph 
should  be  compared  with  the  following  extract  from  Barrow  (Lect.  XI,  19), 
piece  by  piece. 

"Again,  let  AMB  be  a  curve  of  which  the  axis  is  AD  and  let  BD  be 
perpendicular  to  AD;  also  let  KZL  be  another  line  such  that,  when  any  point 
M  is  taken  in  the  curve  AB,  and  through  it  are  drawn  MT  a  tangent  to  the 
curve  AB,  and  MFZ  parallel  to  DB,  cutting  KZ  in  Z  and  AD  in  F,  and  R  is 
a  line  of  given  length,  TF:  FM  =  R :  FZ.  Then 
the  space  ADLK  is  equal  to  the  rectangle  con- 
tained by  R  and  DB. 

For,  if  DH  =  R  and  the  rectangle  BDHI 
is  completed,  and  MN  is  taken  to  be  an  indefi- 
nitely small  arc  of  the  curve  AB,  and  MEX, 
NOS  are  drawn  parallel  to  AD;  then  we  have 
NO :  MO  =  TF :  FM  =  R :  FZ ; 

NO.FZ  =  MO.R    and    FG.FZ-ES.EX. 

Hence,  since  the  sum  of  such  rectangles  as 
FG.FZ  differs  only  in  the  least  degree  from 

the  space  ADLK,  and  the  rectangles  ES.EX  form  the  rectangle  DHIB,  the 
theorem  is  quite  obvious. 


T       A 

fi 

Z. 

F 

L 
U----  R         . 

\ 

^ 

N 

0 

0                H 
S               x 

N. 

E.              x 

M 

V 

B                | 

436 


THE   MONIST. 


at  once  a  method  for  finding  the  length  of  a  curve  by  means  of 

XX 

some  quadrature ;  e.  g.,  in  the  case  of  the  parabola,  if  y=-^~ ,  then  we 


have  d\'= ,  and  hence  iC  zC=- 

a  a 


the  ordinate  of  the  hyperbola  Vaa  +  •*'•*"  is  to  the  constant  line  a; 

I   r        

that  is,  -    I   dx^aa  +  xx  ,   a   straight   line   equal   to  the  arc   of   a 

a  J 

parabola,  depends  on  the  quadrature  of  the  hyperbola,  as  has  already 
been  found  by  others ;  and  thus  we  can  derive  by  the  calculus  all 
the  most  beautiful  results  discovered  by  Huygens,  Wallis,  van 
Huraet,  and  Neil.70 

I  said  above  that  t : y  ::  dx\dy;  hence  we  have  t dy  =  y Ax,  and 
therefore  §t  dy=  §ydx.  This  equation,  enunciated  geometrically, 
gives  an  elegant  theorem  due  to  Gregory.71  namely  that,  if  BAF  is  a 
right  angle,  and  AF  =  BG,  and  FG  is  parallel  to  AB  and  equal  to 
BT,  that  is,  1F1G  =  1B1T,  then  ftdy,  or  the  sum  of  the  rectangles 
contained  by  t  (e.g.,  4F4G  or  4B4T)  and  dy  (3F4F  or  3D4C)  is 
equal  to  the  rectangles  4F3G  +  3F2G  +  2F  jG  +  etc.,  or  the  area  of  the 

70  All  the  things  given  are  to  be  found  in  Barrow,  but  his  name  is  not  even 
mentioned. 

71  This  is  the  strangest  coincidence  of  all !     For,  Barrow  also  quotes  this 
very  same  theorem  of  Gregory,  and  no  other  theorem ;  also  it  occurs  in  this 
very  same  Lect.  XI  that  has  been  referred  to  already !    Leibniz  does  not  give 
a  diagram;  nor  from  his  enunciation  could  I  complete  the  figure  required,  until 
I  had  referred  to  the  figure  given  by  Barrow !! !     The  two  diagrams  are  given 
below  for  comparison,  Barrow's  figure  being  the  one  referred  to  in  the  note 
above.    Query,  is  Leibniz's  figure  taken  from  Gregory's  original,  which  I  have 
not  been  able  to  see,  or  is  it  the  Leibnizian  variation  of  Barrow's? 


THE  MANUSCRIPTS  OF  LEIBNIZ.  437 

figure  A  4F  4GA  is  equal  to  fy  dx,  that  is,  to  the  figure  A  4B  4CA ; 
or  generally,  the  figure  AFGA  is  equal  to  the  figure  ABCA. 

Again,  other  things,  which  are  immediately  evident  on  inspec- 
tion, from  a  figure,  are  readily  deduced  by  the  calculus ;  for  instance, 
in  the  case  of  the  trilinear  figure  ABCA,  the  figure  ABCA  together 
with  its  complementary  figure  AFCA  is  equal  to  the  rectangle 
ABCF,  for  the  calculus  readily  shows  that  fydx+fxdy  =  xy. 

If  it  is  required  to  find  the  volume  of  the  solid  formed  by 
rotation  round  an  axis,  it  is  only  necessary  to  find  Cy-  dx ;  for  the 
solid  formed  by  a  rotation  round  the  base,  $x-dy ;  for  the  moment 
about  the  vertex,  fyxdx;  and  these  things  serve  to  find  the  center 
of  gravity  of  a  figure,  and  also  give  the  frusta  of  Gregory  St. 
Vincent,  and  all  that  Pascal,  Wallis,  De  Laloubere,  and  others  have 
found  out  about  these  matters. 

For,  if  it  is  required  to  find  the  centers  of  lines,  or  the  surfaces 
generated  by  their  rotation,  e.  g.,  the  surface  generated  by  the  rota- 
tion of  the  line  AC  about  AB,  it  is  only  necessary  to  find 


J  y  V  dx.  dx  +  dy.  dy 


or  the  sum  of  every  PC  applied  to  the  axis  at  the  point  B  that 
corresponds  to  it,  (thus  »P  2C  will  be  applied  perpendicular  to  the 
axis  AB  at  2B),  producing  in  this  way  a  figure  of  which  the  above 
represents  the  area.  Thus  the  whole  thing  will  immediately  reduce 
to  the  quadrature  of  some  plane  figure,  if,  instead  of  y  and  dy,  their 
values,  obtained  from  the  nature  of  the  ordinates  and  the  tangents 
to  the  curve,  are  substituted.  Thus,  in  the  case  of  the  parabola, 

if  y  is  equal  to  ~\/2ax,  then  dy=  -  -  (as  will  be  seen  directly)  ; 
hence  we  get 

aa  r      I  r      I 

dxdx  +  —  dxdx  or  J  dx^Ayy  +  aa  or  j  dx^l  2ax  +  aa  , 

which  depends  on  the  quadrature  of  the  parabola  (for  every 
-\/2ax  +  aa  or  PC  can  be  applied  to  a  parabola,  if  it  is  supposed  that 
AC  is  the  parabola,  and  AB  its  axis,  provided  in  that  case  the 
figure  is  changed  and  the  curve  turns  its  concavity  toward  the 
axis)  ;72  and  this  may  be  obtained  by  ordinary  geometry,  and  there- 

72  The  Latin  here  is  rather  ambiguous ;  query,  a  misprint.  But  I  think  I 
have  correctly  rendered  the  argument.  It  is  to  be  noted  that  the  parabola 
was  at  this  period  always  thought  of  in  the  form  we  should  now  denote  by 
the  equation  y  =  xz,  and  the  figure  referred  to  by  Leibniz  is  that  which  Wallis 
calls  the  complement  of  the  semiparabola. 


438  THE   MONIST. 

fore  also  a  circle  will  be  found  equal  to  the  surface  of  the  parabolic 
conoid ;  but  this  is  not  the  place  to  deduce  it  at  full  length. 

Now  these,  which  may  seem  to  be  great  matters,  are  only  the 
very  simplest  results  to  be  obtained  by  this  calculus ;  for  many 
much  more  important  consequences  follow  from  it,  nor  does  there 
occur  any  simple  problem  in  geometry,  either  pure  or  applied  to 
mechanics,  that  can  altogether  evade  its  power.  Now  we  will  ex- 
pound the  elements  of  the  calculus  itself. 

The  fundamental  principle  of  the  calculus. 

Differences  and  sums  are  the  inverses  of  one  another,  that  is 
to  say,  the  sum  of  the  differences  of  a  series  is  a  term  of  the  series, 
and  the  difference  of  the  sums  of  a  series  is  a  term  of  the  series ; 
and  I  enunciate  the  former  thus,  §dx-x,  and  the  latter  thus, 
d^x-x. 

Thus,  let  the  differences  of  a  series,  the  series  itself,  and  the 
sums  of  the  series,  be,  let  us  say, 

Diffs.  1       2      3      4      5     dx 

Series  0       1       3      6       10     15      x 

Sums  0       1       4      10    20    25      . .    $x 

Then  the  terms  of  the  series  are  the  sums  of  the  differences,  or 
x=(dx;  thus,  3=1  +  2,  6=1+2  +  3,  etc.;  on  the  other  hand,  the 
differences  of  the  sums  of  the  series  are  terms  of  the  series,  or 
d^x-x\  thus,  3  is  the  difference  between  1  and  4,  6  between 
4  and  10. 

Also  da  =  Q,  if  it  is  given  that  a  is  a  constant  quantity,  since 
a-a  =  0. 

Addition  and  Subtraction. 

The  difference  or  sum  of  a  series,  of  which  the  general  term 
is  made  up  of  the  general  terms  of  other  series  by  addition  or  sub- 
traction, is  made  up  in  exactly  the  same  manner  from  the  differ- 
ences or  sums  of  these  series ;  or 


x  +  y - v  =  §dx  +  dy-  dv,  §x  +  y-v  =  §x  +  fy-  fv. 

This  is  evident  at  sight,  if  you  take  any  three  series,  set  out  their 
sums  and  their  differences,  and  take  them  together  correspondingly 
as  above. 


THE  MANUSCRIPTS  OF  LEIBNIZ.  439 

Simple  Multiplication. 

Here  dxy  =  xdx  +  ydy,  or  xy=fxdx+fydy. 

This  is  what  we  said  above  about  figures  taken  together  with  their 
complements  being  equal  to  the  circumscribed  rectangle.  It  is 
demonstrated  by  the  calculus  as  follows: 

dxy  is  the  same  thing  as  the  difference  between  two  successive 
xy's;  let  one  of  these  be  .vy,  and  the  other  x  +  dx  into.y  +  dy;  then 
we  have 


dxy  =  x  +  dx  .  y  +  dy-  xy  =  xdy  +  y  dx  +  dx  dy ; 

the  omission  of  the  quantity  dx  dy,  which  is  infinitely  small  in  com- 
parison with  the  rest,  for  it  is  supposed  that  dx  and  dy  are  infinitely 
small  (because  the  lines  are  understood  to  be  continuously  increas- 
ing or  decreasing  by  very  small  increments  throughout  the  series 
of  terms),  will  leave  xdy  +  ydx;  the  signs  vary  according  as  y  and  x 
increase  together,  or  one  increases  as  the  other  decreases ;  this 
point  must  be  noted. 

Simple  Division. 

y      x  dy—  y  dx 

Here  we  have  d  -  —  - . 

x  xx 

.y          y  +  dy         y        x  dy  —y  dx          ,  .  ,    ,  , . , 

For,  d  -    =   -  .   _  i  =  _  _       which  becomes   (if  we 

x         x+dx        x         xx  +  x  dx 

write  xx  for  xx  +  xdx,  since  xdx  can  be  omitted  as  being  infinitely 

small  in  comparison  with  xx}  equal  to  — - — - — —  ;  also,  if  y  =  aa. 

xx 

then  dy  =  0,  and  the  result  becomes ,  which  is  the  value  we 

xx 

used  a  little  while  before  in  the  case  of  the  tangent  to  the  hyper- 
bola. 

From  this  any  one  can  deduce  by  the  calculus  the  rules  for 
Compound  Multiplication  and  Division;  thus, 
dxvy  =  xy  dv  +  xv  dy  +  yv  dx, 

,  y  _  xv  dy  —yv  dz — yz  dv 
a—  — ; 

vz  vv.zz 

as  can  be  proved  from  what  has  gone  before ;  for  we  have 

dv  =  x  dy—y  dx . 

X  XX 

hence,  putting  zv  for  x,  and  sdv  +  vdz  for  dx  or  dzv  in  the  above, 
we  obtain  what  was  stated. 


44°  THE    MONIST. 

Powers  follow:  dxz  =  2xdx,  dx*  =  Zx*dx,  and  so  on.  For,  putting 
y  =  x,  and  v=x,  we  can  write  dxz  for  dxy,  and  this  is  (from  above) 
equal  to  xdy  +  ydx,  or  (if  x  =  y,  and  consequently  dx  =  dy)  equal 
to  2xdx.  Similarly,  for  dxs  we  write  dxyv,  that  is  (from  above) 
xydv  +  xvdy  +  yvdx,  or  (putting  x  for  y  and  v  and  d.r  for  cfy  and 
dv)  equal  to  3x*dx.  Q.  E.  D.  By  the  same  method,  in  general, 
dx'  =  e.x'—  dx,  as  can  easily  be  proved  from  what  has  been  said. 

.1             h  dx 
Hence  also,  a -— *  = A+T  • 

For,  if  —  =  jv',  then  e=  —  h,  and  x'~l—  ->— , ,  as  is  well  known  to 

any  one  who  understands  the  nature  of  the  exponents  in  a  geo- 
metrical progression.  The  same  thing  will  do  for  fractions.  The 
procedure  is  the  same  for  irrationals  or  Roots.  dt\/xh  —  dx''r, 
(where  by  h:r  I  mean  h/r,  or  h  divided  by  r),  or  dxe  (taking  e 
equal  to  h/r),  or  e.x~  dx,  by  what  has  been  said  above,  or  (by 


substituting  once  more  h :  r  for  e,  and  h-r:r  for  e  -  \ )  — .  x*-rr  .dx; 

and  thus  finally  we  get  the  value  of  d>\/xH. 

Moreover,  conversely,  we  have 

f 

r 


f.t*Wxm~,    f-e<tx=—=-l ,    f<i 

J  e  +  V  J  xe  e-1.*-1 J 


«jr*i±Z. 

These  are  the  elementary  principles  of  the  differential  and 
summatory  calculus,  by  means  of  which  highly  complicated  formu- 
las can  be  dealt  with,  not  only  for  a  fraction  or  an  irrational  quan- 
tity, or  anything  else ;  but  also  an  indefinite  quantity,  such  as  x  or  y, 
or  any  other  thing  expressing  generally  the  terms  of  any  series, 
may  enter  into  it. 

§  XVII. 

The  next  manuscript  bears  no  date;  but  this  can  be 
easily  assigned  to  a  certain  extent,  from  internal  evidence. 
It  is  for  one  thing  later  than  the  publication  in  the  A  eta 
Eruditorum  of  Leibniz's  first  communication  to  the  world 
of  his  calculus  in  1684.  The  manuscript  is  an  answer,  or 
rather  the  first  rough  draft  probably  of  such  an  answer, 
to  the  animadversions  of  Bernhard  Nieuwentijt  against 
the  idea  of  the  infinitesimal  calculus.  The  latter  stated 
that  (i)  Leihniz  could  explain  no  more  than  Barrow  or 


THE  MANUSCRIPTS  OF  LEIBNIZ.  44! 

Newton  how  the  infinitely  small  differences  differed  from 
absolute  zero;  (ii)  it  was  not  clear  how  the  differentials 
of  higher  order  were  obtained  frorh  those  of  the  first 
order;  (iii)  the  differential  method  cannot  be  applied  to 
exponential  functions.  Leibniz  answers  the  first  point  skil- 
fully, fails  over  the  second  through  erroneous  work,  which 
I  think  he  afterward  perceived;  for  he  has  a  note  that  the 
whole  thing  is  to  be  carefully  revised  before  publication. 
It  almost  seems  that  he  was  not  quite  confident  in  his  own 
powers  of  completely  answering  these  objections,  for  he 
also  notes  that  the  rudeness  of  language  in  which  the 
answer  is  commenced  must  be  mollified. 

On  the  third  point  he  is  silent;  in  the  later  written 
Historia,  we  have  seen  he  is  able  to  get,  not  over,  but  round 
the  difficulty  of  the  exponential  function;  but  the  silence 
here  would  seem  to  say  that  Leibniz  could  not  manage  ex- 
ponentials as  yet. 

The  success  of  the  answer  to  the  first  point  is  due  to 
the  underlying  principle  that  the  ratio  dy :  dx  ultimately 
becomes  a  rate;  when  this  idea  is  muddled  by  an  admixture 
of  the  infinitesimal  idea  in  the  last  paragraph  the  result 
is  almost  disastrous.  Leibniz,  however,  looked  on  his  cal- 
culus as  a  tried  tool  more  than  anything  else. 

When  my  infinitesimal  calculus,  which  includes  the  calculus  of 
differences  and  sums,  had  appeared  and  spread,  certain  over-precise 
veterans  began  to  make  trouble ;  just  as  once  long  ago  the  Sceptics 
opposed  the  Dogmatics,  as  is  seen  from  the  work  of  Empicurus 
against  the  mathematicians  (i.  e.,  the  dogmatics),  and  such  as 
Francisco  Sanchez,  the  author  of  the  book  Quod  nihil  scitur,  brought 
against  Clavius  ;  and  his  opponents  to  Cavalieri,  and  Thomas  Hobbes 
to  all  geometers,  and  just  lately  such  objections  as  are  made  against 
the  quadrature  of  the  parabola  by  Archimedes  by  that  renowned 
man,  Dethlevus  Cluver.  When  then  our  method  of  infinitesimals, 
which  had  become  known  by  the  name  of  the  calculus  of  differences, 
began  to  be  spread  abroad  by  several  examples  of  its  use,  both  of 
my  own  and  also  of  the  famous  brothers  Bernoulli,  and  more  espe- 


442  THE   MONIST. 

cially  by  the  elegant  writings  of  that  illustrious  Frenchman,  the 
Marquis  d'Hopital,  just  lately  a  certain  erudite  mathematician, 
writing  under  an  assumed  name  in  the  scientific  Journal  de  Trevoux, 
appeared  to  find  fault  with  this  method.  But  to  mention  one  of 
them  by  name,  even  before  this  there  arose  against  me  in  Holland 
Bernard  Nieuwentiit,  one  indeed  really  well  equipped  both  in 
learning  and  ability,  but  one  who  wished  rather  to  become  known 
by  revising  our  methods  to  some  extent  than  by  advancing  them. 
Since  I  introduced  not  only  the  first  differences,  but  also  the  second, 
third  and  other  higher  differences,  inassignable  or  incomparable 
with  these  first  differences,  he  wished  to  appear  satisfied  with 
the  first  only;  not  considering  that  the  same  difficulties  existed 
in  the  first  as  in  the  others  that  followed,  nor  that  wherever  they 
might  be  overcome  in  the  first,  they  also  ceased  to  appear  in  the 
rest.  Not  to  mention  how  a  very  learned  young  man,  Hermann 
of  Basel,  showed  that  the  second  and  higher  differences  were 
avoided  by  the  former  in  name  only,  and  not  in  reality ;  moreover, 
in  demonstrating  theorems  by  the  legitimate  use  of  the  first  differ- 
ences, by  adhering  to  which  he  might  have  accomplished  some 
useful  work  on  his  own  account,  he  fails  to  do  so,  being  driven  to 
fall  back  on  assumptions  that  are  admitted  by  no  one ;  such  as 
that  something  different  is  obtained  by  multiplying  2  by  m  and  by 
multiplying  m  by  2 ;  that  the  latter  was  impossible  in  any  case  in 
which  the  former  was  possible;  also  that  the  square  or  cube  of  a 
quantity  is  not  a  quantity  or  Zero. 

In  it,  however,  there  is  something  that  is  worthy  of  all  praise, 
in  that  he  desires  that  the  differential  calculus  should  be  strength- 
ened with  demonstrations,  so  that  it  may  satisfy  the  rigorists ;  and 
this  work  he  would  have  procured  from  me  already,  and  more 
willingly,  if,  from  the  fault-finding  everywhere  interspersed,  the 
wish  had  not  appeared  foreign  to  the  manner  of  those  who  desire 
the  truth  rather  than  fame  and  a  name. 

It  has  been  proposed  to  me  several  times  to  confirm  the  essen- 
tials of  our  calculus  by  demonstrations,  and  here  I  have  indicated 
below  its  fundamental  principles,  with  the  intent  that  any  one  who 
has  the  leisure  may  complete  the  work.  Yet  I  have  not  seen  up 
to  the  present  any  one  who  would  do  it.  For  what  the  learned 
Hermann  has  begun  in  his  writings,  published  in  my  defence  against 
Nieuwentiit,  is  not  yet  complete. 

For  I  have,  beside  the  mathematical  infinitesimal  calculus,  a 
method  also  for  use  in  Physics,  of  which  an  example  was  given  in 


THE  MANUSCRIPTS  OF  LEIBNIZ. 


443 


the  Nouvelles  de  la  Republique  des  Lettres;  and  both  of  these  I 
include  under  the  Law  of  Continuity ;  and  adhering  to  this,  I  have 
shown  that  the  rules  of  the  renowned  philosophers  Descartes  and 
Malebranche  were  sufficient  in  themselves  to  attack  all  problems 
on  Motion. 

I  take  for  granted  the  following  postulate: 

In  any  supposed  transition,  ending  in  any  terminus,  it  is  per- 
missible to  institute  a  general  reasoning,  in  which  the  final  terminus 
may  also  be  included. 

For  example,  if  A  and  B  are  any  two  quantities,  of  which  the 
former  is  the  greater  and  the  latter  is  the  less,  and  while  B  remains 
the  same,  it  is  supposed  that  A  is  continually  diminished,  until  A 
becomes  equal  to  B ;  then  it  will  be  permissible  to  include  under  a 
general  reasoning  the  prior  cases  in  which  A  was  greater  than  B, 
and  also  the  ultimate  case  in  which  the  difference  vanishes  and  A 
is  equal  to  B.  Similarly,  if  two  bodies  are  in  motion  at  the  same 
time,  and  it  is  assumed  that  while  the  motion  of  B  remains  the 
same,  the  velocity  of  A  is  continually  diminished  until  it  vanishes 
altogether,  or  the  speed  of  A  becomes  zero ;  it  will  be  permissible 
to  include  this  case  with  the  case  of  the  motion  of  B  under  one 
general  reasoning.  We  do  the  same  thing  in  geometry,  when  two 


C 

i 

(O 


straight  lines  are  taken,  produced  in  any  manner,  one  VA  being 
given  in  position  or  remaining  in  the  same  site,  the  other  BP  passing 
through  a  given  point  P,  and  varying  in  position  while  the  point  P 
remains  fixed ;  at  first  indeed  converging  toward  the  line  VA  and 
meeting  it  in  the  point  C;  then,  as  the  angle  of  inclination  VGA 
is  continually  diminished,  meeting  VA  in  some  more  remote  point 
(C),  until  at  length  from  BP,  through  the  position  (B)P,  it  comes 


444  THE  MONIST. 

to  fiP,  in  which  the  straight  line  no  longer  converges  toward  VA, 
but  is  parallel  to  it,  and  C  is  an  impossible  or  imaginary  point. 
With  this  supposition  it  is  permissible  to  include  under  some  one 
general  reasoning  not  only  all  the  intermediate  cases  such  as  (B)P 
but  also  the  ultimate  case  (3P. 

Hence  also  it  comes  to  pass  that  we  include  as  one  case  ellipses 
and  the  parabola,  just  as  if  A  is  considered  to  be  one  focus  of  an 
ellipse  (of  which  V  is  the  given  vertex),  and  this  focus  remains 
fixed,  while  the  other  focus  is  variable  as  we  pass  from  ellipse  to 
ellipse,  until  at  length  (in  the  case  when  the  line  BP,  by  its  inter- 
section with  the  line  VA,  gives  the  variable  focus)  the  focus  C 
becomes  evanescent73  or  impossible,  in  which  case  the  ellipse  passes 
into  a  parabola.  Hence  it  is  permissible  with  our  postulate  that  a 
parabola  should  be  considered  with  ellipses  under  a  common  rea- 
soning. Just  as  it  is  common  practice  to  make  use  of  this  method 
in  geometrical  constructions,  when  they  include  under  one  general 
construction  many  different  cases,  noting  that  in  a  certain  case  the 
converging  straight  line  passes  into  a  parallel  straight  line,  the 
angle  between  it  and  another  straight  line  vanishing. 

Moreover,  from  this  postulate  arise  certain  expressions  which 
are  generally  used  for  the  sake  of  convenience,  but  seem  to  con- 
tain an  absurdity,  although  it  is  one  that  causes  no  hindrance, 
when  its  proper  meaning  is  substituted.  For  instance,  we  speak  of 
an  imaginary  point  of  intersection  as  if  it  were  a  real  point,  in  the 
same  manner  as  in  algebra  imaginary  roots  are  considered  as  ac- 
cepted numbers.  Hence,  preserving  the  analogy,  we  say  that,  when 
the  straight  line  BP  ultimately  becomes  parallel  to  the  straight  line 
VA,  even  then  it  converges  toward  it  or  makes  an  angle  with  it, 
only  that  the  angle  is  then  infinitely  small ;  similarly,  when  a  body 
ultimately  comes  to  rest,  it  is  still  said  to  have  a  velocity,  but  one 
that  is  infinitely  small ;  and,  when  one  straight  line  is  equal  to 
another,  it  is  said  to  be  unequal  to  it,  but  that  the  difference  is 
infinitely  small ;  and  that  a  parabola  is  the  ultimate  form  of  an 
ellipse,  in  which  the  second  focus  is  at  an  infinite  distance  from  the 
given  focus  nearest  to  the  given  vertex,  or  in  which  the  ratio  of 
PA  to  AC,  or  the  angle  BCA,  is  infinitely  small. 

Of  course  it  is  really  true  that  things  which  are  absolutely 
equal  have  a  difference  w^hich  is  absolutely  nothing ;  and  that 
straight  lines  which  are  parallel  never  meet,  since  the  distance 

73  The  term  is  here  used  with  the  idea  of  "vanishing  into  the  far  distance." 


THE  MANUSCRIPTS  OF  LEIBNIZ.  44$ 

between  them  is  everywhere  the  same  exactly ;  that  a  parabola  is 
not  an  ellipse  at  all,  and  so  on.  Yet,  a  state  of  transition  may  be 
imagined,  or  one  of  evanescence,  in  which  indeed  there  has  not  yet 
arisen  exact  equality  or  rest  or  parallelism,  but  in  which  it  is 
passing  into  such  a  state,  that  the  difference  is  less  than  any  assign- 
able quantity ;  also  that  in  this  state  there  will  still  remain  some 
difference,  some  velocity,  some  angle,  but  in  each  case  one  that  is 
infinitely  small ;  and  the  distance  of  the  point  of  intersection,  or 
the  variable  focus,  from  the  fixed  focus  will  be  infinitely  great, 
and  the  parabola  may  be  included  under  the  heading  of  an  ellipse 
(and  also  in  the  some  manner  and  by  the  same  reasoning  under  the 
heading  of  a  hyperbola),  seeing  that  those  things  that  are  found  to 
be  true  about  a  parabola  of  this  kind  are  in  no  way  different,  for 
any  construction,  from  those  which  can  be  stated  by  treating  the 
parabola  rigorously. 

Truly  it  is  very  likely  that  Archimedes,  and  one  who  seems 
so  have  surpassed  him,  Conon,  found  out  their  wonderfully  elegant 
theorems  by  the  help  of  such  ideas ;  these  theorems  they  completed 
with  reductio  ad  absurdum  proofs,  by  which  they  at  the  same  time 
provided  rigorous  demonstrations  and  also  concealed  their  methods. 
Descartes  very  appropriately  remarked  in  one  of  his  writings  that 
Archimedes  used  as  it  were  a  kind  of  metaphysical  reasoning 
(Caramuel  would  call  it  metageometry),  the  method  being  scarcely 
used  by  any  of  the  ancients  (except  those  who  dealt  with  quad- 
ratrices)  ;  in  our  time  Cavalieri  has  revived  the  method  of  Archi- 
medes, and  afforded  an  opportunity  for  others  to  advance  still 
further.  Indeed  Descartes  himself  did  so,  since  at  one  time  he 
imagined  a  circle  to  be  a  regular  polygon  with  an  infinite  number 
of  sides,  and  used  the  same  idea  in  treating  the  cycloid ;  and  Huy- 
gens  too,  in  his  work  on  the  pendulum,  since  he  was  accustomed 
to  confirm  his  theorems  by  rigorous  demonstrations ;  yet  at  other 
times,  in  order  to  avoid  too  great  prolixity,  he  made  use  of  infini- 
tesimals ;  as  also  quite  lately  did  the  renowned  La  Hire. 

For  the  present,  whether  such  a  state  of  instantaneous  transi- 
tion from  inequality  to  equality,  from  motion  to  rest,  from  con- 
vergence to  parallelism,  or  anything  of  the  sort,  can  be  sustained 
in  a  rigorous  or  metaphysical  sense,  or  whether  infinite  extensions 
successively  greater  and  greater,  or  infinitely  small  ones  successively 
less  and  less,  are  legitimate  considerations,  is  a  matter  that  I  own 
to  be  possibly  open  to  question ;  but  for  him  who  would  discuss 
these  matters,  it  is  not  necessary  to  fall  back  upon  metaphysical 


446  THE   MONIST. 

controversies,  such  as  the  composition  of  the  continuum,  or  to 
make  geometrical  matters  depend  thereon.  Of  course,  there  is  no 
doubt  that  a  line  may  be  considered  to  be  unlimited  in  any  manner, 
and  that,  if  it  is  unlimited  on  one  side  only,  there  can  be  added 
to  it  something  that  is  limited  on  both  sides.  But  whether  a  straight 
line  of  this  kind  is  to  be  considered  as  one  whole  that  can  be  re- 
ferred to  computation,  or  whether  it  can  be  allocated  among  quan- 
tities which  may  be  used  in  reckoning,  is  quite  another  question 
that  need  not  be  discussed  at  this  point. 

It  will  be  sufficient  if,  when  we  speak  of  infinitely  great  (or 
more  strictly  unlimited),  or  of  infinitely  small  quantities  (i.  e.,  the 
very  least  of  those  within  our  knowledge),  it  is  understood  that 
we  mean  quantities  that  are  indefinitely  great  or  indefinitely  small, 
i.  e.,  as  great  as  you  please,  or  as  small  as  you  please,  so  that  the 
error  that  any  one  may  assign  may  be  less  than  a  certain  assigned 
quantity.  Also,  since  in  general  it  will  appear  that,  when  any  small 
error  is  assigned,  it  can  be  shown  that  it  should  be  less,  it  follows 
that  the  error  is  absolutely  nothing;  an  almost  exactly  similar  kind 
of  argument  is  used  in  different  places  by  Euclid,  Theodosius  and 
others ;  and  this  seemed  to  them  to  be  a  wonderful  thing,  although 
it  could  not  be  denied  that  it  was  perfectly  true  that,  from  the 
very  thing  that  was  assumed  as  an  error,  it  could  be  inferred  that 
the  error  was  non-existent.  Thus,  by  infinitely  great  and  infinitely 
small,  we  understand  something  indefinitely  great,  or  something 
indefinitely  small,  so  that  each  conducts  itself  as  a  sort  of  class, 
and  not  merely  as  the  last  thing  of  a  class.  If  any  one  wishes  to 
understand  these  as  the  ultimate  things,  or  as  truly  infinite,  it  can 
be  done,  and  that  too  without  falling  back  upon  a  controversy  about 
the  reality  of  extensions,  or  of  infinite  continuums  in  general,  or 
of  the  infinitely  small,  ay,  even  though  he  think  that  such  things 
are  utterly  impossible;  it  will  be  sufficient  simply  to  make  use  of 
them  as  a  tool  that  has  advantages  for  the  purpose  of  the  calcula- 
tion, just  as  the  algebraists  retain  imaginary  roots  with  great  profit. 
For  they  contain  a  handy  means  of  reckoning,  as  can  manifestly  be 
verified  in  every  case  in  a  rigorous  manner  by  the  method  already 
stated. 

But  it  seems  right  to  show  this  a  little  more  clearly,  in  order 
that  it  may  be  confirmed  that  the  algorithm,  as  it  is  called,  of  our 
differential  calculus,  set  forth  by  me  in  the  year  1684,  is  quite 
reasonable.  First  of  all,  the  sense  in  which  the  phrase  "dy  is  the 


THE  MANUSCRIPTS  OF  LEIBNIZ.  447 

element  of  r,"  is  to  be  taken  will  best  be  understood  by  considering 
a  line  AY  referred  to  a  straight  line  AX  as  axis. 

Let  the  curve  AY  be  a  parabola,  and  let  the  tangent  at  the 
vertex  A  be  taken  as  the  axis.  If  AX  is  called  x,  and  AY,  y,  and 
the  latus-rectum  is  a,  the  equation  to  the  parabola  will  be  xx  =  ay, 
and  this  holds  good  at  every  point.  Now,  let  A  J£  =  x,  and  1XlY  =  y 


and  from  the  point  jY  let  fall  a  perpendicular  XYD  to  some  greater 
ordinate  2X  2Y  that  follows,  and  let  jX  2X,  the  difference  between 
A  XX  and  A  2X,  be  called  dx ;  and  similarly,  let  D  2Y,  the  difference 
between  tX  XY  and  2X  ,Y,  be  called  dy. 

Then,  since  y  =  xx :  a,  by  the  same  law,  we  have 

y  +  dy  =  xx  +  2x  dx  +  dx  dx, :  a ; 

and  taking  away  the  y  from  the  one  side  and  the  xx:a  from  the 
other,  we  have  left 

dy :  dx  =  2x  +  dx  :a  ; 

and  this  is  a  general  rule,  expressing  the  ratio  of  the  difference  of 
the  ordinates  to  the  difference  of  the  abscissae,  or,  if  the  chord  tY  2Y 
is  produced  until  it  meets  the  axis  in  T,  then  the  ratio  of  the  ordinate 
XX  jY  to  T  tX,  the  part  of  the  axis  intercepted  between  the  point 
of  intersection  and  the  ordinate,  will  be  as  2x  +  dx  to  a.  Now, 
since  by  our  postulate  it  is  permissible  to  include  under  the  one 
general  reasoning  the  case  also  in  which  the  ordinate  2X  oY  is  moved 
up  nearer  and  nearer  to  the  fixed  ordinate  iX  iY  until  it  ultimately 
coincides  with  it,  it  is  evident  that  in  this  case  rf.r  becomes  equal  to 
zero  and  should  be  neglected,  and  thus  it  is  clear  that,  since  in  this 
case  T  tY  is  the  tangent,  ,X  ^Y  is  to  T  XX  as  2x  is  to  a. 

Hence,  it  may  be  seen  that  there  is  no  need  in  the  whole  of  our 
differential  calculus  to  say  that  those  things  are  equal  which  have 
a  difference  that  is  infinitely  small,  but  that  those  things  can  be 
taken  as  equal  that  have  not  any  difference  at  all.  provided  that 
the  calculation  is  supposed  to  be  general,  including  both  the  cases 
in  which  there  is  a  difference  and  in  which  the  difference  is  zero ; 


448 


THE    MONIST. 


and  provided  that  the  difference  is  not  assumed  to  be  zero  until  the 
calculation  is  purged  as  far  as  is  possible  by  legitimate  omissions, 
and  reduced  to  ratios  of  non-evanescent  quantities,  and  we  finally 
come  to  the  point  where  we  apply  our  result  to  the  ultimate  case. 
Similarly,  if  x'A  =  aay,  then  we  have 

x3  -f  3xx  dx  +  3x  dx  Ax  +  dx  d.v  dx  -  aay  +  aa  dy, 
or  cancelling  from  each  side, 

3xx  dx  +  3x  dx  dx  -i  dx  dx  dx  -  aa  dy, 
or  3xx  +  2>x  dx  +  dx  dx,  :  aa  -  dy  :  dx  =  XX  tY  :  T  tX  ; 

hence,  when  the  difference  vanishes,  we  have 


But  if  it  is  desired  to  retain  dy  and  dx  in  the  calculation,  so  that 
they  may  represent  non-evanescent  quantities  even  in  the  ultimate 
case,  let  any  assignable  straight  line  be  taken  as  (dx),  and  let  the 
straight  line  which  bears  to  (dx)  the  ratio  of  y  or  aX  tY  to  aXT  be 
called  (dy)  ;  in  this  way  dy  and  dx  will  always  be  assignables 
bearing  to  one  another  the  ratio  of  D  2Y  to  D  1Y,  which  latter  vanish 
in  the  ultimate  case. 

[Leibniz  here  gives  a  correction  for  a  passage  in  the  Ada 
Eruditorum,  which  is  unintelligible  without  the  context.] 

On  these  suppositions,  all  the  rules  of  our  algorithm,  as  set 
out  in  the  A  eta  Eruditorum  for  October  1684,  can  be  proved  without 
much  trouble. 


Let  the  curves  YY,  VV,  ZZ  be  referred  to  the  same  axis  AXX ; 
and  to  the  abscissae  A  tX  (=*)  and  A  2X  (=x  +  dx)  let  there  cor- 
respond the  ordinates  1X1'Y(=y)  and  2X  2Y  (=y  +  dy),  and  also 
the  ordinates  1XlV(=v)  and  2X2V  (=v  +  dv),  and  the  ordinates 


THE  MANUSCRIPTS  OF  LEIBNIZ.  449 

lX1Z(=rr)  and2X8Z(=s  +  <k).  Let  the  chords  /Y-jY,  1V2V,  ^Z, 
when  produced  meet  the  axis  AXX  in  T,  U,  W.  Take  any  straight 
line  you  will  as  (d)x,  and,  while  the  point  4X  remains  fixed  and 
the  point  2X  approaches  jX  in  any  manner,  let  this  remain  constant, 
and  let  (d}y  be  another  line  which  bears  to  (d}x  the  ratio  of  3;  to 
jXT,  or  of  dy  to  dx;  and  similarly,  let  (d)v  be  to  (d)x  as  v  to  jXU 
or  dv  to  rf.r;  also  let  (d}z  be  to  (d}x  as  r  to  tXW  or  dz  to  rfjr; 
then  (d)x,  (d}y,  (d}z,  (d)w  will  always  be  ordinary  or  assignable 
straight  lines. 

Nor  for  Addition  and  Subtraction  we  have  the  following: 

If  y-s  =  v,  then  (d)y-  (d)s  =  (d)v. 

This  I  prove  thus:  y +  dy-s-ds=v  +  dv,  (if  we  suppose  that  as  v 
increases,  2  and  v  also  increase ;  otherwise  for  decreasing  quantities, 
for  2  say,  -ds  should  be  taken  instead  of  ds,  as  I  mentioned  once 
before)  ;  hence,  rejecting  the  equals,  namely  y-s  from  one  side, 
and  v  from  the  other,  we  have  dy-dz-dv,  and  therefore  also 
dy  -  dz :  dx  =  dv :  dx.  But  dy :  dx,  dz :  dx,  dv :  dx  are  respectively 
equal  to  (d)y:(d).v,  (d}z:(d}x,  and  (d}v:(d)x.  Similarly,  (d~)s 
:(d)y  and  (d}v:  (d)y  are  respectively  equal  to  dz:dy  and  dv.dy. 
Hence,  (d)y-(d)s,  :(d)x  =  (d)v  :(d}x\  and  thus  (d}y-(d)z  is 
equal  to  (d}v,  which  was  to  be  proved ;  or  we  may  write  the  result 
as  (d)v  :(d)y=l-  (d}z  :(d}y. 

This  rule  for  addition  and  subtraction  also  comes  out  by  the 
use  of  our  postulate  of  a  common  calculation,  when  tX  coincides 
with  ,X,  and  ,YT,  jYU,  ,YW  are  the  tangents  to  the  curves  YY, 
VV,  ZZ.  Moreover,  although  we  may  be  content  with  the  assign- 
able quantities  (eOy,  (d)v,  (d}z,  (d}x,  etc.,  since  in  this  way  we 
may  perceive  the  whole  fruit  of  our  calculus,  namely  a  construction 
by  means  of  assignable  quantities,  yet  it  is  plain  from  what  I  have 
said  that,  at  least  in  our  minds,  the  unassignables  dx  and  dy  may  be 
substituted  for  them  by  a  method  of  supposition  even  in  the  case 
when  they  are  evanescent :  for  the  ratio  dy :  dx  can  always  be 
reduced  to  the  ratio  (d)v  '-(d)x,  a  ratio  between  quantities  that 
are  assignable  or  undoubtedly  real.  Thus  we  have  in  the  case  of 
tangents  dv :  dy  =  1  -dz :  dx,  or  dv=dy-  dz. 

Multiplication.     Let  ay  =  xv,  then  a(d)y  =  x(d)v  +  v (d)x. 
Proof.          ay  +  ady  =  x  +  dx,  v  +  dv=xv  +  xdv  +  vdx 
and,  rejecting  the  equals  ay  and  xy  from  the  two  sides, 


450  THE    MONIST. 

a  dy  -  xdv  +  v  dx  +  dx  dv, 
or 

a  dy       x  dv 
— £  =  _—  +  v  +  dv ; 
dx         dx 

and  transferring  the  matter,  as  we  may,  to  straight  lines  that  never 
become  evanescent,  we  have 

a(d)y  ,  x(d}y 

~TT\ r  '/j\      +  v  +  dv; 

(d)x         (d)x 

so  that,  since  it  alone  can  become  evanescent,  dv  is  superfluous, 
and  in  the  case  of  the  vanishing  differences,  as  in  that  case  dv  =  Q, 
we  have 

a(d}y  =  x(d}v  +  v(d}x,  as  was  stated, 
or  (d}y  :  (d}x  =  x  +  v, :a. 

Also,  since  (d)y  \(d}x  always  -dy.dx,  it  will  be  allowable  to  sup- 
pose this  is  true  in  the  case  when  dy,  dx  become  evanescent,  and  to 
say  that  dy  :dx  =  x  +  v:a,  or  a  dy  =  x  dv  +  v  dx. 

Division.     Let  s:  a  -v.x,  then   (d}z:  a-v(d}x-x(d}y,  :xx. 
Proof  z  f  dz:  a-v  +  dv, :  ,x  +  dx; 

or  clearing  of  fractions,  xz  f  xds  +  sdx  +  dsdx  -av  +  adv ;  taking  away 
the  equals  xs  and  av  from  the  two  sides,  and  dividing  what  is  left 
by  dx,  we  have 

adv-  x  dS)  :dx  =  z  +  ds, 
or  a(d}v-x(d)z, :d.v  =  s  +  ds; 

and  thus,  only  ds,  which  can  become  evanescent,  is  superfluous. 
Also,  in  the  case  of  vanishing  differences,  when  XX  coincides  with 
2X,  since  in  that  case  ds  =  Q,  we  have 

a  (d)v  -  x(d)z, :  (d)x  =  s  -  av :  x ; 

whence,  (as  was  stated)   (d}s  =  ax(d}v-av(d}x,\ xx, 
or  (d)z:  (d}x=  (a:.r)  (d)v:  (d)x-av\xx. 

Also,  since  (d}z:(d}x  is  always  equal  to  dz\dx,  on  all  other 
occasions,  it  is  allowable  to  suppose  this  to  be  so  also  when  dz,  dv, 
dx  are  evanescent,  and  to  put 

dz :  dx  =  ax  dv  -  av  dx,  :xx 
For  Powers,  let  the  equation  be  aa^x'^y"  f  then 

(d)y      *•*?• 
(d)x      n.yn-^' 


THE  MANUSCRIPTS  OF  LEIBNIZ.  451 

and  this  I  will  prove  in  a  manner  a  little  more  detailed  than  those 
above,  thus: 

MXJL      -         €     /_  i     •          €  \C  ~~~  A      *  _*»    *      •  C\€  ~~  J  .c       w     f  —  -\     r      f      t 

a  "-',  -*'  +     ^  rf*  +  -r-^~  xf=?dxdx  +  -    ..    '    -  x-^  dxdxdx 

JL  JL  J  i~  J  i^c> 

(and  so  on  until  the  factor  e-e  or  0  is  reached) 

,    W-.o-i   .  7Z,W  —  1     __2    ,  #,M  —  1,W  —  2     „_,     . 

=  j  y"  +  i^—  dy  +  —  j-y-  y".2<tydy  +  -     1  2'3  --  >>V  ### 

(and  so  on  until  the  factor  n-n  or  0  is  reached)  ; 
take  away  from  the  one  side  a"-f  xe  ,  and  from  the  other  side  vn, 
these  being  equal  to  one  another,  and  divide  what  is  left  by  dx, 
and  lastly,  instead  of  the  ratio  dy  :  dx,  between  the  two  quantities 
that  continually  diminish,  substitute  the  ratio  that  is  equal  to  it, 
(d)y:(d)x,  a  ratio  between  two  quantities,  of  which  one,  (d}x, 
always  remains  the  same  during  the  time  that  the  differences  are 
diminishing,  or  while  2X  is  approaching  the  fixed  point  jX  and 
we  have 

€       f  __  i         €  •  €  ™""~  JL         *  _  o      .  CtC  ~~"  '  -L  >  c  ~~~*  w        .  _  -i     ,       _ 

r-  x    '  +  -V^~  ^^  ^  +     —         —  j;—  rf^rf^  +  etc. 

\~  I  jW  J  »  «-  )*J 


.  «.«  —  !,«—  2    «_  ,    , 

=  f  ^  (i)i~w~^  ?Si+  Tw~'^  W"i**  *  etc' 

Now,  since  by  the  postulate  there  is  included  in  this  general  rule 
the  case  also  in  which  the  differences  become  equal  to  zero,  that 
is  when  the  points  2X,  2Y  coincide  with  the  points  jX,  jY  respec- 
tively ;  therefore,  in  that  case,  putting  dx  and  dy  equal  to  0,  we  have 

e       ,      n   n-i(dly 

I*       ~\y      (d)x' 

the  remaining  terms  vanishing,  or  ((f)y  :  ('/)^  =  e.x*~l  :  n  y^L. 
Moreover,  as  we  have  explained,  the  ratio  (d)y:(d}x  is  the  same 
as  the  ratio  of  v,  or  the  ordinate  XX  ,Y,  to  the  subtangent  iXT, 
where  it  is  supposed  that  Tj  Y  touches  the  curve  in  tY. 

This  proof  holds  good  whether  the  powers  are  integral  powers 
or  roots  of  which  the  exponents  are  fractions.  Though  we  may 
also  get  rid  of  fractional  exponents  by  raising  each  side  of  the 
equation  to  some  power,  so  that  e  and  n  will  then  signify  nothing 
else  but  powers  with  rational  exponents,  and  there  will  be  no  need 
of  a  series  proceeding  to  infinity.  Moreover,  at  any  rate,  it  will  be 
permissible,  by  means  of  the  explanation  given  above,  to  return  to 
the  unassignable  quantities  dy  and  dx,  by  making  in  the  case  of 
evanescent  differences,  as  in  all  other  cases,  the  supposition  that 
the  ratio  of  the  evanescent  quantities  dy  and  dx  is  equal  to  the  ratio 


452 


THE   MONIST. 


of  (d)y  and  (rf).r,  because  this  supposition  can  always  be  reduced 
to  an  undoubtable  truth. 

Thus  far  the  algorithm  has  been  demonstrated  for  differences 
of  the  first  order:  now  I  will  proceed  to  show  that  the  same  method 
will  hold  good  for  the  differences  of  the  differences.  For  this 
purpose,  take  three  ordinates,  iXiY,  2X  2Y,  3X3Y,  of  which  jX/Y 
remains  constant,  but  ,X  2Y  and  3X  3Y  continually  approach  jX.^Y 
until  finally  they  both  coincide  with  it  simultaneously ;  which  will 
happen  if  the  speed  with  which  3X  approaches  jX  is  to  the  speed 
with  which  2X  approaches  ,X  is  in  the  ratio  of  !X3X  to  tX,,X. 
Also  let  two  straight  lines  be  assigned,  (d).v  always  constant  for 
any  position  of  2X,  and  .2(d}x  for  any  position  of  3X ;  also  let  (d)y 
always  be  to  (rf).r  as  D2Y  is  to  jX  2X,  or  as  v  (i.  e.,  jX^)  is  to 
,XT;  thus,  while  (d}x  remains  always  the  same,  (d)y  will  be 
altered  as  2X  approaches  jX ;  similarly,  let  2(d)v  be  to  *(d}x  as 
2D3Y  to  2X  3X  or  as  y+  dy  (i.  e.,  2X  2Y)  to  2X  2T  ;  thus  while  2(</).r 
remains  constant,  2(rf)v  will  be  altered  as  3X  approaches  tX. 

Also  let  (d)y  be  always  taken  in  the  varying  line  ,X  2Y,  and 
let  2X  !<w  be  equal  to  (d) y,  and  similarly  take  2(d}y  in  the  line  3X  3Y, 
and  let  3X  .,w  be  equal  to  2(d}y.  Thus,  while  2X  and  3X  continually 
approach  to  the  straight  line  jX  jY,  2X  tw  and  3X  2o>  continually 
approach  it  also,  and  finally  coincide  with  it  at  the  same  time  as 


2X  and  3X.  Further,  let  the  point  in  the  ordinate  jX  1Y,  which  jw 
continually  approaches  and  with  which  it  at  last  coincides,  be 
marked,  and  let  it  be  Q ;  then  jXft  is  the  ultimate  (d)y,  which  bears 
to  (rf).f  the  ratio  of  the  ordinate  jX  jY  to  the  subtangent  jXT, 
where  it  is  supposed  that  T  XX  touches  the  curve  in  XY,  because 
then  indeed  XY  and  2Y  coincide.  Now,  since  all  this  can  be  done, 


THE  MANUSCRIPTS  OF  LEIBNIZ.  453 

no  matter  where  ,Y  may  be  taken  on  the  curve,  it  is  evident  that 
•a  curve  Qft  will  be  produced  in  this  way,  which  is  the  differentrix 
of  the  curve  YY ;  just  as,  conversely,  the  curve  YY  is  the  summatrix 
curve  of  QQ,  as  can  be  readily  demonstrated. 

By  this  method,  the  calculus  may  be  demonstrated  also  for  the 
differences  of  the  differences. 

Let  ,X  ,Y,  ,X  2Y,  3X:!Y  be  three  ordinates,  of  which  the  values 
are  v,  y  +  dy,  y  +  dy  +  ddy,  and  let  tX2X  (dx)  and  2X  3X  (dx  +  ddx) 
be  any  distances,  and  D.,Y(dy)  and  2D3Y  (dy+ddy)  the  differ- 
ences. Now  the  difference  between  (d)y  and  2(d)y,  or  between 
,X«  and  ,X2n  is  8,Q,  and  that  between  iXoX  and  2X3X  is  ddx; 
also  let 

(d)dx:  (d)x  =  dx:  .,(d).v,     74     and  similarly  let 
(d)dv:(rf).v  =  an8:1X2X  or  ^O^XT. 

Now,  for  the  sake  of  example,  let  us  take  ay  =  xv.  Then  we 
have  ady  =  xdv  +  vdx  +  dxdv,  as  has  been  shown  above;  and  simi- 
larly, 

ady  +  addy  =  (x  +  dx}  (dv  +  ddv)  +  (v  +  dv)  (dx  +  ddx)       " 

+  (dx+ddx)  (dv  +  ddv) 
=  x  dv  +  x  ddv  +  dx  dv  +  dx  ddv  +  vdx  +  v  ddx 
+  dvdx  +  dv  ddx  +  d.v  dv  +  dx  ddv 

+  ddx  dv  +  ddx  ddv. 

Taking  away  a  dy  from  one  side,  and  x  dx  +  v  dx  +  dx  dv  from  the 
other,  there  will  be  left  in  any  case 

ddy  _  xddy      v      2  dxdv      2^dv      2  dx  ddx       ddv 
ddx      a  ddx      a      a   ddx          a  a  ddx  a 

In  this  it  is  evident  that  the  ratio  between  ddy  and  ddx  can  be 
expressed  by  the  ratio  of  the  straight  line  (d)dy  to  (d)x,  the  straight 
line  assumed  above,  which  we  have  supposed  to  remain  constant 
as  2X  and  3X  approach  ,X.  Also,  since  (d)dx,  (since  it  bears  an 
assignable  ratio  to  (d)x,  however  nearly  2X  approaches  to  XX,  or 

74  This  makes  (d)dx  an  inassignable.    It  may  be  a  misprint  due  to  a  slip 
of  Leibniz,  or  of  Gerhardt  in  transcription ;  for  there  is  no  similarity  between 
it  and  the  statement  in  the  next  line.     I  cannot  however  offer  any  feasible 
suggestion  for  correction. 

75  This  is  quite  wrong.     Leibniz  has  evidently  substituted  x  -f-  dx  for  x, 
etc.;   which   is  not  legitimate  unless  SXSY   is  taken  as  y  -f  dy  -f-  d(y-\-  dy), 
and  so  on ;  even  then  fresh  difficulties  would  be  introduced.    As  it  stands,  this 
line  should  read 

o  dy  +  o  ddy  =  x(dv  -f-  ddv)  +  v(dx  -f  ddx)  -f  (dx  -f  ddx)  (dv  +  ddv). 

On  account  of  this  error  and  that  noted  above,  there  is  not  much  profit  in 
considering  the  remainder  of  this  passage. 


454  THE  MONIST. 

however  much  dx,  the  difference  between  the  abscissae,  is  dimin- 
ished), is  not  evanescent,  even  when,  finally,  d.v  and  ddx,  dv  and 
ddv,  are  all  supposed  to  be  zero.  In  the  .same  way,  the  ratio  of 
ddv  to  ddx  may  be  expressed  by  the  ratio  of  an  assignable  straight 
line  (d)dv  to  the  assumed  constant  (d)x;  and  even  the  ratio  of 
dvdx  to  a  ddx  may  be  so  expressed;  for,  since  dv:  dx=(d}v:(d}x, 
therefore  dvdx:>dxdx=(d}v:(d}  x.  Henoe,  i  f  a  new  straight 
line,  (dd)x,  is  assumed  to  be  such  that  a  ddx:  dx  dx=(dd}xs(d}x, 
then  the  new  straight  line  will  be  assignable,  even  though  dx,  ddx, 
etc.  become  evanescent.  Since  therefore  dvdx:dxdx=(d}v:(d)x 
and  dvdx:addx=(d)x:(A&)x,  it  follows  that  dv  dx  \  a  ddx  =(d)v  : 
(dd)x,  an  thus  at  length  there  is  prouced  an  equation  that  is  freed 
as  far  as  possible  from  those  ratios  that  might  become  evenescent, 
namely, 

(d)dy  _  x(<T)dy      y      2  (d)y      2  dv       2dx  (</)  dy       ddv 
(d)dx  ~  a  (d)dx      a       (dd)x         a  a    (d)dx        a 

Thus  far  all  the  straight  lines  have  been  considered  to  be  assign- 
able so  long  as  jX  and  2X  do  not  coincide  ;  but  in  the  case  of  coin- 
cidence, dv  and  ddv  are  zero,  and  we  have 

(d}dy  _x(d}dv      v       2  (d)y      0       2  (d*)dv  0       0 
(d~)dx      a  (d)dx      a       (dd)x       a        (d~)dx    a       a  ' 

or,  omitting  terms  equal  to  zero, 

(d~)dy    i  x  (d)dv      v 


a  (d)dx      a       (dd~)x 

Hence,  if  dx,  ddx,  dv,  ddv,  dy,  ddy,  are  by  a  certain  fiction  imagined 
to  remain,  even  when  they  become  evanescent,  as  if  they  were  in- 
finitely small  quantities  (and  in  this  there  is  no  danger,  since  the 
whole  matter  can  be  always  referred  back  to  assignable  quantities), 
then  we  have  in  the  case  of  coincidence  of  the  point  tX  and  2X  the 
equation 

ddv       x  ddy       v       2  dx  dy 

ddx       a  ddx       a       a    ddx 

J.  M.  CHILD. 
DERBY,  ENGLAND. 


LIBRA: 

THE  ETERNAL  BALANCE  OF  GOOD  AND  ILL. 
I. 

FROM  everlasting  is  the  Universe, 
And  unto  everlasting  shall  extend; 
Without  beginning  is  it;  without  end 
Its  morrows  ever  yesterdays  rehearse; 
Not  first  nor  last  but  only  midst  it  knows; 
As  never  young,  so  never  old  it  grows. 

ii. 

Yet  is  the  secret  of  its  permanence 

Not  rest  but  striving,  not  a  dead  repose, 
No  peace  of  mutually  slaughtered  foes, 

Nor  truce  of  wearied,  but  a  strife  intense, 

Deathless,  of  powers  that  charge  and  countercharge 
Ever,  yet  never  may  their  bounds  enlarge. 

in. 

Not  progress  is  the  secret  of  the  sky, 

And  not  decay  the  withering  doom  of  earth ; 
Though,  out  of  star-mist,  systems  round  to  birth, 

And  a  dead  moon  mirrors  earth's  destiny, 

The  star  shall  sink  in  darkness  whence  it  came, 
And  earth's  grim  desert  be  reborn  in  flame. 

IV. 

Tt  is  the  wave  with  endless  rise  and  fall, 


456  THE   MONIST. 

It  is  the  tide  with  ceaseless  ebb  and  flow, 
The  changing  moons,  the  hours  with  gloom  and  glow, 
That  hold  the  mystery  of  each  and  all,— 
The  rhythmic  secret,  wherein  man  has  part 
Even  from  the  first  pulsation  of  his  heart. 

v. 

The  pendulum  with  its  untiring  swing 
Not  only  metes  out  time,  but  it  reveals, 
Babbling,  the  word  eternity  conceals, 

Though  to  men  deaf  with  their  own  questioning; 
The  lilting  ripple  of  the  poet's  song 
Itself  contains  the  clue  he  sought  life-long. 

vr. 

Nothing  can  be  unfolded  but  has  first 
Been  folded  in,  and  shall  be  so  again ; 
Nor  yet  can  aught  in  equipoise  remain, 

But  ever  driveth  toward  the  best  or  worst; 
Nature  keeps  neither  full  nor  empty  cup, 
And  the  half-filled  she  drains  or  fills  it  up. 

VII. 

Yet  what  had  no  besrinninq-  always  is 

«Z>  C5  f 

And  never  can  become;  no  inward  change, 
However  wide  its  outward  motions  range, 
Can  touch  its  heart;  despite  man's  fantasies, 
The  Universe  exists,  not  merely  seems 
An  everlasting  see-saw  of  extremes. 

vm. 

These  two  extremes  man  knows  as  More  and  Less, 
As  Good  and  111,  lastly  as  Right  and  Wrong; 
Feels  them  as  Love  or  Hate  his  pulses  throng; 


THE  ETERNAL  BALANCE  OF  GOOD  AND  ILL.  457 

Sees  them  with  Beauty  clothed  or  Ugliness, 

And  names  them  from  their  power  to  bless  or  ban 
God  and  Devil,  Ormuzd  and  Ahriman. 

IX. 

The  righteous  Paul  lamented  in  his  heart 
The  Good  by  Evil  thwarted.     So  in  thine 
The  False  and  True,  the  Cruel  and  Benign, 

The  Pure  and  Impure  make  thee  what  thou  art 
And  what  the  All  is:  tiger,  dove,  and  man, 
Seraph  and  fiend,  are  fashioned  on  one  plan. 

x. 

Even  as  the  Universe,  mid  seeming  change, 

Really  is  locked  in  iron  permanence, 

So,  everywhere,  despite  our  cheated  sense, 
From  one  self-nature  may  it  never  range: 

One  is  it,  one  in  body  and  the  soul, 

And  every  part  is  parcel  of  the  whole. 

XI. 

Behind  all  forces  hides  the  primal  Force, 
The  Unconditioned,  which  is  bad  and  good 
Impartially,  and  its  divided  mood 

The  single  spirit  of  the  Universe, 

Of  you  and  me  and  all  men  and  the  earth 
And  all  the  worlds  Infinify  wheels  forth. 

,* 

XTT. 

But  mortal  life  displays  not  one  but  two, 
Shows  Good  all-perfect  warring  against  111, 
Which  yet  abides  unconquerable  still, 

And  in  this  duel  sets  for  man  a  part, 

And  teaches  he  must  choose  the  side  of  Good, 
Or  rank  below  the  cleft,  insensate  wood. 


458  THE    MONIST. 

XIII. 

Had  it  been  destined  to  be  otherwise, 

Long  since  it  would  have  been  so;  nay,  for  we 
Deal  not  with  time  but  with  eternity, 

It  would  have  been  so  always;  had  our  skies 
Been  fated  to  o'erarch  a  perfect  earth, 
They  would  have  overarched  it  from  their  birth. 

XIV. 

This  is  the  revelation;  this  alone 

Rained  ever  from  the  Milky  Way  adown, 

Or  flamed  from  Vega  and  the  Northern  Crown, 

Even  this  that  written  on  my  heart  I  own. 
Not  ours  to  ask  if  unto  me  or  you 
The  word  be  welcome,  but  if  it  be  true. 

xv. 

What  then  must  be  the  Universe,  ideal? 

Never  and  nowhere;  but  endurable, 

A  place  where  on  the  whole  'tis  fairly  well, 
Where  at  least  men  can  live;  in  short,  the  real. 

Had  it  been  more,  there  were  no  need  to  ask; 

Had  it  been  less,  not  ours  had  been  the  task. 

XVI. 

If  this  be  true,  as  Life  forbids  to  doubt, 

Is  low  then  one  with  high,  is  conscience  vain? 

Forever  no!    But,  though  I  shall  not  gain 
After  short  strife  a  glorious  mustering  out, 

My  privilege  more  glorious  is  to  be 

A  soldier  of  the  Right  eternally. 

XVII. 

Yet  what  avails  my  battle  for  the  Right, 


THE  ETERNAL  BALANCE  OF  GOOD  AND  ILL.  459 

You  ask,  if  through  eternity  shall  still 
Be  kept  the  balance  between  Good  and  111? 
Me  much  avails  it,  for  'tis  mine  to  fight 

On  the  Lord's  side,  being  birthmarked  with  his  seal; 
My  joy,  my  life  is  in  that  battle  peal. 

XVIII. 

More  can  I  ask?    Shall  some  far  eon  see 
The  Evil  quelled,  the  Good  supreme  prevail? 
Not  if  our  world  have  told  us  a  true  tale. 

But  can  we  hear  and  judge  it  rightfully? 
Our  torch  is  feeble;  but  at  least  its  light 
Reveals  us  friend  and  foeman  in  the  fight. 

XIX. 

The  rest  is  God's.    Yet  who  would  change  that  could 
Doom  so  divine,  which  loftiest  souls  must  bear, 
Though  archangelic? — in  all  worlds  to  share 

The  warfare  of  the  soldiers  of  the  Good, 
Though  marching  under  orders  ever  sealed, 
And  battling  ever  on  a  doubtful  field! 

HARRY  LYMAN  KOOPMAN. 

PROVIDENCE,  R.  I. 


CRITICISMS  AND  DISCUSSIONS. 

LOGIC  AND  PSYCHOLOGY. 

The  nature  and  purpose  of  symbolic  or  mathematical  logic, 
which  began  to  be  developed  by  Leibniz  and  was  continued  quite 
independently  by  Boole  and  others,  is  tolerably  well  known  by  now. 
Logical  reasoning  is  translated  by  it  into  what  Leibniz  called  a  "real 
characteristic"  which  is  very  analogous  to  ordinary  algebra,  and 
helps  swiftness  and  accuracy  of  reasoning — even  complicated  rea- 
soning— in  much  the  same  way  as  the  signs  in  algebra  do.  This 
tendency  culminated  in  the  very  ingenious  and  useful  "mathematical 
logic"  of  Peano.  Peano's  system  was  far  more  complete  than 
Boole's,  for  the  whole  of  a  piece  of  reasoning  which  included 
algebraic  formulas  and  equations  could  be  put  into  a  symbolical 
form  in  which  ordinary  words — which  are  not  part  of  a  "real  char- 
acteristic"— are  not  used.  In  this  direction  Peano's  system  met  the 
much  earlier  system  devised  by  Frege.  However,  Frege's  system 
was  not  thought  out  so  much  with  a  view  to  rapidity  of  reasoning 
and  convenience  of  writing  as  with  a  view  to  emphasizing  slight 
and  important  logical  distinctions  in  very  similar  concepts  and 
deductions  and  consequently  a  scrupulous  accuracy  in  deductions. 
It  may  thus  be  noticed,  by  the  way,  that  the  purpose  of  Frege's 
symbolism  was  different  from  that  of  all  previous  symbolisms  in 
logic  and  mathematics,  for  Frege  wished  to  lay  stress  upon  the 
differences  in  various  analogous  ideas  and  deductions  rather  than 
upon  their  analogies.  Broadly  speaking,  Russell  and  Whitehead's 
work  may  be  characterized  by  saying  that  it  is  formed  under  the 
influence  of  a  combination  of  the  two  tendencies  represented  by 
Frege  and  Peano.  The  convenient  symbolism  of  Peano  is  retained 
wherever  possible  and  the  superior  analysis  and  subtlety  of  Frege 
is  fully  used.  We  ought  to  add  also  that  nearly  all  of  Frege's  dis- 


CRITICISMS  AND  DISCUSSIONS.  461 

coveries  were  made  independently  by  Russell  himself,  Frege's  great 
work  having  been  neglected  by  philosophers  and  mathematicians. 

There  is  one  great  point  in  which  Russell's  works  differs  much 
from  that  of  Frege:  full  use  is  made  of  the  enormously  important 
researches  of  Georg  Cantor  on  transfmite  numbers.  While  putting 
on  a  firm  basis  the  treatment  of  infinite  classes  and  numbers,  Can- 
tor's work  led  to  the  recognition  of  forms  of  a  paradox  absolutely 
fundamental  in  logic.  After  many  vain  attempts  by  various  mathe- 
maticians and  philosophers,  this  paradox  has  been  satisfactorily 
solved  by  the  thorough  remoulding  of  logic  given  in  Whitehead  and 
Russell's  Principia  Mathematica. 

From  Peano's  various  Fonnitlaires  to  the  work  last  mentioned 
the  subject-matter  is  principally  the  collection  of  truths  which  we 
can  reach  by  logical  deduction  from  logical  principles.  This  body 
of  truths  is  not  a  description  of  psychological  methods  of  discovery 
or  psychological  results,  but  is  of  course  reached  by  psychical 
processes,  like  most  other  discoveries  in  a  purely  intellectual  domain. 
It  is  then  simply  irrelevant  to  complain  that  there  is  no  place  in 
the  Fonnulaires  or  Principia  for  that  "intuition"  which  brings  about 
mathematical  discoveries.  It  would  be  just  as  much  to  the  point 
to  complain  that  in  what  is  excavated  we  do  not  discover  the  tools 
used  for  excavating  or  the  method  of  excavation.  And  yet  this  is 
what  the  rather  superficial  and  amusing  discussions  of  Henri  Poin- 
care  are  mostly  about.  And  these  discussions  are  what  Prof.  J.  B. 
Shaw  in  the  number  of  The  Monist  for  July,  1916,  refers  to  (p. 
397)  as  Poincare's  "successful  attacks  on  logistic."  We  might 
reasonably,  it  seems  to  me,  have  expected  that  Professor  Shaw 
should  make  some  reference  to  the  reply  by  Louis  Couturat  to 
Poincare  which  was  translated  in  The  Monist  for  October,  1912, 
and  which  is  quite  conclusive  on  so  many  points.  Professor  Shaw, 
in  his  eloquent  and  somewhat  inaccurate  (both  from  the  points  of 
view  of  history  and  logic)  attack  on  mathematical  logic,  urges 
what  are,  at  bottom,  the  very  same  irrelevant  arguments.  I  shall 
try  to  point  out  some  of  these  inaccuracies,  both  because  they  are 
fairly  common  even  now  among  mathematicians,  and  because  it 
is  surely  the  duty  of  every  one  to  contribute  as  far  as  he  can  to 
the  clarification  of  notions  in  America  above  all  other  countries ; 
for  it  is  from  America  that  we  expect  an  exceedingly  large  pro- 
portion of  the  work  of  the  intellect  in  future  now  that  Europe  has 
deliberately  handicapped  herself. 


462  THE   MONIST. 

Professor  Shaw's  slighting  remark  on  the  impotence  and  boast- 
ing power  of  logistic  (p.  411)  is  the  result  of  a  strange  miscon- 
ception. Logistic  deals  with  logical  entities  and  deductions  which 
are  fundamental  to  mathematics,  and  it  is  unjust  to  try  to  make 
people  believe  that  logistic  ever  claimed  to  be  the  overlord  of  mathe- 
matics. There  seems,  in  fact,  to  be  a  note  almost  of  personal  dislike 
for  logistic  in  those  mathematicians  who  attack  it.  And  yet  the 
question  is  wholly  concerned  with  logical  facts,  and  is  not  to  be 
answered  by  rhetorical  appeals  to  prejudice  or  sentiment.  If  logic 
is  more  fundamental  than  mathematics,  why  should  there  be  any 
objection  to  the — successful  as  it  happens — attempt  to  define  mathe- 
matical entities  in  terms  of  logical  ones?  If  mathematics  is  more 
fundamental  than  logic,  the  first  thing  to  do  is  to  draw  up  a  scheme 
showing  that  logical  entities  can  be  deduced  from  specifically  math- 
ematical ones.  Until  this  is  done,  and  certain  objections  to  it  are  at 
once  obvious,  it  is  quite  unconvincing  to  disparage  cultivators  of 
logistic.  After  all,  logisticians  are  working  at  mathematics  in  much 
the  same  way  that  other  mathematicians  are.  They  are  concerned 
with  more  fundamental  problems  and  problems  which  do  not  so 
easily  appeal  to  the  public,  as,  say,  a  proof  of  Fermat's  great 
theorem  would,  but  they  discover  truths  just  as  much  as  any  other 
mathematicians.  They  introduce  conceptions  to  work  with.  We 
may  mention  the  idea  of  pro  positional  function  actually  mentioned 
by  Professor  Shaw  in  terms  of  commendation  (p.  411),  which  was 
introduced  implicitly  by  Boole  and  MacColl — both  early  mathemat- 
ical logicians  —  and  explicitly  by  Frege,  Peano  and  Russell  —  all 
logisticians.  A  small  acquaintance  with  such  a  work  as  that  of 
Frege  will  give  plenty  of  examples  of  other  powerful  new  ideas 
introduced.  And  then  as  to  truths  discovered  by  logisticians,  we 
may  remind  Professor  Shaw  that  the  solution  of  ''the  paradoxes 
of  logic"  is  wholly  due  to  them,  while  mathematicians  who  were 
unacquainted  with  logistic  hopelessly  floundered  in  the  search  for  a 
solution.  Twelve  years  ago  I  was  one  of  these  flounderers  myself, 
and  my  "solution"  had  been  accepted  as  satisfactory  by  many 
mathematicians. 

The  real  fact  is  that  these  results  of  logistic  do  not  strike 
some  mathematicians  as  nearly  so  important  as  some  of  the  results 
of  the  theory  of  functions,  for  instance.  I  think  they  forget  that 
it  is  only  in  virtue  of  all  truths  being  really  of  equal  "nobility" 
that  Jacobi  was  right  in  claiming  that  a  theorem  in  the  theory  of 


CRITICISMS  AND  DISCUSSIONS.  463 

numbers  was  just  as  fine  as  a  very  striking  result  in  mathematical 
astronomy. 

'Perhaps  the  greatest  mistake  made  by  Professor  Shaw  is  the 
extraordinary  statement  about  the  nature  of  truth  near  the  top  of 
page  409.  It  is  surely  quite  evident  that  truths  themselves  do  not 
develop.  That  twice  two  are  four  was  just  as  true  last  year  as  it 
will  be  next  year, — even  if  no  people  at  all  are  left  alive  on  the 
earth  next  year.  Professor  Shaw  finds  fault  with  something  I 
wrote  because  he  thinks  that  I  maintained  that  ideas  are  not  created 
by  man.  It  is  quite  evident  from  what  I  said  in  the  context  that 
I  only  held  that  truths  are  not  created,  though  I  certainly  said  in 
a  slipshod  and  inaccurate  way  that  "we  do  not  really  create  anything 
in  science."  Really  Professor  Shaw  shows  afterward  that  he  agrees 
with  me  that  truth  itself  is  not  created,  and  his  remark  that  doubt- 
less I  thought  that  words  and  ideas  waited  in  the  mines  of  thought 
for  the  lucky  prospector  does  not  appear  to  be  either  logical  or  a 
good  guess  (see  pp.  409-411).  However,  at  the  top  of  page  409 
he  remarks  that  the  world  of  universals  changes  in  time.  I  suppose 
that  he  means  that  our  ideas,  say  of  an  "integral"  or  "continuity" 
have  changed ;  but  I  hardly  think  that  he  ought  to  have  fallen  into 
the  error  of  mistaking  the  thing  itself  for  a  result  of  our  groping 
after  the  thing.  I  take  it  also  that  he  does  not  intend  to  say  that 
truth  evolves,  for  that  rests  on  a  confusion  between  a  proposition 
and  a  propositional  function,  such  as  in  thinking  that  such  a  func- 
tion as  "Dr.  Wilson  is  President  of  the  United  States"  is  a  propo- 
sition and  not  a  function  of  the  time  which  becomes  a  proposition 
when  any  instant  is  specified  and  is  then  constantly  true  or  false 
eternally.  What  is  the  case  seems  to  me  to  be  that  in  logic  and  mathe- 
matics the  world  we  are  concerned  with  is  a  world  of  facts,  not  of 
conceptions.  Conceptions  are  formed  by  us  for  the  purpose  of 
stating  truths,  and  in  the  world  of  pure  mathematics  we  only  come 
across  facts  and  form  and  variables.  In  this  I  think  that  I  shall 
have  the  support  of  one  at  least  among  philosophers:  I  refer  to 
Dr.  Cams,  who  has  always  maintained  that  mathematics  is  essen- 
tially concerned  with  the  ideas  of  form  and  "anyness." 

We  now  come  to  the  last  inaccuracy  in  Professor  Shaw's  paper 
that  I  shall  deal  with.  This  is  the  question  about  the  logic  of  in- 
finity. The  inaccuracy  of  the  statements  on  page  412  appears 
clearly  if  we  give  a  short  statement  of  the  facts  in  the  treatment 
of  infinity  by  mathematicians  and  logicians.  Georg  Cantor,  in  a 


464  THE    MONIST. 

series  of  works  dating  from  1871  to  1897,  succeeded  in  founding 
a  new  and  immensely  important  theory  of  transfinite  numbers. 
The  use  of  the  lowest  transfinite  cardinal  numbers  did  not  audioes 
not  present  any  difficulty  whatever  to  mathematicians  or  even  logi- 
cians ;  but,  as  Burali-Forti,  Russell,  and  others  noticed  in  various 
forms  the  whole  series  of  transfinite  numbers  presents  difficulties 
which  were  later  found  to  be  fundamental  logical  difficulties  of  the 
same  nature  as  that  of  the  Cretan  who  said  that  all  Cretans  were 
liars.  Such  problems  were  discussed  at  length  in  Russell's  Prin- 
ciples of  Mathematics  of  1903  and  in  the  years  after  the  publication 
of  this  book  were  satisfactorily  solved  by  him  and  Whitehead. 
These  solutions  may  be  found  in  the  Principia  of  1910,  and  in  the 
almost  wholly  symbolical  form  of  the  book  last  mentioned  it  is 
naturally  impossible,  even  if  it  were  not  superfluous,  that  the  claims 
made  in  the  earlier  work  should  be  repeated.  Thus  it  is  unjust  to 
conclude  (p.  404)  that  the  Principia  is  an  abandonment  of  the 
claims  of  the  Principles,  brought  about  because  of  the  difficulties 
found  in  Cantor's  work.  One  might  just  as  well  conclude  that  the 
difficulties  of  a  solution  of  the  great  difficulty  of  "Cantorism"  had 
made  Russell  give  up  joking,  for  there  are  many  jokes  in  the  Prin- 
ciples and  only  one  in  the  Principia.  There  is  one  more  point.  It 
is  only  what  we  may  call  a  "boundary  problem"  about  Cantor's 
numbers  that  gives  rise  to  difficulty:  the  resolve  that  any  object 
about  which  we  talk  or  reason  must  be  defined  in  a  finite  number 
of  words  (p.  413)  does  not  succeed  in  putting  out  of  court  all 
classes  that  have  an  infinite  number  of  members.  Infinite  classes 
of  objects  each  of  which  can  be  finitely  defined  can  be  defined  in 
a  finite  number  of  words,  or  better  symbols  of  a  "real  characteristic." 
The  class  of  prime  numbers  is  such  a  class.  If  indeed  we  may  use 
the  notion  of  any  (which  is  represented  by  one  word)  or  the 
notion  of  a  variable  in  general,  we  cannot  avoid  admitting  definitions 
of  infinite  classes  by  a  definite  number  of  words.  If  also  we  may 
use  a  sign  for  a  variable,  there  is  no  earthly  difficulty  in  giving  a 
general  rule  for  correspondence  in  a  way  that  is  denied  by  Professor 
Shaw  on  page  413.  The  rule,  for  example,  if  n  is  an  integer,  given 
in  the  formula  n  +  p,  where  p  is  another  integer,  indicates  precisely 
another  class  of  integers  which  is  correlated  to  the  whole  class  of 
integers  considered  first. 

There  is  a  small  logical  error  committed  by  Professor  Shaw, 
at  least  if  he  considers  philosophy  to  be  the  same  thing  as  meta- 


CRITICISMS  AND  DISCUSSIONS.  465 

physics,  which  may  explain  why  he  is  so  satisfied  with  himself  for 
ignoring  philosophy.  On  page  409  he  characterizes  a  certain  as- 
sumption as  "philosophical"  and  explicitly  divides  "philosophy" 
from  "mathematics."  On  page  414  he  agrees  with  Lord  Kelvin 
that  "mathematics  is  the  only  true  metaphysics."  Thus  he  would 
seem  to  hold  that  there  is  no  such  thing  as  philosophy  at  all ;  this 
would  certainly  explain  why  philosophical  assumptions  are  so  little 
worth  serious  discussion.  Such  discussion  would  in  fact  be  as 
foolish  a  problem  as  to  investigate  the  birthplace  of  Jack  the  Giant 
Killer's  hen.  But  if  we  try  to  take  a  somewhat  broader  view,  and 
are  not  satisfied  with  dividing  our  knowledge  into  arbitrary  water- 
tight compartments  labeled  "Philosophy,"  "Mathematics,"  and  so 
on,  we  see  that  there  are  certain  logical  questions  which  can  be  and 
have  been  solved  by  symbolical  methods  which  strongly  remind  us 
of  algebra,  which  are  absolutely  fundamental  in  mathematics,  and 
which  when  formulated  in  ordinary  language  sound  so  like  what 
professional  philosophers  have  often  talked  about  that  many  are 
tempted  to  hurry  them  out  of  sight  into  the  "philosophical"  compart- 
ment. These  are  some  of  the  questions  with  which  logistic  deals. 
Logistic  never  claimed  to  be  able  to  run  without  the  guidance  of  a 
human  intellect  (see  p.  411)  any  more  than  the  sciences  of  mathe- 
matics or  logic  or  chemistry  did.  What  it  does  claim  to  do  is,  like 
ordinary  mathematics,  to  save  our  minds  the  labor  of  performing 
again  each  elementary  reasoning  which  requires  no  talent  but  only 
memory — often  a  prodigious  memory  when  the  reasoning  is  compli- 
cated ;  so  that  we  can  reserve  all  the  talents  we  may  possess  for 
overcoming  those  obstacles  to  a  discovery  of  truth  that  have  not  been 
hitherto  overcome.  Then  again,  unlike  ordinary  mathematics,  logis- 
tic seeks  to  point  out  differences  in  analogous  ideas  and  reasonings 
which  play  an  even  greater  part  then  analogies  when  we  come  to 
consider  really  subtle  reasoning.  Thus  the  analogy  between  impli- 
cation between  propositions,  inclusion  between  classes,  and  inclusion 
between  relations  breaks  down  in  certain  cases,  and  we  see  that 
Russell  in  his  later  work  forsakes  the  identical  form  of  the  symbols 
expressing  these  relations.  Peano,  as  we  know,  kept  to  the  same 
symbol  on  account  of  the  very  close  analogy  between  the  relations 
spoken  of. 

If  we  are  content  to  accept  without  examination  the  arbitrary 
classification  of  people  who  were  unacquainted  with  modern  logic 
into  exclusive  "mathematical"  and  "philosophical"  compartments, 


466  THE   MONIST. 

we  must  be  prepared  to  think  we  see  what  we  think  are  the  rigid 
foundations  of  mathematics  being  eaten  into  by  philosophy,  and  if 
we  wish  still  to  maintain  that  the  foundations  of  mathematics  are 
rigid  we  shall  have  continually  to  give  the  new  name  of  "philosophy" 
to  parts  of  what  were  hitherto  considered  to  be  mathematical.  This 
state  of  things  was  actually  brought  about  by  what  Poincare  called 
"Cantorism" :  truths  which  were  hitherto  considered  solid  and  math- 
ematical seemed  to  be  thrown  into  doubt  by  the  advance  of  philos- 
ophy. Of  course  this  was  not  really  so:  the  logical  questions  at 
the  foundation  of  mathematics  are  capable  of  scientific  investigation 
just  as  much  as  the  theory  of  numbers  of  the  differential  calculus, 
and  it  is  unnecessary  and  ridiculous  to  narrow  the  scope  of  our 
investigations  because  we  shall  meet  logical  difficulties  if  we  do  not. 
What  would  be  thought  of  a  tradesman  who  thought  he  could  calm 
the  mind  of  his  assistants  by  maintaining  that  the  ravages  of  a  bull, 
although  they  seemed  to  be  in  his  own  china  shop,  were  really  in 
a  drapery  department  which  had  somehow  extended  into  that  part 
of  his  shop  where  plates  were  sold?  This  is  what  those  mathe- 
maticians do  who  dismiss  awkwardness  to  "philosophy"  and  think 
that  thereby  they  have  kept  mathematics  pure  and  free  from  all 
"metaphysical"  discussions. 

Miss  Dorothy  Wrinch  has  sent  the  following  comments  on 
Professor  Shaw's  article  in  The  Monist  for  July,  1916: 

"The  chief  thing  that  I  quarrel  with  in  Professor  Shaw's  article 
is  his  idea  of  one:  selecting  one  pencil  from  a  pile  is  really  rather 
different  from  considering  the  class  whose  members  are  the  classes 
'living  kings  of  England,'  'fathers  of  A,'  etc.  Further,  it  would  be 
difficult  to  give  a  definition  of  one  or  ttvo  which  is  not  a  statement 
in  which  one  or  two  appears:  he  does  not  attempt  to  say  that  the 
other  constituents  of  this  'statement'  have  not  been  defined,  or  that 
the  definiendum  is  not  unique.  These  could  be  his  only  grounds  for 
attacking  a  definition,  which  is  merely  a  statement  in  symbolic 
form  of  cases  in  which  the  number  one  or  the  number  two  appears. 
Also  it  seems  a  pity  (line  9,  p.  407)  that  he  should  fall  into  the 
error  that  he  deplores  in  mathematical  logicians,  viz.,  the  error 
of  introducing  the  notion  of  truth  (and  truth  value)  when  'in  no 
place. . . .  they  are  defined.' 

"I  suppose  that  it  was  'in  the  intoxication  of  the  moment'  that 
Professor  Shaw  called  a  prepositional  function  of  two  variables 
a  relation  (p.  404,  line  14),  and  let  out  of  the  bag  the  existence  of 


CRITICISMS  AND  DISCUSSIONS.  467 

a  difference — hitherto,  apparently,  kept  dark  by  mathematicians — 
between  the  properties  of  the  roots  of  a  quadratic  equation  and 
the  properties  of  quadratic  functions  of  x. 

"Professor  Shaw  makes  some  strange  remarks  on  page  413. 
If  the  collection  of  all  integers  does  not  exist  (line  8)  it  seems 
hardly  necessary  to  refute  the  proposition  that  it  is  possible  to  cor- 
relate the  collection  of  all  integers  to  some  other  infinite  collections. 

"It  seems  rather  unsportsmanlike  to  rely  upon  people's  short 
memories  and  call  Poincare's  attacks  on  logistic  successful.  Might 
it  not  be  well  to  remind  people  of  the  conclusions  to  which  M. 
Couturat  came  at  the  end  of  his  article  in  The  Monist  for  October, 
1912:  'Admitting  the  principles  and  primitive  ideas  of  the  logisticians, 
M.  Poincare  has  maintained  that,  setting  out  from  these  data,  they 
cannot  build  up  mathematics  without  another  postulate — an  appeal 
to  intuition  or  a  synthetic  a  priori  judgment ;  and  he  has  thought 
that  he  has  discovered  in  their  logical  construction  certain  paral- 
ogisms (beggings  of  the  question  or  vicious  circles).  I  believe  that 
I  can  conclude  from  the  above  discussion  that  not  one  of  these 
theses  is  proved,  and  that,  in  particular,  the  logisticians  have  not 
committed  any  of  the  logical  errors  that  are  so  lightly  imputed 
to  them/" 

PHILIP  E.  B.  JOURDAIN. 

FLEET,  HANTS,  ENGLAND. 


THE  CAL-DIF-FLUK  SAGA. 
EDITORIAL  INTRODUCTION. 

Mr.  J.  M.  Child  has  given,  in  the  following  "Saga,"  an  amusing 
description  of  the  results  he  has  arrived  at  in  his  book  on  Barrow, 
just  published  in  the  series  of  "Open  Court  Classics."  The  closing 
lines  represent  the  opinion  he  has  formed  from  a  consideration  of 
the  manuscripts  of  Leibniz,  an  annotated  translation  of  which  has 
been  appearing  in  current  numbers  of  The  Monist,  beginning  with 
October,  1916,  and  continued  in  the  April  number  and  the  present 
one. 

The  saga  evidently  refers  to  the  question  of  the  invention  of 
the  infinitesimal  calculus.  Isa-Roba  is  Barrow,  Isa-Tonu  is  Newton, 
Zin-BH  is  Leibniz,  while  Cavalieri  is  mentioned  under  the  name  of 
Ler-a-Cav.  Gen-Tan-Agg  stands  for  Barrow's  Gen-eral  method  of 

gents  and  of  ^^-regates ;  while  Shun-Fluk  and  Cal-Dif  ob- 


468  THE   MONIST. 

viously  refer  to  the  methods  of  Newton  and  Leibniz.  Batnac  is 
the  ordinary  abbreviation  of  the  Latin  for  Cambridge,  Cantab.,  with 
its  letters  reversed ;  and  the  allusion  in  the  next  line  is  to  Barnwell 
Pool,  where  it  is  stated  that  an  undergraduate  whose  boat  had 
overturned  was  saved  from  drowning,  but  died  soon  afterward 
from  blood-poisoning!  Terangel  is  a  transformation  of  Angleterre, 
i.  e.,  England.  Ris-Pah  is  Paris,  where  Leibniz  lived  at  the  time 
of  the  invention  of  the  calculus. 

In  the  second  stanza,  the  allusions  to  ''burning  midnight  oil," 
the  quill  pen,  incandescent  gas  mantle,  and  the  electric  light  are  all 
fairly  obvious ;  while  the  Swan  may  be  taken  to  refer  to  a  well- 
known  make  of  fountain  pen.  Stanza  5  refers  to  the  publication  of 
a  book.  The  archery  in  the  first  method  of  training  alludes  to  the 
ancient  definitions  of  a  tangent  and  a  normal  to  a  curve ;  and  the 
sword-play  recalls  Euc.  I,  10  and  Euc.  I,  1,  while  the  allusions  in 
the  second  are  easily  referred  to  the  method  of  indivisibles. 

In  Stanza  9.  the  dagger  refers  to  the  differential  triangle,  which 
Barrow  only  included  in  the  first  edition  of  his  work  on  the  advice 
of  Newton ;  the  knobs  on  the  hand-grip  refer  to  Newton's  "dot" 
notation. 

The  two  weapons  of  Zin-Bli  are  the  signs  invented  by  him  for 
differentiation  and  integration.  Lastly,  Li-Nu-Ber  is  John  Ber- 
noulli, who  stated  that  Leibniz  got  the  whole  of  his  fundamental 
ideas  from  Barrow,  whereas  Leibniz  himself  denied  any  indebtedness 
to  Barrow. 

THE  CAL-DIF-FLUK  SAGA.1 

1.  Saga  of  sons  of  a  Goddess,  of  Thought  and  Learning  the  fountain, 
(Haply  in  that  which  I  sing,  a  real  historical  meaning. 
Wrapped  in  a  fanciful  garb,  and  oddly  disguised  as  a  saga. 
Those  who  are  skilled  in  lore,  and  erudite  more  than  their  fellows, 
Knowing  the  facts  of  the  case,  if  they  diligently  seek  may  discover.) 
Dwelt  She,  She  dwells  upon  Earth,  and  henceforth  for  ever  and  ever 
Dwell  so  She  will  among  mortals.  'Tis  thus  decreed  by  the  All- Wise. 

2.  Oil  from  the  Midnight  Lamp  the  sacrifice  burned  on  her  altars, 
Plumes  from  the  wing  of  the  Goose  her  now  peculiar  token ; 

Not  so  at  first  was  it  thus,  and  not  in  the  times  that  are  coming 

1  From  a  manuscript  found  in  1916  A.  D.,  while  searching  an  ancient 
tumulus  or  "barrow,"  and  made  out  from  the  original  by  J.  M.  Child. 


CRITICISMS  AND  DISCUSSIONS.  469 

Will  it  be  Oil  and  Plume ;  I  see  with  the  eye  of  the  seer 
Wondrous  visions  of  Light,  enwrapped  in  a  Mantle  resplendent, 
Torn  from  the  heart  of  a  stone,  the  essential  soul  of  the  Sun-god 
Prisoned  for  ages  therein ;  and  globes  of  crystal  translucent 
Glowing  with  filaments  bright  kept  hot  by  the  Spirit  of  Lightning, 
Swan  of  the  Golden  Beak  instead  of  the  Goose  for  her  token. 

3.  Sent  upon  Earth  to  dwell  with  mortals  by  will  of  the  All-Wise, 
Children  divine  to  bear  to  those  who  Her  fancy  might  capture. 
Ardent  and  long  was  the  wooing,  both  strong  and  patient  the  lover, 
Ere  he  received  his  reward,  or  ere  She  presented  him  offspring. 
Else  as  a  mark  of  Her  love  to  him  She  had  chosen  to  honor, — 
Chosen  for  womanly  whim,  for  some  unaccountable  reason 
Honored  above  all  else,  who  never  had  courted  her  favor — 

Sent  She  on  lighting  wings  the  soul  of  Her  heart,  Inspiration. 

4.  Children  of  fathers  of  Earth,  but  endowed  with  the  life  of  the 

Mother, 

Destined  as  Heroes  to  wage  perpetual  warfare  on  all  things 
Troubling  the  minds  of  men  desiring  to  widen  the  limits 
Set  on  the  realm  of  We-Know,  by  the  race  of  the  children  gigantic, 
Issue  of  Never-Before  out  of  We-Xever-Heard-of  the-Method. 
Children  begotten  from  Her  are  known  by  the  names  of  their  fathers, 
More  by  the  deeds  of  the  sons  are  the  fathers  so  held  up  to  honor ; 
Accurate  records  are  kept;  thus  long  through  the  ages  that  follow, 
Known  by  the  deeds  of  the  sons  are  the  fathers  so  held  in  remem- 
brance. 

Rightly  was  this  the  Law,  for  responsible  he  for  the  training, 
Fitting  the  son  for  the  fight  for  freedom  and  fuller  perception. 

5.  Till  'twas  such  time  as  was  meet,  the  custom  obtained  that  in 

secret 

(Jealous  that  others  might  see  not  fully  developed  the  power 
Promising  greatness  to  come),  this  fatherly  training  continued 
Day  after  day  for  an  eon ;  until  with  a  flourish  of  trumpets, 
Front  of  the  eyes  of  all,  tattooed  with  the  symbols  of  Learning, 
Clad  in  a  mantle  of  calf -skin,  bearing  on  back  and  on  bosom 
Plainly  for  all  to  observe,  in  resplendent  gold  letters,  his  title, 
Son  of  the  Goddess  of  Thought,  was  he  set  as  a  champion  of 

Knowledge. 


47O  THE   MONIST. 

6.  The  methods  of  training  were  two,  at  least  only  two  were  ac- 

counted. 

Oldest  and  best  known  of  all  was  the  method  derived  from  the  An- 
cients, 

Cumbrous,  exhaustive  and  long ;  horizontal  and  parallel  bar  work, 
Drawing  of  cord  of  the  bow,  and  the  rings  were  considered  essential ; 
Accurate  hand  and  eye  were  developed  by  shooting  an  arrow, 
Grazing  the  cheek  of  a  figure,  or  forth  from  it  standing  erected ; 
Cleaving  a  bar  into  twain,  so  each  part  as  to  balance  the  other 
( Nought  but  two  measuring  swings  ere  the  cut  was  delivered  allowed 

him). 

Such  like  in  days  of  old  had  fitted  the  Heroes  for  battle. 
Founded  on  this  was  the  second,  but  strangely  unlike  it  in  practice ; 
Suppleness  rathe.r  than  strength  was  the  object  and  creed  of  the 

trainer. 
Straight-edged  still  was  the  sword ;  with  it  blocks  were  sliced  into 

shavings, 
Shavings  were  sliced  into  threads,  and  threads  were  chopped  into 

pieces, 

Parts  of  ineffable  smallness,  divisible  reckoned  no  further. 
Masonry  part  of  the  course,  in   which  arches   with  bricks  were 

fashioned, 

Leaving  the  corners  undressed  ;  as  the  pupil  advanced  in  his  training, 
Smaller  and  smaller  the  bricks,  indivisible  finally  counted. 
Specially  fitted  for  Heroes,  prepared  for  attack  on  the  giant 
Clans  of  A-Re-A  and  Vol-Yum,  the  brood  of  Cur-Va-Rum  and 

Mez-zur. 

Failed  jf  the  fatherly  training,  the  Goddess  in  sorrowful  anger 
Took  from  the  child  his  soul,  the  gift  which  at  birth  She  had  given, 
Worthier  father  to  bless,  if  ever  another  such  won  Her. 

7.  Once  in  the  days  now  gone,  there  lived  on  the  banks  of  the  Batnac, 
Renowned  for  its  smells  and  its  mud,  where  pollution  enters  at  Well- 
Barn 

(Truly  not  then  was  this  fame,  nor  yet  at  the  time  of  this  writing 

Thus  had  it  won  a  repute,  'tis  a  prophecy  sure  that  I  utter), 

Land  of  Terangel  within,  a  mortal  yclept  Isa-Roba. 

Many  and  varied  his  loves,  his  fickleness  surely  a  drawback ; 

Truly  a  wonder  it  was  that  the  Goddess  e'er  let  him  approach  Her. 

Bare  She  however  a  son,  Isa-Roba  undoubted  the  father, 

Fair  both  in  face  and  in  form,  a  divine  conception  befitting ; 


CRITICISMS  AND  DISCUSSIONS.  47! 

Ne'er  such  a  babe  before  was  born  with  so  splendid  a  future ; 
Seemed  that  the  soul  of  his  Mother  had  enter'd  the  Child  at  his 

birth-time : 

Best  that  She  had  to  give,  best  that  She  can  give  for  all  time, 
Gave  She  this  son  of  Her  heart ;  Gen-Tan- Agg  Isa-Roba  did  name  it. 

8.  Trained  he  the  boy  in  a  manner  that  savored  of  that  of  the  An- 

cients, 

Discipline  rigorous  keeping,  yet  toned  with  a  method  that  fore-time 
Ler-A-Cav  brought  to  perfection,  a  mingling  of  first  and  of  second 
Systems  of  training  recounted  ;  however  'twas  foredoomed  to  failure. 

9.  Hercules  never  so  strong  as  the  youth  Gen-Tan-Agg,  no,  nor 

Samson. 

Armed  with  his  two-handed  weapon  he  met  many  giants  in  combat ; 
Numerous  clans  he  defeated,  by  slaying  their  general  doughty. 
Nevertheless  were  his  muscles  too  stiffened  by  reason  of  rigor, 
Due  to  the  manner  in  which  Isa-Roba  conducted  his  training. 
Love  for  the  two-handed  broadsword,  with  which   Isa-Roba  had 

armed  him, 

Made  him  neglect  the  superior  weapon  that  hung  at  his  waist-belt, 
Sharper  by  far  than  the  sword-blade,  a  steel  of  superior  temper ; 
Seems  Gen-Tan-Agg  only  used  it  preparing  the  shafts  of  his  arrows  ; 
Nigh  came  to  leave  it  at  home  as  he  set  out  upon  his  first  journey, 
Girding  it  on  at  the  last,  not  perceiving  in  it  that  a  weapon 
Ready  to  hand  he  had  got  against  which  no  armor  of  mortals 
Could  for  a  moment  prevail ;  for  piercing  the  joints  of  the  harness. 
Off' ring  no  passage  to  sword-blade,  it  reached  his  opponent's  main 

vitals ; 
Forced  him  to  give  up  his  treasure,  the  secret  protected  for  ages. 

10.  Happened  it  thus  that  a  Hero,  high-blessed  by  the  Goddess  his 

Mother, 

Spoiled  by  the  weapon  mistaken  his  anxious  sire  recommended, 
Fame  and  renown  and  great  honor  did  miss  for  ever  and  all  time, 
Losing  the  chance  that  was  offered,  a  name  and  a  high  reputation. 
Lastly,  by  father  discarded  (who  fickly  returned  to  a  first  love), 
Languished  and  nigh  came  to  perish,  unhonored,  unsung  and  neg- 
lected. 

11.  Some  of  the  records  of  giants  the  youth  Gen-Tan-Agg  had  de- 

feated 


472  THE   MONIST. 

Chanced  Isa-Roba,  however,  had  told  to  a  friend  Isa-Tonu ; 
Agile  by  nature,  the  latter  immediate  saw  that  the  dagger, 
Superior  far  to  the  broadsword,  was  a  weapon  of  magical  value ; 
('Twas  Isa-Tonu's  advising  that  just  at  the  very  last  moment 
Caused  Isa-Roba  to  add  to  his  offspring's  armor  the  dagger).    . 
Pity,  perhaps,  for  the  youth,  or  a  covetous  eye  for  the  poignard, 
Caused  Isa-Tonu  to  take  neath  his  fostering  care  the  young  stripling, 
Freeing  his  father  from  trouble,  unhampered  to  follow  his  fancy. 
Thus  Isa-Roba  the  story  departs  from,  unhonored  for  all  time ; 
Save  and  if  only  in  future,  this  tomb  may  be  opened  by  some  one 
Trying  to  find  out  the  truth  of  the  Hero's  father  and  birthtime. 
Under  the  fostering  care  of  a  trainer  less  hide-bound  by  nature, 
Slowly  at  first,  then  apace,  did  the  Hero  recover  his  power. 
Changed  was  his  armor,  the  sword  altogether  replaced  by  the  dagger, 
Changed  was  the  dagger  in  form,  for  a  knob,  sometimes  two,  on  the 

hand-grip 

Gave  it  a  far  better  balance.    Obsessed  by  his  special  requirements, 
Secretly  long  Isa-Tonu  did  bind  Gen-Tan-Agg  to  his  service. 
Later  ungratefully  hiding  the  name  of  the  Hero  who  served  him, 
Swearing  that  all  had  been  done  by  his  own  bastard  offspring,  young 

Shun-Fluk. 

12.  Thus  once  again  was  the  Hero  discarded  and  left  for  to  languish, 
Shun-Fluk  attaining  the  fame  that  should  his  have  been  truly  and 

rightly. 

Nemesis,  son  of  old  Equity,  sternest  of  Gods  and  the  justest, 
Saw  Isa-Tonu's  deception,  and  straightway  the  Goddess  of  Learning 
Sought  He  and  told  Her  the  story.    In  sorrowful  anger  the  Goddess 
Listened  with  eyes  that  flamed  at  the  failure  that  followed  Her  off- 
spring, 

Due  to  his  father's  bad  training,  and  then  Isa-Tonu's  enslavement ; 
Listened  and  cursed  the  first,  for  the  other  a  punishment  thought  out. 

13.  "Punishment  dreadful  and  dire !"    So  she  spake,  the  while  Neme- 

sis listened, 

Listened  and  nodded  and  smiled,  as  approved  He  the  plan  She  sug- 
gested. 

"Lives  there  a  mortal  in  Ris-Pah,  who  long  has  courted  my  favor; 

Often  of  late  have  I  thought  that  at  last  I'd  rewrard  his  devotion. 

Lacks  he  but  one  little  thing,  only  one  thing  to  render  him  fitting 


•    CRITICISMS  AND  DISCUSSIONS.  473 

Trainer  of  offspring  of  mine;  but  the  lack  mean  I  now  to  forgive 

him. 

Never  again  could  I  bear  such  a  child  as  I  bore  Isa-Roba ; 
Certain  is  that ;  but  immortal  the  soul  that  at  birth-time  I  gave  him, 
Breath  of  my  life,  Inspiration,  again,  Gen-Tan-Agg  expiring, 
Can,  if  I  will  it,  enlighten  the  child  which  I'll  offer  to  Zin-Bli. 
Thus  is  he  called  by  mortals,  an  inventor  of  weapons  and  symbols. 
One  has  he  fashioned  already,  in  shape  like  a  chopper  for  fire-wood, 
Straight  in  the  shaft,  with  a  hand-stop  to  stay  it  from  slipping, 
Circular  edge  to  the  axe-blade,  to  shaft  is  it  fastened  by  bolt-head ; 
Much  like  the  symbol  that  mortals  set   fourth  in  the  lower-case 

system. 

This  shall  he  teach  my  offspring  to  use  to  more  delicate  purpose. 
Binds  he  his  sticks  all  together  with  cord  made  out  of  the  sum-omn ; 
Lurking  however  in  thought  is  the  germ  of  a  better  invention. 
Rod  with  curl  at  each  end,  slightly  bent,  so  that  clipped  round  the 

bundle, 

Binding  the  whole  into  one,  he  is  able  to  thus  grasp  it  firmly. 
Armed  with  each  of  these  twain,  shall  his  offspring  forth  stand  as 

a  Hero." 
Spake  She,  and  Nemesis  nodding  to  all  His  approval,  it  was  so. 

14.  Cal-Dif  named  Zin-Bli  the  child,  and  he  trained  him  these  weap- 

ons to  master ; 

Speed,  at  all  rates,  with  the  first  he  created  new  records  completely, 
Nor  did  he  stay  at  that ;  with  the  second,  the  brood  of  the  giants, 
Laid  he  them  low  in  the  dust,  so  that  never  again  should  they  trouble. 
All  that  the  Goddess  had  said,  so  performed  She;  the  credit  of 

Cal-Dif 
Famed  through  the  kingdoms  of  mortals,  became  a  renown  for  the 

father, 
Ne'er  to  be  equalled  till  Earth  is  devoid  of  reasoning  mankind. 

15.  Swelled  as  to  head  by  renown,  though  Zin-Bli  well  knew  Inspi- 

ration 

(Could  he  forget  this?)  had  wrought  in  a  magical  manner  the  marvel, 
Yet  could  not  bear  it  for  others  to  know  whence  the  source  of  his 

wisdom ; 

Denied  he  the  source  whence  it  came,  Isa-Roba's  offspring  discarded. 
Nemesis  saw  what  he  did,  and  he  stirred  up  the  folk  of  Terangel, 


474  THE  MONIST. 

Shun-Fluk  to  accuse  him  of  stealing  and  sending  him  forth  as  his 

Dif-Cal. 
None  seemed  to  have  guessed  the  truth,  save  a  man  by  the  name  of 

Li-Nu-Ber. 

16.  Ye  who  perchance  may  consider  this  saga  in  future  far  ages, 
Know  now  the  truth  ye  may ;  that  the  soul  of  the  Goddess  of  Learn- 
ing 

Entered  at  first  Gen-Tan-Agg,  but  he  languished  for  lack  of  good 
training  ; 

Afterwards,  renamed  Shun-Fluk,  he  recovered  some  of  his  birth- 
right ; 

Dying,  his  soul  was  then  given  to  an  ordinary  child  of  a  mortal, 

Rendering  its  face  and  its  form  like  one  of  divine  conception. 

17.  Accepted  as  such  by  all,  till  the  day  that  this  saga's  discovered, 
Haply  e'en  then,  for  foretell  I  that  Cal-Dif 

Unfortunately,  the  manuscript,  which  consists  of  another  couple 
of  sheets  that  were  outermost  in  the  roll,  here  becomes  indecipher- 
able through  being  destroyed  by  damp;  it  would  have  been  inter- 
esting, and  useful  in  the  light  of  judging  of  the  truth  of  the  facts 
given,  to  have  verified  how  far  the  prophecies  were  fulfilled  by 
events  since  the  time  at  which  they  were  written  down  and  the 
manuscript  hidden  in  this  old  burial-mound. 

J.  M.  CHILD. 

DERBY,  ENGLAND. 


NOTES   ON   DE   MORGAN'S   BUDGET   OF   PARADOXES. 

In  a  work  requiring  the  large  amount  of  reading  involved  in 
editing  a  book  like  the  Budget  of  Paradoxes,  and  particularly  in  the 
condensing  of  the  results  to  the  proper  proportions  for  footnotes 
to  aid  the  reader,  it  was,  of  course,  inevitable  that  a  certain  number 
of  inaccuracies  would  occur.  It  is  also  evident  that  many  more 
notes  might  profitably  have  been  added  to  elucidate  the  meaning 
of  the  text,  or  to  correct  the  original  where  this  would  be  warranted. 

De  Morgan  was  a  careless  writer  and  many  of  his  errors  are 
mentioned  in  the  footnotes ;  but  numerous  others  exist,  some  of 
which  are  patent  to  any  reader  and  others  of  which  might  profitably 


CRITICISMS  AND  DISCUSSIONS.  475 

have  been  set  forth  by  the  editor.  It  is  also  a  serious  question  as 
to  whether  the  translation  of  common  phrases  is  not  more  of  a 
hindrance  than  a  help  to  even  the  casual  reader,  and  whether  the 
space  used  by  such  translation  might  not  have  been  more  profitably 
devoted  to  a  further  elucidation  of  the  text.  This  is  the  feeling  of 
one  or  two  critics. 

Since  the  work  was  published,  several  friends  have  called 
attention  to  a  few  misprints,  a  few  generous  critics  have  suggested 
helpful  changes,  and  one  or  two  others  have  objected  to  certain  of 
the  notes.  It  therefore  seems  proper  to  present  a  few  emendata 
and  errata  which  may  assist  the  reader  of  the  work. 

In  the  matter  of  emendata  to  De  Morgan's  text  itself  and  of 
suggestions  as  to  further  helpful  notes  I  am  indebted  chiefly  to 
Prof.  A.  E.  Taylor  of  St.  Andrews,  Scotland,  who  has  gone  over 
the  work  with  great  care  and  has  kindly  given  the  Open  Court 
Publishing  Company  the  benefit  of  his  reading.  The  following 
notes  on  De  Morgan's  text  are  due  to  him. 

Vol.  I,  page  3.  De  Morgan  should  not  have  attributed  to 
Spinoza  the  anonymous  Philosophia  sanctae  scripturae  interpres.  It 
was  probably  the  work  of  his  friend  and  physician  Lodowick  Meyer. 

Vol.  I,  page  41.  De  Morgan's  version  of  the  passage  from  the 
commentary  of  Eutocius  on  the  tract  by  Archimedes  on  the  meas- 
ure of  the  circle  is  not  satisfactory.  The  Cerii  of  Porus  should 
be  the  Ceria  (/ojpia,  honey  combs)  of  Sporus.  He  probably  used 
the  Wallis  edition  of  Eutocius  and  quoted  only  the  first  four  words 
of  the  passage  (Archimedis  Opera  Omnia,  III,  p.  300,  of  the  1881 

edition  of  Heiberg)  :  «s  dKpi/^eorepovs  apidfiovs  dyayetv  TWV  W     'Ap^i- 

fHj8ov^  clprjfifvwv,  rov  re.  £"  <f>r)fj.l  KOI  r<av  i  oa".  The  restoration  adopted 
by  Heiberg  makes  the  statement  of  Eutocius  correct:  "a  more  ac- 
curate evaluation  than  that  of  Archimedes,  i.  e.,  than  the  fractions 
^4  and  l%\"  According  to  Sporus,  Philo  of  Gadara  had  found 
closer  limits.  Archimedes  had  given  3%  as  the  upper  limit  and 
31(%i  as  the  lower  limit  of  TT,  the  £"  and  oa"  representing  merely  the 
fractional  parts. 

Vol.  I,  page  96.  De  Morgan's  language  seems  to  imply  that 
the  Convocation  of  the  University  of  Oxford  is,  or  was,  a  body  of 
ecclesiastics  of  the  Anglican  Church,  but  it  is  not  an  ecclesiastical 
body  at  all.  It  consists  of  all  masters  of  arts  who  qualify  by  the 
regular  payment  of  their  university  dues.  Professor  Taylor  suspects 
that  De  Morgan  may  have  confused  the  Convocation  of  Oxford 
with  the  Convocation  of  the  Clergy  of  the  Province  of  Canterbury. 


4/6  THE   MONIST. 

Vol.  II,  page  274.  For  De  Morgan's  translation  of  oviAov  /ue'Aos, 
read  "a  song  of  bale"  (oAoov  /xe'Aos). 

Vol.  II,  page  277.  De  Morgan  overlooks  the  true  reason  why 
Pope  scans  Mathcsis  as  Mdthesis,  namely,  that  like  all  writers  of 
his  day  he  pronounced  Greek  names  according  to  their  accent,  not 
as  we  now  do  with  an  adjustment  of  the  stress  accent  to  the  quantity 
of  the  vowels. 

Vol.  II,  page  322.  De  Morgan  is  incorrect  in  his  statement  as 
to  Bohme's  division  of  Mercurius.  Bohme  divides  it  Mer-cu-ri-us, 
not  Merc-u-ri-us. 

Vol.11,  page  340.  It  would  be  interesting  to  know  whether  De 
Morgan's  complaint  that  Walter  Scott  did  not  know  what  "Napier's 
bones"  were  is  well  founded. 

Professor  Taylor  suggests  various  other  interesting  notes  re- 
lating to  the  text,  and  of  course  such  a  list  could  easily  be  extended. 

In  the  extensive  bibliography  given  in  the  notes  it  was  inevitable 
that  certain  slips  of  the  pen  should  have  occurred.  In  Vol.  I,  page 
105,  I  followed  Bierens  de  Haan  in  giving  the  spelling  "Johannem 
Pellum."  ^My  friend  Herr  Enestrom  has  a  copy  of  the  edition  in 
question  and  the  spelling  there  given  is  "loannem  Pellivm."  He 
also  calls  my  attention  to  the  proof  given  in  the  Bibliotheca  Mathe- 
matica  recently  that  Mydorge  was  not  the  author  of  the  Recreations 
mathematiques  as  published  in  Boncompagni's  Bullettino. 

Among  the  slips  of  the  pen  which  I  have  noticed  since  the  work 
appeared  is  the  name  of  D'Alembert  for  that  of  De  Lalande  in 
Vol.  I,  page  41  ;  "condemned"  for  "contemned"  on  page  92 ;  and, 
in  Vol.  II,  "blata"  for  "beata"  on  page  61. 

Professor  Taylor  calls  attention  to  the  further  slips  of  "fellow 
of  Cambridge"  for  "fellow  of  Trinity  College,  Cambridge"  and  of 
"Derion"  for  "Denon"  (Vol.  I,  page  76)  ;  "Viscount  of  Palmerston" 
for  "Viscount  Palmerston"  (page  290)  ;  "closed"  for  "classed"  (in 
the  text,  Vol.  II,  page  148)  ;  "tolo"  for  "toto"  in  the  text  (page 
344)  ;  and  ±  for  ±  1  in  the  text  (page  368). 

I  am  also  indebted  to  Professor  Taylor  for  several  suggestions 
of  betterment  of  the  translations,  matters  which  should  have  been 
attended  to  by  me  in  the  preparation  of  these  particular  notes  even 
though  I  entrusted  this  work  to  another.  The  following  changes 
are  not  to  be  attributed  to  him,  although  changes  (sometimes  more 
extended)  were  suggested  by  him. 

In  Vol.  I,  page  3,  for  "what  it  was"  read  "that  it  was" ;  page 
40,  for  "its  appointed  path"  read  "the  appointed  path" ;  for  the  free 


CRITICISMS  AND  DISCUSSIONS.  477 

translation  in  verse  on  pages  53-54,  for  "And  lacking  nothing  but  a 
start,  and  lacking  nothing  but  an  end,"  read  "The  only  one  without 
a  start,  the  only  one  without  an  end" ;  page  339,  for  "think  himself 
to  die"  read  "feel  himself  dying." 

In  Vol.  II,  page  23,  n.  4,  for  "He  was  wont  to  indulge  in"  read 
"He  has  a  habit  of  refreshing  his  reader  by";  page  151,  for  "con- 
demned soul"  (literal)  read  "hack"  (colloquial)  ;  page  154,  change 
the  translation  of  the  familiar  legal  phrase  to  bring  out  the  pun 
upon  J.  S.,  "Summum  J.  S.  (for  jus)  summa  injnria"  (the  height 
of  law — J.  S. — the  height  of  wrong)  ;  page  200,  change  "sleeping 
power"  to  "sleep-producing  power" ;  page  228,  translate  8io<>  ei/xi  rj 
i7pas,  as  "of  Zeus  I  am,  or  Hera,"  and  ^  /uWa  as  "mass" ;  page  260, 
translate  the  quotation  from  Acts  xix.  38,  as  "the  courts  are  sitting"  ; 
page  262,  for  "according  to  which"  read  "relatively" ;  page  283,  for 
the  manifest  error  in  the  note  on  "ab  ovo"  read  "from  the  egg," 
probably  relating  to  the  passage  in  Horace,  "nee  gemino  bellutn 
Trojanmn  ordititr  ab  oro,''  or  possibly  to  "ab  oro  usque  ad  mala" ; 
page  365  for  "slayst"  (misprint  for  "slayest")  read  "keepst." 

Professor  Taylor  also  suggests  that  Hobbes  lived  only  about 
eleven  years  in  France  (Vol.  I,  page  105)  ;  that  Burnet  left  England 
to  avoid  being  involved  in  the  ruin  of  the  Whigs  (page  107)  ;  that 
Street  acted  in  accord  with  the  law  (page  124)  ;  and  that  there  was 
nothing  strange  in  Laud's  patronage  of  Palmer  (page  145).  The 
details  of  these  emendata  and  certain  other  suggestions  of  change 
would  trespass  too  much  upon  the  space  which  the  editor  of  The 
Monist  has  kindly  allowed  me. 

DAVID  EUGENE  SMITH. 

TEACHERS  COLLEGE,  NEW  YORK. 


BOOK  REVIEWS  AND  NOTES. 


REFLECTIONS  ON  VIOLENCE.  By  Georges  Sorel.  Translated,  with  an  introduc- 
tion and  bibliography,  by  T.  E.  Hulme.  London,  George  Allen  &  Unwin, 
1916.  7s.  6d.  net. 

SoreFs  book  is  exceedingly  difficult  to  discuss  in  a  short  review.  Its  sub- 
stance is  a  very  acute  and  disillusioned  commentary  upon  nineteenth-century 
socialism,  and  upon  the  politics  of  the  French  democracy  for  the  last  twenty- 
five  years.  It  contains  also  two  elements  which  must  not  be  confused,  Sorel's 
own  political  propaganda  (if  he  would  allow  it  to  be  so  called)  and  his  phi- 
losophy of  history  formed  under  the  influence  of  Renan  and  Bergson.  And  it 
expresses  that  violent  and  bitter  reaction  against  romanticism  which  is  one 
of  the  most  interesting  phenomena  of  our  time.  As  an  historical  document, 
Sorel's  Reflections  gives,  more  than  any  other  book  that  I  am  acquainted  with, 
an  insight  into  what  Henri  Gheon  calls  "our  directions." 

Doubtless  many  readers  will  be  disposed  to  consider  the  book  under  its 
first  aspect  only.  But  the  study  of  Sorel's  political  observations  requires  an 
accurate  knowledge  of  government  and  parliamentary  activities  since  the  Drey- 
fus trial,  and  does  not  in  itself  make  the  work  of  importance  to  the  English 
and  American  public.  What  Sorel  wants  is  not  a  political,  but  a  social  form. 
One  must  remember  that  his  creed  does  not  spring  from  the  sight  of  wrongs 
to  be  redressed,  abuses  to  be  cured,  liberties  to  be  seized.  He  hates  the  middle 
classes,  he  hates  middle-class  democracy  and  middle-class  socialism;  but  he 
does  not  hates  these  things  as  a  champion  of  the  rights  of  the  people,  he  hates 
them  as  a  middle-class  intellectual  hates.  And  the  proletarian  general  strike 
is  merely  the  instrument  with  which  he  hopes  to  destroy  these  abominations, 
not  a  weapon  by  which  the  lower  classes  are  to  obtain  political  or  economic 
advantages.  His  motive  forces  are  ideas  and  feelings  which  never  occur  to 
the  mind  of  the  proletariat,  but  which  are  highly  characteristic  of  the  present- 
day  intellectual.  At  the  back  of  his 'mind  is  a  scepticism  which  springs  from 
Renan,  but  which  is  much  more  terrible  than  Renan's.  For  with  Renan  and 
Sainte-Beuve  scepticism  was  still  a  satisfying  point  of  view,  almost  an  esthetic 
pose.  And  for  many  of  the  artists  of  the  eighties  and  nineties  the  pessimism 
of  decadence  fulfilled  their  craving  for  an  attitude.  But  the  scepticism  of  the 
present,  the  scepticism  of  Sorel,  is  a  torturing  vacuity  which  has  developed 
the  craving  for  belief. 

And  thus  Sorel,  disgusted  with  modern  civilization,  hopes  "that  a  new  culture 
might  spring  from  the  struggle  of  the  revolutionary  trades  unions  against  the  em- 


BOOK  REVIEWS  AND  NOTES.  479 

ployers  and  the  state."  He  sees  that  new  political  disturbances  will  not  evoke  this 
culture.  He  is  representative  of  the  present  generation,  sick  with  its  own  knowl- 
edge of  history,  with  the  dissolving  outlines  of  liberal  thought,  with  humanita- 
rianism.  He  longs  for  a  narrow,  intolerant,  creative  society  with  sharp  divisions. 
He  longs  for  the  pessimistic,  classical  view.  And  this  longing  is  healthy.  But 
to  realize  his  desire  he  must  betake  himself  to  very  devious  ways.  His  Berg- 
sonian  "myth"  (the  proletarian  strike)  is  not  a  Utopia  but  "an  expression  of  a 
determination  to  act."  The  historian  knows  that  man  is  not  rational,  that 
"lofty  moral  convictions"  do  not  depend  upon  reasoning  but  upon  a  "state  of 
war  in  which  men  voluntarily  participate  and  which  finds  expression  in  well- 
defined  myths."  It  is  not  surprising  that  Sorel  has  become  a  Royalist. 

Mr.  Hulme  is  also  a  contemporary.     The  footnotes  to  his  introduction 
should  be  read.  i? 


THE  NEW  INFINITE  AND  THE  OLD  THEOLOGY.  By  Cassius  J.  Keyser.  Yale 
University  Press,  New  Haven,  1915.  Price  75  cents. 

In  this  essay  Dr.  Keyser  shows  many  interesting  ways  in  which  some  of 
the  most  difficult  problems  of  theology  may  be  partly  or  wholly  overcome  by 
mathematical  means. 

The  relation  between  religion  and  science  is  discussed,  the  author  showing 
that  while  science  belongs  to  the  middle  zone,  or  rational  world,  religion 
belongs  to  the  over-world  or  superrational.  Then  follows  a  brief  discussion 
of  the  relation  of  theology  to  religion,  theology  being  primarily  a  science,  in 
a  word  "the  science  of  idealization."  From  the  purely  theological  standpoint, 
"God  is  an  hypothesis."  In  all  definitions  of  God  the  notion  of  infinity  is 
foremost.  Therefore  the  essay  develops  the  mathematical  concept  of  infini- 
tudes and  through  many  examples  makes  clear  the  denumerable  type  of  in- 
finite manifolds ;  then  far  surpassing  this  in  glory,  the  continuum  type,  and 
points  to  types  of  even  higher  orders.  "The  infinite  of  theology  is  the  limit 
of  the  endless  sequence  of  more  and  more  embracing  infinitudes  presented 
by  science.'' 

The  contradictions  of  theology  are  of  two  kinds,  foreign  and  domestic. 
Theology  may  rid  herself  of  the  foreign  variety  by  casting  out  all  illegitimate 
postulates.  In  the  world  of  infinitudes  the  part  of  a  group  may  be  just  as 
numerous  as  the  whole  group.  So  in  the  realm  of  theology,  the  seemingly 
contradictory  ideas  of  omniscience  and  freedom  may  be  reconciled;  for  the 
dignity  of  omniscience  is  as  great  as  omniscience  itself.  The  same  line  of 
reasoning  is  applied  to  the  doctrine  of  the  Trinity.  The  essay  closes  with  a 
reference  to  the  so-called  domestic  difficulties,  and  shows  that  a  being  may 
have  many  contradictory  aspects  and  yet  viewed  in  a  large  way  all  these 
aspects  may  be  true;  just  as  in  comparing  different  systems  of  geometry  built 
on  various  foundations  the  mathematician  finds  contradictory  facts,  yet  does 
not  doubt  the  truth  of  any  of  these  facts. 

Dr.  Keyser's  careful,  earnest  style  of  writing  makes  it  a  pleasure  to  read 
his  works,  and  any  one  who  has  the  "mathematical  spirit  which  is  simply  the 
spirit  of  logical  rectitude"  will  enjoy  this  unusual  essay. 

EMMA  K.  WHITON. 


480  THE   MONIST. 

THE  STUDY  OF  RELIGIONS.  By  Stanley  A.  Cook,  M.A.,  Ex-Fellow  and  Lec- 
turer in  the  Comparative  Study  of  Religions  and  in  Hebrew  and  Syriac, 
Gonville  and  Cains  College,  Cambridge.  London,  A.  and  C.  Black,  1914. 
Price,  7s.  6d.  net. 

Mr.  Cook  is  very  long-winded,  but  in  spite  of  dryness  and  abstractness  of 
style  he  has  written  a  valuable  book.  Much  thought  has  evidently  gone  into 
it,  and  its  defects  are  due  to  a  difficult  manner  of  exposition,  not  to  poverty  of 
ideas.  This  is  not  an  "Introduction"  of  the  type  of  Jevons's  book ;  it  gives  no 
data  for  the  beginner,  nor,  as  one  is  apt  to  expect  from  the  title,  does  it  deal 
chiefly  with  primitive  religion.  It  is  rather  the  comments  of  a  scholar — Mr. 
Cook  is  a  recognized  authority  in  his  field — on  the  aims  and  methods  of  his 
study.  He  has  a  great  deal  to  say,  and  much  that  is  extremely  good,  on  the 
evolution  of  religion — as  is  indicated  by  several  chapter  headings :  Survivals, 
The  Environment  and  Change,  Development  and  Continuity.  "The  doctrine 
of  survivals,"  Mr.  Cook  says,  "is  entirely  inadequate  when  it  forgets  that  we 
are  human  beings  and  do  not  accept  beliefs  merely  because  they  happen  to  lie 
within  our  reach.  The  doctrine  of  survivals,  is,  in  fact,  a  very  handy  and 
cheap  explanation  of  some  one  else's  beliefs  and  practices — hardly  of  our 
own !"  Survivals  are  not  simply  "left  behind,"  they  are  subconsciously  se- 
lected. Mr.  Cook  warns  very  wisely  against  arguing  from  the  part  to  the 
whole,  against  constructing  a  hypothetical  system  into  which  every  survival 
must  fit.  He  warns  also  against  confusing  the  evolution  of  beliefs  with  the 
evolution  of  environments,  in  judging  apparent  retrogressions.  On  the  crit- 
ical attitude,  on  the  acceptance  of  data,  Mr.  Cook  has  some  excellent  observa- 
tions, and  on  the  historical  versus  the  religious  importance  of  critical  revisions. 
He  holds  that  the  present  is  a  time  of  religious  unrest,  though  like  most  of 
us,  he  cannot  point  to  any  definite  theology  for  the  future.  His  conclusion  is 
as  follows :  "The  unbiased  student  of  religions  can  hardly  escape  the  conviction 
that  the  Supreme  Power,  whom  we  call  God,  while  enabling  man  to  work  out, 
within  limits,  his  own  career,  desires  the  furtherance  of  those  aims  and  ideals 
which  are  for  the  advance  of  mankind."  *? 


Just  as  we  are  going  to  press  we  receive  two  additional  notes  from  Dr. 
W.  B.  Smith  to  be  inserted  in  his  article  as  indicated  respectively  on  pages 
330  and  337. 

Page  330:  "For  which  Rutherford's  'nucleus  theory,'  apparently  required 
by  the  facts  in  the  scattering  of  'alpha  rays'  (of  helium  atoms)  in  passing 
through  laminae,  substitutes  a  positive  electric  core,  extremely  minute,  for 
gold  only  one  trillionth  of  an  inch  in  diameter,  in  volume  one  billionth  of  the 
atom  itself.  It  would  seem  that  the  negative  electron  is  nearly  six  thousand 
million  times  as  large  as  the  positive  hydrogen  core.  For  Thomson's  later 
views  see  Philos.  Mag.,  1913,  p.  892." 

Page  337:  "Why  do  the  members  fall  together  to  the  center  as  their 
energies  are  dissipated  in  electric  radiation?  Bohr  (Philos.  Mag.,  1913,  pp. 
1,  476,  854)  invokes  Planck's  'Quantum'-hypothesis  in  solving  this  riddle.'' 


VOL.  XXVII.          OCTOBER,  1917  NO.  4 


THE  MONIST 


WHAT  IS  A  DOGMA?* 

EDITORIAL  INTRODUCTION. 

The  primary  significance  of  a  dogma  is  not  its  speculative  con- 
tent, but  the  speculative  truth  of  dogma  is  expressed  in  terms  of 
action.  Such  is  the  proclamation  of  a  Roman  Catholic  thinker 
which  has  evoked  a  lively  discussion,  and  although  his  work  has 
been  placed  on  the  Index,  this  has  evidently  been  for  other  reasons 
than  any  connected  with  the  charge  of  heresy.  For  this  thesis  de- 
fines the  general  concept  of  dogma  in  the  expressions  of  the  well- 
known  philosophy  of  action  originated  by  Maurice  Blondel  and 
published  in  his  book  L'action  which  appeared  in  1893,  and  as  far 
as  we  know  his  book  was  not  placed  on  the  Index.  "Perhaps," 
writes  Father  E.  Bernard  Allo,  O.P.,  "the  thesis  sketched  by  Le  Roy 
is  not  so  different,  perhaps  the  divergencies  are  less  in  idea  than 
in  expression,  in  the  significat  itself  than  in  the  modus  significandi" 
(Foi  et  systeme,  Paris:  Bloud  et  Cie.,  180,  181),  and  this  is  con- 
firmed by  Le  Roy  himself  in  a  footnote  on  page  70  of  his  Dogme  et 
critique.  A.  Houtin  in  his  history  of  Catholic  modernism  mentions 
the  Rev.  A.  D.  Sertillanges  as  expressing  the  same  opinions  in  the 
referendum  on  Le  Roy's  article  on  dogma  as  Father  Allo,  and  so 
far  as  we  can  ascertain,  th:ir  writings  have  not  been  placed  on  the 
Index.  Further,  for  a  book  to  be  placed  on  the  Index  does  not  mean 
that  it  is  condemned,  but  the  authorities  intend  to  say  that  for 
some  reason  hie  ct  nnnc  the  book  is  not  to  be  generally  read. 

This  article  of  M.  Edouard  Le  Roy  entitled  "Qu'est-ce  qu'un 
dogme?"  has  even  been  looked  upon  with  favor  in  some  quarters 
by  representative  ecclesiastical  authorities;  and  being  of  great  im- 
portance, not  only  for  Roman  Catholicism,  but  also  for  Protestant- 

*  Translated  by  Lydia  G.  Robinson  from  the  sixth  French  edition  of  tht 
author's  book  Dogme  et  critique. 


482  THE  MONIST. 

ism,  yea  generally  for  all  religion,  we  take  pleasure  in  rendering  it 
accessible  to  English  readers. 

It  first  appeared  in  the  French  fortnightly  journal  La  Quinsaine 
of  April  16,  1905,  where  it  was  accompanied  by  an  editorial  note 
as  follows:  "Without  expressing  any  decision  on  our  own  part 
with  regard  to  the  opinions  of  M.  Le  Roy  it  seems  to  us  both  inter- 
esting and  useful  to  take  a  text  from  his  work  by  which  to  invite 
theologians  to  furnish  the  public  with  the  elucidation  he  asks  for. 
Hence  we  address  a  special  invitation  to  all  the  authorized  special- 
ists in  Catholic  theology,  to  the  professors  of  our  liberal  universities 
and  of  the  larger  seminaries,  to  religious  orders,  and  to  the  priests." 

The  invitation  was  eagerly  accepted,  and  seven  later  numbers 
of  La  Quinzaine  contained  communications  of  varying  importance 
on  the  subject.  But  these  formed  only  a  small  part  of  the  discussion 
raised  by  this  striking  article.  Its  publication  was  followed  by  a 
vast  array  of  controversial  writings  which  continued  with  increasing 
violence  throughout  an  entire  year.  Twenty  or  more  other  journals 
opened  their  pages  to  the  subject ;  not  only  such  distinctly  clerical 
journals  as  Etudes,  Revue  thomiste,  Revue  du  clerge  frangais,  La 
Croix,  etc.,  but  also  general  philosophical  reviews,  La  Pensee  con- 
temporaine,  Revue  de  philosophic,  and  such  liberal  journals  as  La 
Justice  sociale,  Le  Peuple  fran^ais,  and  La  Verite  frangaise.  And 
not  only  these  religious  and  critical  periodicals  devoted  their  pages 
to  the  subject  but  a  well-organized  opposition  to  the  offending 
article  rushed  into  print  through  the  daily  press. 

Still  the  question  which  the  author  put  to  the  clergy  in  def- 
erence to  them  as  being  officially  charged  with  the  instruction  of 
the  people  did  not  receive  a  satisfactory  answer.  Many  heaped 
M.  Le  Roy  with  malicious  calumnies,  and  many  honestly  misunder- 
stood him.  Many  too  misjudged  him  because  they  knew  of  the 
article  only  through  garbled  reports  or  hostile  criticisms.  He  there- 
fore considered  it  necessary  to  put  the  article  in  permanent  form, 
and  so  he  published  it  in  a  book  entitled  Dogme  et  critique  (in  the 
series  Etudes  de  philosophic  et  de  critique  religieuse  with  Librairie 
Bloud  et  Cie.)  together  with  his  published  replies  to  the  most  im- 
portant of  his  adversaries,  a  careful  bibliography  of  the  contro- 
versy and  a  more  detailed  development  of  the  most  significant  points 
of  his  thesis  in  fourteen  brief  additional  chapters. 

*       *      * 
Religion  is  a  practical  affair,  and  its  main  purpose  is  to  serve 


WHAT  IS  A  DOGMA?  483 

us  as  a  guide  through  life.  Religion  as  a  sentiment  is  practically 
universal  and  we  may  consider  it  to  be  innate.  It  is  a  panpathy  or 
all-feeling  which  produces  in  every  individual  a  deep-felt  longing 
to  be  at  one  with  the  whole  universe  of  which  each  is  a  part.1  As 
every  material  particle  is  an  embodiment  of  gravitation  in  propor- 
tion to  its  weight,  and  is  possessed  of  a  well-apportioned  pressure 
somehow  and  bent  some  whither,  so  the  souls  of  things  existent 
feel  themselves  parts  of  the  great  whole  in  which  they  live  and  move 
and  have  their  being. 

This  panpathy  in  its  historical  development  under  definite  con- 
ditions assumes  a  definite  form,  and  so  religion  leads  necessarily 
and  naturally  to  church  life  and  church  formation,  with  dogmas 
and  regulations  of  conduct. 

The  dogmas  of  the  church  are  collected  in  what  has  been  called 
the  symbolical  books  which  accordingly  contain  the  several  con- 
fessions of  faith.  They  are  called  symbolic  because  they  served  as 
symbols,  or  tokens  of  recognition  to  the  members  of  the  church. 
The  man  who  could  recite  the  symbol  was  welcome  in  the  congrega- 
tion as  a  brother  who  cherished  the  same  faith,  having  found  the 
same  solutions  of  the  world  problem  as  the  whole  church  and  hav- 
ing accepted  the  same  formulation  of  it. 

The  dogma  is  a  symbol,  but  it  is  more  than  a  symbol ;  it  is  an 
appropriate  symbol.  It  is  a  statement  satisfactory  to  the  whole 
congregation  and  in  so  far  as  it  is  satisfactory  to  the  whole  con- 
gregation it  has  become  to  them  a  truth. 

Dogmas  are  truths.  Being  religious  truths  they  are  holy  truths, 
and  since  they  are  taken  seriously,  they  have  often  become  the 
cause  of  much  controversy  and  have  led  to  quarrels  and  bloodshed, 
to  persecution  and  warfare,  to  the  establishment  of  the  inquisition 
and  denunciation  of  heretics.  We  now  learn  that  the  intellectual 
feature  of  the  dogma  is  derived  from  the  main  and  essential  feature, 
its  practical  value.  This  is  an  enormous  gain,  for  it  introduces  into 
the  nature  of  dogma  a  condemnation  of  all  intolerance  and  estab- 
lishes an  unlimited  freedom  of  interpretation  without,  however, 
detracting  a  hair's  breadth  from  the  practical  significance  of  the 
dogma.  Not  one  jot  or  one  tittle  shall  pass  from  it,  but  a  thinker 
is  allowed  to  construct  its  meaning  as  best  he  can,  provided  he 
recognizes  and  holds  on  to  its  practical  application. 

God  is  our  father;  he  is  called  upon  in  prayer  as  a  personality 

1  For  a  more  complete  definition  of  religion   in   its   several   phases   see 
Carus's  Dawn  of  a  New  Era,  pp.  96-97. 


484  THE  MONIST. 

— not  a  human  personality,  but  a  divine  personality.  The  inter- 
pretation of  personality  is  a  problem  by  itself,  but  the  significance 
of  the  dogma  "God  is  a  person"  means  that  we  should  adjust  our 
relation  to  God  in  such  a  way  as  to  make  it  a  personal  relation, 
and  this  practical  application  constitutes  the  primary  and  underived 
significance  of  the  dogma. 

This  view  is  not  a  loose  way  of  treating  the  dogma ;  for  the 
freedom  of  interpretation  gives  much  liberty  of  speculation,  but 
not  an  unlimited  license.  It  is  restricted  and  allows  the  dogma  to 
stand  and  remain  unalterable  as  the  only  possible,  the  only  allow- 
able, expression  of  a  truth.  Though  the  dogma  is  not  absolute  it 
is  definite,  and  any  other  formulation  of  it  would  be  wrong  and 
must  be  rejected.  Thus  the  view  of  dogma  here  represented  by 
M.  Le  Roy  remains  as  uncompromising  as  ever  and  would  not 
allow  any  dillydallying  for  the  benefit  of  speculative  minds. 

It  will  be  sufficient  to  characterize  the  author's  effort  and  the 
misunderstandings  created  in  the  broad  problem  in  his  own  words. 
They  will  show  first  the  sincerity  of  his  undertaking  and  explain 
the  situation  of  his  own  mind,  and  secondly  they  will  describe  his 
critics  and  their  inability  to  grasp  M.  Le  Roy's  point  of  view.  A 
faithful  Catholic's  understanding  of  the  nature  of  dogma  is  char- 
acterized by  the  article  itself  and  for  a  summary  of  this  phase  of 
religious  thought  it  is  fully  sufficient. 

This  is  what  our  author  says  in  speaking  of  himself  (Dogme 
et  critique,  pp.  v-x)  : 

"On  April  16,  1905,  I  published  in  the  Quinsaine  an  article 
entitled  'What  is  a  Dogma?'  in  which,  speaking  as  a  philosopher 
who  desires  to  think  his  religion,  I  addressed  various  questions  to 
theologians  and  apologists. 

"Why  did  I  use  the  form  of  interrogation  instead  of  a  direct 
exposition?  In  deference  to  those  who  have  official  charge  of  in- 
struction. It  seemed  to  me  desirable  that  the  reply  should  come 
from  them.  In  this  way  I  hoped  to  manifest  my  intention  to  act 
always  in  conformity  with  the  hierarchical  principle  divinely  estab- 
lished in  the  church.  Although  I  have  scarcely  been  able  to  con- 
gratulate myself  on  the  reserve  and  courtesy  I  thus  showed,  since 
some  have  been  pleased  to  see  in  it  only  a  caution  lacking  in  cour- 
age and  candor,  still  I  retain  to-day  the  same  way  of  looking  at 
things.  But  be  assured  this  does  not  in  the  least  mean  that  I 
experience  the  slightest  difficulty  for  my  own  part  in  reconciling 


WHAT  IS  A  DOGMA?  485 

faith  and  reason,  nor  that  I  hesitate  or  doubt  the  least  bit  in  the 
world  with  regard  to  my  duty  as  a  Catholic. 

"My  aim  was,  briefly,  to  expose  certain  facts  which  I  had  had 
the  opportunity  to  observe  around  me,  and  also  to  report  an  ex- 
perience I  had  had  in  my  relations  with  the  unbelieving  intellectual 
world.  It  was  for  the  theologians,  I  thought,  to  declare  them- 
selves after  discussing  the  plan  which  I  submitted  to  them.  As  for 
myself,  I  was  only  a  witness  testifying  to  what  he  had  seen  and 
come  in  touch  with,  a  Christian  soul  relating  some  of  the  steps  it 
had  taken. 

"This  attitude  has  been  misunderstood.  It  has  been  regarded 
as  craftiness  or  malice,  as  a  challenge  or  an  irony.  Some  one  spoke 
with  reference  to  it  of  a  question  'irreverently  and  even  imper- 
tinently stated.'2  Was  not  'importunately'  meant  instead,  without 
daring  to  say  it,  or  admitting  it?  For,  I  beg  to  inquire,  how  may 
one  set  about  being  more  deferential  than  I  have  been?  Unless  the 
only  deference  that  is  acceptable  and  sufficient  is  the  deference  of 
an  indifferent  or  heedless  silence.  Is  it  true  that  the  question  asked 
was  indiscreet?  Certain  papers  hastened  to  make  the  claim,  and 
the  Si&cle  for  instance  was  much  diverted  at  the  idea  of  Catholics 
not  being  able  to  agree  on  defining  a  dogma.  These  are  certainly 
not  my  own  sentiments.  In  asking  an  explanation  I  never  intended 
to  be,  nor  do  I  think  I  was,  a  trouble  maker,  disturbing  slumber 
or  ruffling  tranquillity.  But  words  like  those  I  have  mentioned 
tend  to  justify  this  ill-natured  hypothesis,  and  therefore  it  is  they 
which  in  the  final  analysis  I  find  lacking  in  courtesy. 

"For  my  part,  on  re-reading  what  I  have  written  and  feeling 
ready  to  write  again,  I  declare  with  M.  Fonsegrive:3  'Have  we  been 
wrong  in  saying  these  things  out  loud  and,  being  Catholics,  in 
having  enough  confidence  in  our  religion,  in  the  power  of  truth,  to 
dare  speak  frankly,  clearly,  even  vigorously?  Would  we  have 
shown  more  regard  for  our  beliefs  if  we  had  spoken  timidly  and 
feebly  as  one  speaks  at  the  bedside  of  the  dying?'  One  must  indeed 
stand  up  for  oneself.  We  are  neither  dissembling  Protestants  nor 
disreputable  rationalists.  We  are  only  searching  always  for  the 
greatest  religion,  without  concessions  or  haggling.  We  do  not  wish 
in  the  least  to  be  either  rebels  or  even  eccentric  persons.  But  our 
faith  is  firm  enough  for  us  not  to  fear  to  look  the  facts  in  the  face 
and  to  speak  out  clearly  what  they  show  us ;  and  we  attach  enough 

2  La  vtrite  franqaise,  Dec.  20,  1905.         8  Quinsainf,  Jan.  1,  1906,  p.  30. 


486  THE  MONIST. 

value  to  the  divine  word  to  wish  to  think  with  all  the  strength  of 
our  soul,  assured  in  advance  that  there  we  will  find  life  and  light 
without  other  limitations  than  our  own.  Moreover  we  feel  that 
we  are  enough  protected  by  the  living  supremacy  of  the  church  to 
preserve  the  most  complete  internal  peace  throughout  our  most 
venturesome  inquiries.  We  are,  in  fine,  sure  enough  of  our  obedi- 
ence to  legitimate  authority  to  have  no  fear  in  running  the  com- 
mendable risks  which  the  experience  of  life  always  entails.  But 
the  obedience  we  intend  to  render  is  not  a  simple  obedience  of 
formulas  and  motions,  it  is  a  profound  obedience  which  lays  hold 
of  our  whole  being,  heart,  will  and  intelligence — in  short,  an  obedi- 
ence of  reasonable  men  and  free  agents,  not  of  slaves  or  mutes. 

"Nevertheless,  as  soon  as  the  article  'What  is  a  Dogma?'  ap- 
peared a  vast  array  of  controversial  writings  began  which  continued 
with  increasing  violence  during  one  whole  year.  Not  only  did  the 
reviews  take  part,  as  was  their  natural  business,  but  the  daily  papers 
as  well.  For  after  having  reproached  me  for  opening  a  discussion 
on  such  a  subject  before  a  public  which  though  educated  was  not 
professionally  qualified,  they  had  nothing  more  urgent  to  do  than  to 
force  the  discussion  before  the  eyes  of  a  crowd  which  this  time 
had  neither  proficiency  nor  culture.  The  organization  of  the  ex- 
posure was  perfect  and  the  matter  was  abundantly  exploited  by 
those  who  make  orthodoxy  a  monopoly  or  a  standard  and  who  are 
always  to  be  found  upon  the  heels  of  any  one  who  takes  the  liberty 
of  thinking  for  himself. 

"To  polemics  conducted  in  this  way  I  shall  make  no  reply. 
Their  authors,  in  spite  of  the  pretensions  they  parade,  are  repre- 
sentative of  nothing  in  the  church,  and  as,  on  the  other  hand,  they 
do  not  discuss  but  condemn  and  anathematize,  substituting  injury, 
slander  or  denunciations  for  arguments,  they  are  representative  of 
nothing  from  the  intellectual  point  of  view.  What  separates  us 
from  them  is  a  question  of  morality  much  more  than  a  question  of 
critique. 

"Fortunately  other  questioners  have  made  their  voices  heard, 
loyal  and  disinterested  questioners  of  broad  minds  and  upright 
hearts,  striving  to  understand  and  seeking  nothing  but  the  kingdom 
of  God,  the  welfare  of  souls,  the  light  of  truth.  The  present  volume 
is  dedicated  to  them,  to  them  and  all  those,  whether  known  or  un- 
known, who  are  like  them.  Is  there  any  need  of  justifying  oneself 
otherwise  than  by  the  words  of  Fenelon,  which  he  might  have  taken 


WHAT  IS  A  DOGMA?  487 

for  a  motto:  'Every  Christian,  far  from  entering  controversies, 
ought  instead  to  explain  his  position  more  and  more  to  try  to  satisfy 
those  who  have  had  trouble  with  the  first  explanation.'  If  this 
motive  is  not  sufficient  I  may  add  that  I  cannot  remain  indifferent 
in  the  face  of  the  opinions  that  have  been  attributed  to  me.  Too 
many  people  have  become  acquainted  with  my  article  only  through 
incomplete  analyses,  through  prejudiced  reports  or  through  refu- 
tations which  may  well  confuse  them ;  it  is  important  that  I  should 
publish  an  authentic  text  with  comments  made  necessary  by  the 
publicity  the  controversy  has  attained. 

"For  the  rest,  I  still  retain  the  same  attitude  I  had  at  the  be- 
ginning. I  wish  to  put  a  question,  nothing  more.  The  accompany- 
ing comments  and  reflections  are  only  to  elucidate  the  meaning  and 
the  scope ;  to  show  also  that  it  has  not  in  the  least  been  adequately 
answered ;  finally  to  furnish  a  definite  theme  for  discussion  and 
investigation.  Who  would  dare  to  find  occasion  in  this  to  accuse 
me  of  heresy? 

"And  now  I  have  finished  my  task  on  this  point ;  I  have  said 
what  I  had  to  say.  The  question  has  been  asked,  and  nothing  could 
prevent  it  from  being  asked.  Henceforth  the  ideas  will  make  their 
way  of  themselves  and  nothing  will  stop  them.  Let  the  future 
answer.  Perhaps  we  shall  soon  see  what  has  often  happened  be- 
fore, that  what  was  once  regarded  as  bold  and  disgraceful  will  end 
by  being  universally  accepted  as  a  very  simple  and  commonplace 
matter." 


According  to  Le  Roy  the  intellectual  feature  of  the  dogma  is 
not  denied  nor  abrogated.  On  the  contrary  it  remains  in  force  and 
takes  about  the  same  place  in  religion  as  the  laws  of  nature  in  nat- 
ural science  which  formulate  uniformities  of  facts  but  are  not  the 
actual  phenomena  as  experienced.  They  both  have  their  positive 
significance.  It  seems  to  we  that  in  this  way  this  conception  of  the 
dogma  is  helpful  to  educated  people. 

It  is  not  necessary  to  make  the  interpretation  of  religion  be- 
come a  product  of  the  Aristotelian  philosophy.  It  would  change 
theology  into  an  ancilla  of  medieval  thinking  and  deprive  it  of  the 
liberty  to  adopt  the  scientific  spirit. 

While  Le  Roy's  theory  resembles  pragmatism,  one  cannot 
characterize  it  as  purely  pragmatic,  and  we  should  consider  that  the 
papal  decree,  Lamcntabilc  sane  exitu  of  July  3,  1907,  condemns  the 


THE   MONIST. 

views  of  those  who  claim  that  dogma  is  exclusively  a  regula  prae- 
ceptiva  actionis,  and  that  it  is  not  a  regula  fidei.  Nor  is  Le  Roy  an 
agnostic.  He  positively  affirms  that  we  can  know  God  in  relation 
to  ourselves,  and  also  that  we  can  know  him  as  he  is  in  se.  The 
essence  of  the  dogma  according  to  him  is  not  exhausted  in  its 
moral  significance,  but  includes  also  the  cnunciatio  speculative 

The  distinction  between  the  actionists  and  analogists  is  more 
one  of  words  than  of  actual  meaning,  for  both  agree  in  presenting 
the  truth  concerning  God  in  terms  of  intellectual  conception  and  in 
terms  of  action,  and  thus  both  sides  insist  on  a  real  cognition  of 
God,  each  in  his  own  terms.  The  whole  controversy  turns  on  this 
question,  "Is  practical  truth  contained  in  the  speculative,  or  the 
spculative  in  the  practical?"  while  we  might  say  they  are  both  two 
phases  of  the  same.  p.  c. 


THIS  title,  "What  is  a  Dogma?"  is  only  a  simple  ques- 
tion and  by  no  means  does  it  promise  an  answer.  It 
is  a  question  from  the  philosopher  to  the  theologian  calling 
for  an  answer  from  the  theologian  to  the  philosopher. 

It  would  indeed  be  vain  to  pretend  to  give  here  a  com- 
plete and  definite  answer  to  this  complex  question.  Such 
problems  cannot  be  solved  in  a  few  pages.  Therefore  the 
reader  must  not  look  for  a  settled  doctrine  in  the  short 
article  which  is  to  follow,  nor  even  for  categorical  theses 
on  any  point.  If  he  sometimes  find  that  I  speak  in  too 
affirmative  a  tone  let  him  be  kind  enough  to  admit  that  I 
do  so  only  for  the  sake  of  greater  clearness  in  my  questions. 
In  fact  I  wish  to  confine  myself  to  simple  suggestions 
which  I  present  merely  as  rough  drafts  of  solutions  offered 
for  the  criticism  of  those  who  have  authority  to  judge  of 
the  subject.  And  moreover  I  can  justify  this  attitude  of 
mine  by  an  imperative  reason,  namely  that  I  am  not  a  theo- 
logian and  do  not  like  to  decide  matters  in  which  I  am  not 
proficient. 

Perhaps  some  one  will  ask,  why  then  do  I  take  the 


WHAT  IS  A  DOGMA?  489 

trouble  to  treat  a  subject  of  which  I  admit  I  have  no  par- 
ticular knowledge  ?  Here  is  my  reason.  In  our  day  every 
layman  is  called  upon  to  fulfil  the  duty  of  apostleship  in  the 
incredulous  world  in  which  he  lives.  He  alone  can  serve 
efficiently  as  the  vehicle  and  intermediary  of  the  Christian 
message  to  those  who  would  not  trust  the  priests.  There- 
fore it  is  inevitable  that  some  problems  of  apologetics 
should  be  laid  before  him,  problems  whose  solution  is  an 
absolute  necessity  for  him  if  he  does  not  wish  to  fail  in  the 
task  which  the  force  of  circumstances  has  laid  upon  him 
without  possibility  of  escape,  if  he  wishes  to  be  always 
ready,  following  the  counsel  of  the  Apostle,  to  satisfy  those 
who  ask  him  the  reason  for  his  faith.  It  is  only  natural 
therefore  that  I  desire  to  be  informed;  and  if  I  formulate 
my  question  publicly  it  is  because  I  am  not  the  only  one  in 
this  situation,  and  because  there  is  a  general  interest  that 
the  answer  shall  also  be  a  public  one. 

Besides  I  have  another  motive  for  acting  as  I  am.  If 
I  freely  acknowledge  my  incompetence  in  a  matter  which  is 
properly  theological,  yet  on  the  other  hand  I  consider  that 
I  am  well  situated  to  appreciate  correctly  the  state  of  mind 
in  contemporary  philosophers  that  is  opposed  to  the  under- 
standing of  Christian  truth.  And  it  is  to  this  that  I  bear 
witness  in  saying  frankly,  even  brutally  (if  I  must  in  order 
to  be  fully  understood),  what  I  know,  what  I  have  ob- 
served, what  perhaps  are  not  always  sufficiently  compre- 
hended, namely  the  exact  reasons  why  unbelieving  philos- 
ophers of  to-day  repulse  the  truth  that  is  brought  to  them, 
and  the  legitimate  causes  (agreeing  in  this  with  the  Chris- 
tian philosophers  themselves)  why  they  are  not  satisfied 
with  the  explanations  that  are  furnished  them. 

My  ambition  goes  no  farther  than  to  point  out  certain 
opinions,  perhaps  to  suggest  certain  reflections,  especially 
to  particularize  the  statement  of  certain  problems.  If  the 
present  work  bring  a  useful  contribution  to  the  studies  of 


4QO  THE  MONIST. 

religious  philosophy,  if  it  furnish  documents  and  materials 
which  others  can  turn  to  account,  I  shall  have  attained  my 
end.  It  is  not  a  question  of  upholding  a  system  nor  of 
aligning  arguments  for  or  against  this  or  that  school,  but 
only  of  elucidating  certain  fundamental  ideas  whose  con- 
sideration is  imposed  upon  every  system  and  upon  every 
school.  An  effort  toward  light  in  the  bosom  of  Catholic 
truth,  faithfully  accepted  in  its  completeness  and  rigor — 
this  is  what  I  submit  to  the  decision  of  those  who  have  been 
charged  with  the  duty  of  defining  and  interpreting  it. 

What  I  desire  above  all,  I  repeat,  is  to  make  better 
known  the  state  of  mind  of  those  contemporaries  who  think, 
the  nature  of  the  questions  they  ask  themselves,  the  ob- 
stacles that  hinder  them  and  the  difficulties  that  perplex 
them.  It  cannot  be  denied  that  the  classical  replies  no 
longer  satisfy  them;  there  is  no  use  in  disputing  over  so 
obvious  a  fact.  The  experience  of  cultivated  non-Christian 
circles  (I  might  even  say  a  personal  experience)  has  dem- 
onstrated to  me  that  the  proofs  brought  forward  as  tradi- 
tional have  no  effect  on  intellects  accustomed  to  the  dis- 
cipline of  contemporary  science  and  philosophy.  Now  why 
this  new  impotence  of  old  methods  which  have  sufficed  so 
long?  The  reason  appears  to  me  to  be,  at  least  in  great 
measure,  that  the  old  apologetics  assumes  the  greater  part 
of  the  problems  to  be  solved  in  advance  which  the  moderns, 
on  the  other  hand,  judge  to  be  essential  and  primordial. 
The  real  difficulty  for  the  moderns  comes  in  altogether 
before  the  arguments  begin  by  which  the  theologians  flatter 
themselves  they  can  convince  them ;  it  lies  in  the  postulates 
taken  for  granted  and  in  the  very  manner  in  which  the  in- 
vestigation is  approached. 

It  will  be  well  to  see  how  the  questions  ought  to  be  put 
to-day ;  this  should  be  the  first  result  to  be  obtained.  It  is 
the  chief  result,  for  without  it  we  would  never  arrive  at 
anything  serious.  Thus  is  imposed  the  preliminary  task 


WHAT  is  A  DOGMA?  491 

of  coming  in  contact  with  the  minds  whom  one  wishes  to 
address  and  whom  one  claims  to  understand.  It  is  neces- 
sary that  the  various  chapters  of  the  apologetic  should  be 
taken  up  successively  from  this  point  of  view  in  order  to  be 
brought  to  general  attention;  and  in  examining  here  the 
idea  of  dogma1  I  only  give  a  first  example  of  the  kind  of 
work  that  I  think  ought  to  be  generally  undertaken. 

Let  no  one  think  such  a  task  profitless  or  superfluous. 
On  the  contrary,  nothing  is  of  greater  urgency  to-day  nor 
of  more  pressing  necessity.  It  is  strange  and  lamentable 
how  little  we  on  the  Catholic  side  know  or  how  greatly  we 
fail  to  appreciate  the  state  of  mind  of  the  opponents  to 
whom  we  try  to  speak.2  Nor  are  we  listened  to  or  under- 
stood. What  we  say  has  no  response  and  carries  no  weight. 
We  exert  ourselves  in  silence  and  in  a  void  without  even 
giving  rise  to  any  criticism  or  refutation.  In  short  we 
only  reach  those  who  do  not  need  to  be  reached — I  mean 
those  who  are  convinced  beforehand  or  whose  difficulties 
are  not  of  a  theoretical  kind.  We  must  not  deceive  our- 
selves. Catholic  thought  at  the  present  day  is  without 
notable  influence  on  the  various  intellectual  movements 
which  are  developing  around  us.  It  sometimes  follows 
them  at  a  distance  and  after  having  resisted  them  for  a 
long  time ;  but  nowhere  does  it  appear  capable  of  directing 
them,  much  less  of  promoting  them.  There  is  nothing 
more  sad  than  to  confess  so  many  efforts  expended  without 
result  on  the  one  hand,  and  on  the  other  hand  so  many 
sincere  questions  asked  which  remain  unanswered. 

Doubtless  one  might  say,  and  indeed  some  have  said, 
that  there  is  no  need  of  taking  into  account  modern  de- 
mands because  they  proceed  from  a  perverted  and  mis- 
guided judgment.  Wretched  subterfuge!  What  contem- 
poraneous thought  is  asking  for  beyond  what  it  receives 

1 1  will  say  once  for  all  that  by  "dogma"  I  mean  especially  the  "dogmatic 
proposition,"  the  "dogmatic  formula,"  not  at  all  the  reality  which  underlies  it. 
2  I  would  say  the  same,  moreover,  of  our  opponents  with  respect  to  us. 


492  THE   MONIST. 

is  perfectly  legitimate,  and  there  is  no  justification  in  pre- 
tending to  refuse  to  grant  it.  Men  of  to-day  are  within 
their  rights  in  not  consenting  to  be  held  down  to  the  point 
of  view  of  the  thirteenth  century.  It  would  indeed  be 
strange  if  any  one  should  ask  for  a  proof  to  support  a 
truth  of  this  kind.3  After  all,  is  it  not  the  very  mission  and 
the  raison  d'etre  of  apologetics  to  address  itself  to  the 
disordered,  if  such  there  be?  It  must  take  people  as  they 
are  and  not  require  of  them  that  they  first  come  of  their 
own  accord  where  it  may  prefer.  Once  again,  it  would  be 
strange  if  one  had  no  right  to  make  a  cure  except  with 
certain  remedies. 

Hence  there  may  be  some  interest  and  some  profit  in 
the  testimony  of  those  whose  situation  has  put  them  in  a 
position  to  know  the  modern  mind,  its  needs  and  its  re- 
quirements. These  may  try  to  tell  how  they  have  come 
to  think  what  they  believe,  how  they  have  succeeded  in 
practically  overcoming,  and  of  their  own  accord,  the  diffi- 
culties that  they  have  met  like  the  others.  I  do  not  say 
that  we  must  accept  the  conclusions  of  their  experiences 
uncritically;  but  after  all,  these  experiences  offer  the  ad- 
vantage of  furnishing  living  documents,  not  dead  opinions, 
and  that  is  something.  I  here  make  no  further  claim. 

One  more  word  before  I  begin.  Perhaps  the  reader 
will  be  surprised  to  find  so  long  a  preamble  introducing  so 
short  an  article.  The  reason  is  first  of  all  that  I  wished 
to  write  a  sort  of  general  preface  for  other  similar  articles 
intended  to  follow  this  one,  and  also  because  I  wished  in 
this  way  to  forestall  any  possible  misunderstanding.  What- 
ever opinion  may  be  held  on  the  ideas  which  I  shall  put 
forth,  it  must  not  happen  that  any  one  will  try  to  answer 
me  by  charging  me  with  heresy.  I  affirm  nothing  in  this 
work  except  facts  easily  verifiable  by  everybody.  As  to 

3  The  object  of  faith  always  remains  the  same  but  not  the  manner  of 
thinking  it  or  of  complying  with  it. 


WHAT  IS  A  DOGMA?  493 

the  rest,  that  is  to  say  the  sketches  of  theories,  whatever 
the  form  of  the  language  which  I  have  adopted  in  order  to 
make  myself  clear,  I  give  them  expressly  as  simple  inter- 
rogations addressed  to  whomsoever  they  may  concern.  In 
a  word,  I  do  nothing  but  state  some  problems ;  it  is  for  the 
apologists  and  theologians  to  solve  them. 

*       *       * 

We  no  longer  live  in  the  day  of  partial  heresies.  For- 
merly a  purely  logical  and  dialectic  argumentation  might 
suffice  because  certain  common  principles  \vere  always  ad- 
mitted on  both  sides.  But  the  case  is  no  longer  the  same 
to-day,  when  these  principles  go  by  default,  when  the  fun- 
damental difficulty  is  to  establish  a  point  of  departure  upon 
which  both  sides  may  agree.  To-day  denial  does  not  attack 
one  dogma  any  more  than  another.  It  consists  above  all 
in  a  preliminary  and  total  demurrer.  The  question  is  not 
whether  a  proposition  is  a  dogma  or  not;  it  is  the  very 
idea  of  dogma  which  is  repugnant,  which  gives  offense. 
Why  is  that? 

When  we  examine  the  ordinary  motives  of  this  repug- 
nance we  find  four  principal  ones  which  I  shall  briefly 
enumerate,  endeavoring  to  present  them  in  all  their  force : 

i.  A  dogma  is  a  statement  presented  as  being  neither 
proved  nor  provable.4  Those  who  declare  it  to  be  true 
declare  at  the  same  time  that  it  is  impossible  ever  to  arrive 
at  the  point  of  grasping  the  intimate  reasons  of  its  truth. 
Now  modern  thought,  faithful  to  the  precept  of  Leibniz, 
endeavors  more  and  more  to  demonstrate  the  old  so-called 
axioms.  At  least  it  wishes  to  justify  them  with  Kant  by 
a  critical  analysis  which  shows  them  to  be  necessary  con- 
ditions of  consciousness  implied  a  priori  in  every  act  of 
reason.  It  is  distrustful  of  those  evidences,  pretending  to 
be  direct,  which  were  so  numerous  in  former  times.  Often 
enough  it  discovers  in  them  simple  postulates  adopted  for 

4  I  mean  here  to  speak  of  intrinsic  proof. 


494  THE  MONIST. 

an  end  of  practical  utility  more  or  less  unconsciously  per- 
ceived.6 In  short  everywhere  and  always  it  calls  for  long 
and  detailed  discussions  before  believing  itself  authorized 
to  draw  conclusions.  And  it  is  not  just  any  more  or  less 
roundabout  proofs  that  it  thus  demands,  but  direct  specific 
proofs.  It  does  not  like  too  general  arguments  which  look 
upon  vast  assemblages  as  a  whole  and  proceed  by  whole- 
sale demonstrations,  because  it  has  had  experience  too 
many  times  with  the  illusions,  mistakes  and  oversights 
which  they  ordinarily  conceal.  Nor  does  it  like  any  better 
external,  extrinsic  arguments  which  end  in  proofs  of  a 
negative  character,  in  reductiones  ad  absurdwn  founded 
on  judgments  of  contradiction  or  impossibility,  because  it 
has  also  had  experience6  too  many  times  with  their  im- 
prudent and  hazardous  character  to  declare  either  impos- 
sible or  contradictory  a  thing  which  may  appear  so  to  us 
only  from  habit.  Therefore  it  seems  that  in  order  to  re- 
main faithful  to  the  tendencies  which  have  assured  its  suc- 
cess in  all  domains  modern  thought  can  do  no  less  than 
condemn  absolutely  the  very  idea  of  a  strictly  dogmatic 
proposition.  In  what  system  acceptable  to  reason  could 
such  a  proposition  find  room  without  violence?  Is  not  the 
first  principle  of  scientific  method  incontestably,  according 
to  Descartes,  that  it  must  hold  as  true  only  what  clearly 
appears  to  be  true  ?  What  justification  would  there  be  for 
making  an  exception  of  just  those  propositions  which  pass 
as  the  most  important,  the  most  profound  and  the  simplest 
of  all  ?  When  affirmations  are  of  the  greatest  consequence 
and  refer  to  the  most  difficult  and  recondite  subjects  it  is 
certainly  not  fitting  to  show  oneself  less  attentive  to  the 
exactness  of  "the  rules  which  constitute  our  protection 
against  error.  On  the  contrary  it  is  just  then  that  it  would 

5  Compare  the  Philosophic  nouvclle  edition  of  Bergson's  works. 

6  Especially  in  the  sciences. 


WHAT  IS  A  DOGMA?  495 

be  legitimate  to  be  even  more  exacting,  more  scrupulous, 
more  particular  than  usual. 

2.  It  will  doubtless  be  said  that  dogmatic  propositions 
are  never  affirmed  without  proof.  In  fact  an  indirect  dem- 
onstration has  been  attempted  over  and  over  again.  One 
certain  apologetic  which  is  regarded  as  purely  traditional7 
claims  to  prove  that  these  propositions  are  true,  although 
it  realizes  that  it  is  incapable  of  bringing  fully  to  light  the 
how  and  the  why  of  their  truth.  There  is  some  analogy, 
it  seems,  between  such  a  proceeding  and  that  of  the  mathe- 
matician who  limits  himself  at  times  to  the  theorems  of 
simple  existence,  or  that  of  the  physicist  who  often  accepts 
facts  of  which  he  cannot  give  any  theoretical  explanation, 
or  yet  again  of  the  historian  who  always  receives  knowl- 
edge only  by  the  path  of  testimony.  Thus  would  end  the 
first  objection. 

Yes,  here  we  would  have  a  very  simple  solution,  but 
there  is  one  misfortune,  namely  that  the  analogy  pointed 
out  proves  upon  reflection  to  be  absolutely  inaccurate.  The 
difficulty  we  wish  to  avoid  reappears  in  toto  when  we  try  to 
justify  postulates  on  which  the  alleged  indirect  demonstra- 
tion rests.  When  a  mathematician  is  satisfied  with  estab- 
lishing a  theorem  of  simple  existence,,  I  mean  a  theorem 
affirming  the  existence  of  a  solution  inaccessible  in  itself, 
he  reasons  no  less  rigorously  than  in  other  branches  of  his 
science.  Now  here  we  have  nothing  like  that.  It  would 
be  necessary  to  prove  directly  that  God  exists,  that  he  has 
spoken,  that  he  has  said  this  and  that,  that  we  possess  his 
authentic  teachings  to-day.  This  amounts  to  the  same 
thing  as  saying  that  the  problem  of  God,  the  problem  of 
revelation,  of  the  inspiration  of  the  Bible  and  of  the  author- 
ity of  the  church,  must  be  solved  by  a  direct  analysis.  Now 
these  are  questions  of  the  same  kind  as  the  strictly  dogmatic 

7  This  method  of  extrinsic  demonstration  is  regarded,  as  traditional.  Here 
is  a  historical  point  on  which  much  might  be  said,  but  such  a  discussion  is 
foreign  to  my  subject. 


496  THE   MONIST. 

questions,  questions  with  reference  to  which  it  is  indeed  im- 
possible to  produce  arguments  comparable  to  those  of  the 
mathematician.  Likewise  when  a  physicist  accepts  a  fact 
to  which  he  can  give  no  theoretical  explanation  this  fact 
corresponds,  at  least  for  him,  to  certain  definite  experi- 
ences, to  certain  manipulations  that  can  be  practically  car- 
ried out,  in  short  to  a  group  of  motions  of  which  he  has 
direct  knowledge.  What  similarity  is  there  here?  And 
finally  even  the  historian  does  not  consent  to  receive  truth 
by  testimony  except  because  he  is  dealing  with  phenomena 
of  the  same  kind  as  those  of  which  he  has  a  direct  view  by 
some  other  means.  He  still  regards  his  science  as  always 
conjectural  and  uncertain  so  long  as  it  treats  of  somewhat 
profound  causes  or  of  events  that  are  more  or  less  remote. 
How  much  more  ought  one  draw  the  same  conclusion  in 
the  case  of  dogmas  which  reflect  only  facts  that  are  mys- 
terious, strange  and  disconcerting,  and  to  which  no  anal- 
ogy in  our  human  experience  corresponds!  It  has  been 
well  done.  The  alleged  indirect  proof  has  inevitably  for 
its  basis  an  appeal  to  the  transcendence  of  pure  authority. 
It  claims8  to  introduce  the  truth  into  us  fundamentally 
from  the  outside  in  the  fashion  of  a  ready-made  "thing" 
which  might  enter  into  us  forcibly.  Thus  any  dogma  what- 
ever seems  like  a  subservience,  like  a  limit  to  the  rights  of 
thought,  like  a  menace  of  intellectual  tyranny,  like  a  shackle 
and  a  restriction  imposed  from  without  upon  the  liberty  of 
investigation — all  of  which  is  radically  opposed  to  the  very 
life  of  the  spirit,  to  its  need  of  autonomy  and  sincerity,  to 
its  generative  and  fundamental  principle  which  is  the  prin- 
ciple of  immanence. 

Let  us  insist  a  little  upon  this  last  point,  for  the  prin- 
ciple of  immanence  has  not  always  been  rightly  understood. 
Too  often  it  has  been  made  out  a  monster,  whereas  nothing 

8  Or  at  least  appears  to  claim,  which  is  the  form  under  which  it  is  too 
often  presented. 


WHAT  IS  A  DOGMA?  497 

is  more  simple  nor  on  the  whole  more  clear.  We  may  say 
that  to  have  gained  a  clear  consciousness  of  it  is  the  essen- 
tial result  of  modern  philosophy.  Who  refuses  to  admit  it 
is  from  that  time  forth  no  longer  counted  among  the  num- 
ber of  philosophers,  who  does  not  succeed  in  understanding 
it  indicates  thereby  that  he  has  not  the  philosophic  sense. 
And  this  is  what  constitutes  the  principle  of  immanence. 
Reality  is  not  made  of  separate  pieces  put  in  juxtaposi- 
tion, but  everything  is  within  everything  else;  in  the 
smallest  detail  of  nature  or  of  science  analysis  recognizes 
all  of  science  and  all  of  nature.  Each  of  our  states  and 
of  our  actions  comprises  our  entire  soul  and  the  total- 
ity of  its  powers.  Thought,  in  a  word,  is  wholly  in- 
cluded in  each  of  its  moments  or  degrees.  In  short,  there 
is  never  for  us  a  purely  external  fact  like  some  sort  of 
raw  material.  Such  a  fact  indeed  would  remain  abso- 
lutely unassimilable,  unthinkable;  it  would  be  a  nonentity 
to  us,  for  where  could  we  take  hold  of  it?  Experience  it- 
self is  not  in  the  least  an  acquisition  of  "things"  which 
previously  were  entirely  unknown  to  us.  No,  it  is  much 
more  a  transition  from  the  implicit  to  the  explicit,  a  pro- 
found movement  revealing  to  us  the  latent  requirements 
and  actual  abundance  in  the  system  of  knowledge  already 
explained,  an  effort  of  organic  development,  putting  to  use 
its  reserves  or  arousing  needs  which  increase  our  activity. 
Thus  no  truth  ever  enters  into  us  except  as  it  is  postulated 
by  that  which  precedes  it  as  a  more  or  less  necessary  com- 
plement; just  as  an  article  of  food  to  become  valuable  as 
nourishment  presupposes  in  the  one  who  receives  it  certain 
preliminary  dispositions  and  preparations,  for  instance,  the 
appeal  of  hunger  and  the  ability  to  digest.  In  the  same  way 
the  statement  of  a  scientific  fact  presents  this  character, 
no  fact  having  meaning  nor,  consequently,  existing  for  us 
except  by  a  theory  in  which  it  is  born. 

On  these  various  points  a  critical  examination  of  the 


498  THE   MONIST. 

sciences  has  recently  come  to  confirm  the  reflection  of  the 
philosophers.  It  is  obvious  that  I  could  not  enter  here  into 
detail,9  but  the  little  that  I  have  said  will  doubtless  suffice 
to  give  a  glimpse  at  least  of  how  that  which  has  been  called 
extrinsicism10  is  opposed  in  spirit,  attitude  and  method  to 
modern  thought. 

3.  In  spite  of  what  we  have  just  said  let  us  admit,  how- 
ever, the  instruction  of  dogmas  by  simple  affirmation  of  a 
doctrinal  authority  which  is  accepted  almost  without  criti- 
cism. Nevertheless,  in  order  to  be  acceptable  these  dogmas 
would  need  to  be  perfectly  intelligible  in  their  statements, 
leaving  no  room  for  any  ambiguity  of  interpretation  or  any 
possibility  of  error  with  regard  to  their  real  meaning.  Now 
this  is  not  the  case.  In  the  first  place  their  formulas  often 
belong  to  the  language  of  a  particular  philosophical  sys- 
tem which  is  not  always  easily  understood,  which  does 
not  always  escape  the  danger  of  equivocation  or  even  of 
contradiction.  There  is  no  doubt,  for  instance,  that  the 
doctrine  of  the  Word  in  origin  and  context  is  closely  con- 
nected with  Alexandrian  neo-Platonism ;  that  the  theory 
of  substance  and  form  in  the  sacraments  and  that  of  the 
relations  between  substance  and  accidents  in  the  dogma  of 
the  real  presence,  are  really  closely  connected  with  Aristo- 
telian and  scholastic  conceptions.  Now  these  diverse  phi- 
losophies are  sometimes  doubtful  as  to  their  basis  and  ob- 
scure as  to  their  expression.  In  any  event  they  have  long 
been  antiquated,  fallen  into  disuse  among  philosophers  and 
scholars.  Would  it  therefore  be  necessary,  in  order  to  be 
Christians,  to  commence  by  being  converted  to  these  philos- 
ophies ?  This  would  be  a  difficult  undertaking,  before  which 

9  See  the  Bulletin  de  la  societe  franqaise  de  philosophie,  meeting  of  Febru- 
ary 25,  1904. 

10  Blondel  uses  the  term  extrincesisme  together  with  historicisme  to  denote 
two  kinds  of  apologetics  which  he  condemns.     See  his  article  on  "Histoire  et 
dogme"  in  La  Quinzaine  of  Jan.  15,  Feb.  1  and  Feb.  15  of  1904. 


WHAT  IS  A  DOGMA?  499 

many  believers  themselves  would  feel  strangely  embar- 
rassed. And  moreover  even  this  would  not  suffice,  for  the 
confusion  of  many  languages  resulting  from  heterogeneous 
philosophies  constitutes  still  another  difficulty  no  less 
troublesome  than  the  first. 

But  this  is  not  all.  Aside  from  this,  dogmatic  formulas 
contain  metaphors  borrowed  from  every-day  matters,  for 
instance  when  they  speak  of  the  Divine  Fatherhood  or  Son- 
ship.  It  is  impossible  to  give  an  exact  intellectual  inter- 
pretation of  these  metaphors,  and  consequently  to  deter- 
mine their  precise  theoretical  value.  They  are  images 
which  cannot  be  converted  into  concepts.  It  would  require 
anthropomorphism  to  take  them  literally,  and  at  the  same 
time  it  would  be  difficult  to  give  them  any  deep  significance. 
One  cannot  even  handle  them  without  reserve,  nor  follow 
them  to  a  conclusion  without  arriving  too  quickly  at  ridicu- 
lous consequences  and  absurdities.  Hence  arises  a  great 
uncertainty  that  continues  to  increase  the  confusion  of  im- 
aginative symbols  with  the  abstract  formulas  of  which  we 
were  just  speaking. 

After  all,  the  first  difficulty  with  regard  to  dogmas 
which  many  people  find  to-day  consists  in  the  fact  that  they 
do  not  succeed  in  discovering  a  thinkable  meaning  in  them. 
These  statements  tell  them  nothing,  or  rather  seem  to  them 
to  be  indissolubly  connected  with  a  state  of  mind  which 
they  no  longer  possess  and  to  which  they  think  they  are  no 
longer  able  to  return  without  degenerating.  Moreover 
many  believers  are  virtually  of  the  same  opinion,  and  pre- 
fer to  refrain  from  all  reflection,  foreseeing  certain  ob- 
stacles that  they  would  meet  in  thinking  what  they  believe 
under  the  forms  laid  before  them.  A  contemporaneous 
philosopher  has  said:  "What  would  most  embarrass  the 
greater  number  of  believers  would  be  if,  before  asking 
them  for  a  proof  of  what  they  believe,  one  were  simply  to 


5OO  THE   MONIST. 

call  upon  them  to  define  exactly  what  it  is  they  affirm  and 
what  they  deny."1 

4.  Finally,  let  us  pass  over  these  difficulties.  Even  after 
they  are  disposed  of  there  still  remains  a  last  objection 
which  seems  very  grave,  namely  that  in  any  event  dogmas 
form  a  group  incommensurable  with  the  whole  of  positive 
knowledge.  Neither  by  their  content  nor  by  their  logical 
nature  do  they  belong  to  the  same  system  of  knowledge  as 
other  propositions.  They  therefore  could  not  be  arranged 
with  others  in  a  way  to  form  a  coherent  system,  so  that  if 
one  accepts  them  the  result  is  an  inevitable  breach  of  unity 
in  the  mind,  a  disastrous  necessity  of  playing  a  double  part. 
Being  unalterable  they  appear  foreign  to  progress,  which 
is  the  very  essence  of  thought.  Being  transcendent  they 
exist  without  relation  to  effective  intellectual  life.  They 
bring  no  increase  of  light  to  any  of  the  problems  which 
occupy  science  and  philosophy.  Thus  the  least  reproach 
that  one  can  cast  upon  them  is  that  they  seem  to  be  without 
profit,  to  be  useless  and  barren — a  very  grave  reproach  in 
a  period  when  it  becomes  more  and  more  perceptible  that 
the  value  of  a  truth  is  measured  above  all  by  the  services 
that  it  renders,  by  the  new  results  that  it  suggests,  by  the 
consequences  which  it  brings  forth,  in  short  by  the  vivi- 
fying influence  it  exerts  on  the  entire  body  of  knowledge. 

Such,  briefly  summed  up,  are  the  principal  reasons  why 
the  idea  of  any  dogma  whatever  is  repugnant  to  modern 
thought.  I  have  endeavored  to  present  them  in  all  their 
force,  taking  the  same  point  of  view  in  setting  them  forth 
as  those  who  regard  them  as  conclusive  and  speaking,  so 
to  say,  not  in  my  own  name  but  in  theirs.  It  remains  now 
to  investigate  some  conclusions  and  some  lessons  which  we 
ought  to  be  able  to  derive  from  them. 

11  Belot,  Bibliotheque  du  congrcs  international  de  philosophic  de  7900.  Paris : 
Armand  Colin. 


WHAT  IS  A  DOGMA?  $01 

These  reasons,  it  must  be  recognized,  are  perfectly 
valid.  I  do  not  see  any  legitimate  way  of  refuting  the 
preceding  line  of  argument.12  The  principles  which  it  in- 
vokes seem  to  me  no  more  contestable  than  the  deductions 
which  it  draws  from  them.  In  fact  I  do  not  see  that  it  has 
ever  been  answered  except  by  worthless  subtleties  or  rhe- 
torical artifices.13  But  eloquence  is  not  a  proof,  neither  is 
diplomacy.  Hence  our  only  real  resource  is  to  prove  that 
the  idea  of  dogma  which  is  condemned  and  rejected  by 
modern  thought,  is  not  the  Catholic  idea  of  dogma. 

Perhaps  it  will  be  found  that  in  speaking  in  this  way 
I  depart  from  the  role  in  which  I  have  promised  to  confine 
myself,  that  this  time  decidedly  I  am  stating  theses  and 
not  asking  questions.  This  would  be  a  mistake.  There  is 
no  doubt  that  I  am  affirming  something  here,  but  what? 
Nothing  but  facts.  It  is  a  fact  that  the  unbelievers  of 
to-day  are  halted  in  the  face  of  dogmas  by  the  foregoing 
objections.  It  is  also  a  fact  that  whoever  (even  among 
believers)  has  truly  comprehended  the  spirit  and  the  meth- 
ods of  contemporary  science  and  philosophy,  cannot  but 
give  his  assent  to  these  objections.  Now  please  note :  those 
very  people  who  submit  most  completely  and  most  cordially 
to  the  authority  established  over  them  could  not  be  affected 
by  it.  No  authority  indeed  could  bring  it  about  or  prevent 
that  I  find  an  argument  valid  or  weak,  nor  especially  that 
this  or  that  notion  has  or  has  not  any  meaning  for  me. 
I  not  only  say  that  no  authority  has  any  right  in  the  world 
to  do  so,  but  that  it  is  absolutely  impossible;  for  after  all 
it  is  I  who  do  the  thinking  and  not  the  authority  that  thinks 
for  me.  No  argument  could  prevail  against  this  fact.  I 
can  neither  force  myself  to  feel  satisfaction  nor  prevent 
myself  from  feeling  it  at  the  evidence  on  one  side  or  an- 

12  I  say  refuting,  but  it  could  be  cut  short  by  destroying  the  postulate  which 
is  its  root. 

13  It  would  be  interesting  to  enter  into  a  detailed  discussion  of  these  an- 
swers, but  there  is  no  room  for  it  here. 


5O2  THE  MONIST. 

other.  To  be  sure  I  admit  that  authority  imposes  upon  me 
this  or  that  belief  with  the  result  that  it  makes  me  follow 
this  or  that  line  of  conduct,  but  how  could  it  compel  me  by 
virtue  of  such  a  proof  to  believe  what  I  do  not  regard  as 
convincing  ?  And  how  would  I  be  able  to  obey  it  if  it  com- 
manded me  to  understand  this  or  that  declaration  which  I 
did  not  understand  at  all?  As  well  might  it  require  me 
to  cease  thinking.  No  reason  can  be  founded  on  faith. 
Here  we  have  an  identity  pure  and  simple.  There  is  no 
such  thing  as  revealed  logic. 

Hence  I  come  back  to  what  I  said  a  while  ago,  and, 
speaking  as  a  philosopher,  I  declare  myself  incapable  of 
thinking  differently  from  our  adversaries  on  the  above- 
mentioned  points. 

Moreover  in  making  this  declaration  I  consider  that  I 
am  doing  nothing  but  stating  a  problem.  The  state  of 
mind  which  I  have  described  exists,  it  is  triumphant  to-day ; 
even  those  who  belie've  the  most  firmly  share  it.  These  are 
the  facts  which  it  is  impossible  not  to  take  into  account  and 
which  constitute,  I  repeat,  the  statement  of  a  question  to 
be  solved.  Let  us  see  exactly  what  this  question  is. 

I  shall  henceforth  regard  it  as  granted  that  the  objec- 
tions summed  up  above  cannot  be  evaded  so  long  as  the 
idea  of  dogma  which  they  contain  is  preserved.  Does  this 
mean  that  we  must  conclude  definitely  that  there  is  an 
absolute  incompatibility  between  the  idea  of  dogma  and 
the  essential  conditions  of  reasonable  thought?  That  in 
order  to  think  as  a  Christian  it  is  necessary  to  cease  think- 
ing altogether?  I  certainly  do  not  believe  so.  But  to 
avoid  the  objections  in  the  case  and  to  obtain  the  desired 
harmony  I  ask  myself  if  it  is  not  the  very  manner  in  which 
the  idea  of  dogma  is  presented  that  is  the  real  cause  of  the 
contention,  and  if  consequently  we  have  not  reason  to 
change  this  manner.1* 

14  I  beg  the  reader  to  give  heed  to  the  limits  within  which  this  question  is 


WHAT  IS  A  DOGMA?  503 

Now  when  we  examine  the  conception  of  dogma  which 
the  four  objections  above  enumerated  assume  and  imply, 
we  are  surprised  to  find  that  it  is  common  to  the  greater 
number  of  Catholics  and  their  opponents.  It  is  a  distinctly 
intellectual  conception.  It  regards  the  practical  and  moral 
meaning  of  the  dogma  as  secondary  and  derived  and  places 
in  the  first  rank  its  intellectual  meaning,  believing  that  this 
constitutes  the  dogma  whereas  the  other  is  merely  a  con- 
sequence of  it.  In  a  word,  it  makes  of  a  dogma  something 
like  the  statement  of  a  theorem — an  intangible  statement 
of  an  undemonstrable  theorem,  but  a  statement  having 
nevertheless  a  speculative  and  theoretical  character  and 
relating  above  all  to  pure  knowledge.  This  is  the  common 
postulate  that  one  discovers  by  analysis  at  the  foundation 
of  both  of  the  two  opposed  doctrines,  the  one  that  accepts 
and  the  one  that  rejects  the  idea  of  dogma.  Here  I  believe 
is  the  crux  of  the  difficulty.  From  this  unexpressed  postu- 
late and  from  the  conception  which  flows  from  it  originate, 
in  my  opinion,  both  the  abuses  to  which  the  idea  of  dogma 
can  give  rise  and  the  conscientious  objections  that  it  raises. 
Indeed  it  is  inevitable  that  one  would  finally  draw  the  con- 
clusion that  all  dogma  was  illegitimate,  for  he  would  at 
the  same  time  define  it  as  a  theoretical  statement  while 
nevertheless  attributing  to  it  characteristics  the  very  oppo- 
site of  those  which  make  statements  correct.  It  is  very 
curious  that  the  apologists  are  not  more  often  informed  of 
a  fact  of  such  great  importance  as  that  their  conception 
of  dogma  would  destroy  in  advance  the  theses  that  they 
wish  to  establish.  On  the  other  hand,  the  same  intellec- 
tualist  idea  of  dogma  leads  to  two  very  regrettable  and  un- 
fortunately very  frequent  exaggerations;  one  consists  of 
confusing  dogmas  properly  so  called  with  certain  opinions 

comprised.  It  does  not  discuss  in  any  way  the  modification  of  the  content 
of  dogma,  nor  even  its  traditional  religious  interpretation,  but  only  the  deter- 
mination of  the  modality  of  the  dogmatic  judgment  and  of  the  qualification  it 
possesses. 


504  THE  MONIST. 

and  certain  theological  systems,  that  is  to  say,  with  intel- 
lectual accessory  representations;  the  other,  in  failing  to 
see  that  a  dogma  could  never  possess  any  scientific  signifi- 
cance and  that  there  are  no  more  dogmas  concerning  for 
instance  biological  evolution  than  there  are  concerning  the 
movements  of  planets  or  the  compressibility  of  gas. 

From  a  thorough  study  of  these  various  points  we  reach 
the  conviction  that  the  problem  of  dogma  is  usually  badly 
stated;16  and  perhaps  we  will  see  at  the  same  time  how  it 
ought  to  be  stated  in  order  to  render  possible  a  satisfactory 

solution. 

*       *       * 

From  this  point  I  enter  at  once  into  the  domain  in  which 
I  must  keep  myself  in  an  interrogatory  attitude.  This  is 
my  definite  intention  although  to  insure  clearness  I  may 
keep  the  didactic  tone.  What  follows  must  be  taken  as  a 
simple  exposition  of  what  I  ordinarily  reply  to  those  who 
ask  me  what  I  think  of  the  idea  of  dogma.  Am  I  wrong 
to  speak  in  this  way  ?  I  am  quite  ready  to  acknowledge  it  if 
any  one  will  show  me  that  it  is  not  the  right  way  in  the 
eyes  of  the  church. 

First  of  all  I  say  that  a  dogma  cannot  be  compared  to 
a  theorem,  of  which  we  only  know  the  statement  without 
its  proof  and  whose  proof  can  only  be  guaranteed  by  the 
assertion  of  a  teacher.  Nevertheless  I  know  that  this  is 
the  most  common  conception.  We  like  to  think  of  God  in 
the  act  of  revelation  as  a  very  wise  professor  whose  word 
we  must  believe  when  he  communicates  to  his  audience 
results  whose  proof  that  audience  is  not  capable  of  under- 
standing. But  this  appears  to  me  to  be  hardly  satisfactory. 
We  say  that  God  has  spoken.  What  does  the  word  "speak" 
mean  in  this  case?  Most  certainly  it  is  a  metaphor.  What 
is  the  reality  which  it  conceals?  Herein  lies  the  whole 
difficulty. 

15  At  least  in  books  in  current  use  and  in  elementary  education. 


WHAT  IS  A  DOGMA?  505 

Without  recurring  to  the  general  considerations  I  have 
already  developed  let  us  take  some  examples  that  will  serve 
to  specify  what  we  have  hitherto  looked  upon  only  in  large 
outlines. 

"God  is  a  person."  Here  we  have  a  dogma.  Let  us 
try  to  see  in  it  a  statement  having  above  all  an  intellectual 
meaning  and  a  speculative  interest,  a  proposition  belong- 
ing first  of  all  to  the  order  of  theoretical  knowledge.  I 
'  pass  over  the  difficulties  aroused  by  the  word  "God,"  but 
let  us  consider  the  word  "person."  How  must  we  under- 
stand it? 

If  we  grant  that  the  use  of  this  word  bids  us  conceive 
the  divine  personality  in  the  form  shown  to  us  by  psycho- 
logical experience  on  the  model  of  what  common  sense 
designates  by  the  same  name,  as  a  human  personality, 
idealized  and  carried  on  to  perfection,  we  have  here  a 
complete  anthropomorphism,  and  Catholics  would  certainly 
agree  with  their  opponents  in  rejecting  such  a  conception. 
Moreover  to  carry  such  a  thought  to  its  extreme  limits  is 
a  very  delicate  thing,  very  likely  to  induce  error  or  at  least 
mere  verbiage,  incapable  in  any  event  of  producing  any- 
thing more  than  very  vague  metaphors  and  perhaps  even 
eventually  contradictory  results. 

Shall  we  limit  ourselves  to  saying  that  the  divine  per- 
sonality is  essentially  incomparable  and  transcendent  ?  Very 
well,  but  if  so  it  is  very  badly  named,  and  in  a  way  which 
seems  made  expressly  to  induce  delusion.  For  if  we  de- 
clare that  the  divine  personality  does  not  resemble  in  any 
respect  that  with  which  we  are  acquainted,  what  right 
have  we  to  call  it  "personality"?  Logically  it  should  be 
designated  by  a  word  which  would  belong  only  to  God, 
which  could  not  be  employed  in  any  other  instance.  This 
word  would  therefore  be  intrinsically  undefinable.  Let  us 
imagine  any  assemblage  whatever  of  syllables  deprived  of 
all  possible  significance.  Let  A  be  this  assemblage.  Then 


506  THE  MONIST. 

by  our  hypothesis  "God  is  a  person"  does  not  have  any 
other  meaning  than  "God  is  A."  Is  this  an  idea? 

The  dilemma  is  unsolvable  for  any  one  who  is  seeking 
an  intellectualist  interpretation  of  the  dogma  "God  is  a 
person."  Either  he  will  define  the  word  "personality,"  and 
then  he  is  fatally  sure  to  fall  into  anthropomorphism;  or 
he  will  not  define  it,  and  then  he  will  fall  none  the  less 
fatally  into  agnosticism.  Here  we  have  a  circle. 

The  same  remarks  hold  with  regard  to  the  propositions 
"God  is  conscious  of  himself ;  God  loves,  wills,  thinks,  etc." 

Let  us  take  another  example,  the  resurrection  of  Jesus. 
If  this  dogma,  whatever  may  be  eventually  its  practical 
consequences,  has  for  its  first  aim  to  increase  our  knowl- 
edge in  guaranteeing  to  us  the  accuracy  of  a  certain  fact, 
if  it  constitutes  before  all  a  statement  of  an  intellectual 
character,  the  question  to  which  it  first  gives  rise  is  this: 
What  precise  meaning  does  it  assume  is  to  be  attached  to 
the  word  "resurrection"  ?  Jesus,  after  having  experienced 
death,  has  once  more  become  alive.  What  does  this  mean 
from  the  theoretical  point  of  view?  Doubtless  nothing 
except  that  after  three  days  Jesus  reappeared  in  a  state 
identical  with  that  in  which  he  was  before  he  was  nailed 
on  the  cross.  Now  the  Gospel  itself  tells  us  exactly  the 
opposite.  The  resurrected  Jesus  was  no  longer  subject  to 
ordinary  physical  or  physiological  laws;  his  "glorified" 
body  was  no  longer  perceptible  in  the  same  conditions  as 
before,  etc.  What  does  this  mean?  The  idea  of  life  has 
not  the  same  content  when  applied  to  the  period  preceding 
the  crucifixion  as  to  that  which  followed  it.  Now  what 
does  the  word  represent  with  relation  to  this  second  period? 
Nothing  that  can  be  expressed  by  concepts.  It  is  simply  a 
metaphor  which  cannot  be  converted  into  specific  ideas. 
Here  again,  to  be  exact,  it  would  be  necessary  to  create  a 
new  word,  a  word  reserved  for  this  single  case,  a  word 


WHAT  IS  A  DOGMA  ?  507 

consequently  to  which  it  would  not  be  possible  to  give  any 
regular  definition. 

Let  us  borrow  a  final  example  from  the  dogma  of  the 
real  presence.  Here  it  is  the  term  "presence"  which  must 
be  interpreted.  What  does  it  usually  signify?  A  being-  is 
said  to  be  present  when  he  is  perceptible,  or  when  though 
he  himself  cannot  be  grasped  by  perception  he  yet  manifests 
himself  by  perceptible  effects.  Now  according  to  the  dogma 
itself  neither  of  these  two  circumstances  is  realized  in  the 
case  in  hand.  The  presence  in  question  is  a  mysterious 
presence,  ineffable,  unique,  without  analogy  to  anything 
that  one  ordinarily  understands  by  that  name.  Now  I  ask 
what  idea  is  there  here  for  us?  A  thing  that  can  neither 
be  analyzed  nor  even  defined  could  not  be  called  an  "idea" 
except  by  an  abuse  of  the  word.  We  wish  a  dogma  to  be  a 
statement  of  an  intellectual  order.  What  does  it  state? 
It  is  impossible  to  say  exactly.  Does  not  this  fact  condemn 
the  hypothesis? 

Finally  the  pretension  of  conceiving  dogmas  as  state- 
ments whose  first  function  would  be  to  communicate  cer- 
tain theoretical  bits  of  knowledge  would  run  against  im- 
possibilities on  every  hand.  It  seems  to  end  inevitably  in 
reducing  dogmas  to  pure  nonsense.  Perhaps  it  must  for 
this  reason  be  resolutely  abandoned.  Let  us  therefore  see 
what  different  kind  of  significance  remains  possible  and 
legitimate. 

*       *       * 

First  of  all,  if  I  do  not  deceive  myself,  a  dogma  has  a 
negative  meaning.  It  excludes  and  condemns  certain  er- 
rors instead  of  positively  determining  the  truth.16 

Let  us  once  more  take  up  our  former  examples.  We 
shall  first  consider  the  dogma  "God  is  a  person."  I  nowhere 
see  in  it  any  definition  of  the  divine  personality.  It  teaches 

16  We  shall  shortly  see  how  dogmas  are  more  and  greater  than  this.  But 
at  the  start  I  shall  place  myself  in  a  strictly  intellectualist  point  of  view. 


508  THE  MONIST. 

me  nothing  about  that  personality.  It  does  not  reveal  its 
nature  to  me  nor  furnish  me  with  any  explicit  idea.  But 
I  see  clearly  that  it  tells  me,  "God  is  not  impersonal" ;  that 
is  to  say,  God  is  not  simply  a  law,  a  formal  category,  an 
ideal  principle,  an  abstract  entity,  any  more  than  he  is  a 
universal  substance,  or  some  unknown  cosmic  force  dif- 
fused throughout  the  world.  In  short,  the  dogma  "God  is 
a  person"  does  not  bring  to  me  any  new  positive  conception 
nor  does  it  any  more  guarantee  to  me  the  truth  of  any  par- 
ticular system  among  those  which  the  history  of  philosophy 
shows  to  have  been  successively  proposed,  but  it  warns  me 
that  this  or  that  form  of  pantheism  is  false  and  ought  to 
be  rejected. 

I  would  say  the  same  with  regard  to  the  real  presence. 
The  dogma  does  not  tell  me  any  theory  about  that  presence, 
it  does  not  even  teach  me  in  what  it  consists ;  but  it  tells  me 
very  clearly  that  it  must  not  be  understood  in  such  or  such  a 
way  as  were  formerly  proposed,  that  for  instance  the  con- 
secrated host  must  not  be  regarded  solely  as  a  symbol  or 
a  figure  of  Jesus. 

The  resurrection  of  Christ  gives  rise  to  the  same  re- 
marks. This  dogma  does  not  teach  me  in  any  degree  what 
was  the  mechanism  of  this  unique  fact  nor  of  what  kind 
the  second  life  of  Jesus  was.  In  short  it  does  not  communi- 
cate a  conception  to  me.  But  on  the  contrary  it  excludes 
certain  conceptions  that  I  might  be  tempted  to  make.  Death 
has  not  put  an  end  to  the  activity  of  Jesus  with  reference 
to  the  things  of  this  world.  He  still  mediates  and  lives 
among  us,  and  not  at  all  merely  as  a  thinker  who  has  dis- 
appeared and  left  behind  a  rich  and  living  influence  and 
whose  work  has  left  results  through  the  ages ;  he  is  literally 
our  contemporary.  In  short,  death  has  not  been  for  him, 
as  it  is  for  ordinary  mortals,  the  definite  cessation  of  prac- 
tical activity.  This  is  what  the  dogma  of  the  resurrection 
teaches  us. 


WHAT  IS  A  DOGMA?  509 

Shall  I  insist  further?  It  does  not  seem  advisable  at 
this  time.  The  foregoing  examples  are  sufficient  to  make 
the  principle  of  interpretation  that  I  have  in  mind  clearly 
understood.  Of  course  long  expositions  would  be  neces- 
sary if  we  would  enumerate  in  detail  all  the  consequences 
of  this  principle  and  all  its  possible  applications,  and  an 
enumerative  study  of  the  different  dogmas  would  therefore 
become  indispensable.  But  this  is  not  my  real  purpose.  I 
wish  to  confine  myself  simply  to  indicating  an  ideal.  This 
is  why  I  do  not  undertake  either  to  multiply  examples  or 
even  to  develop  any  one  of  them  completely. 

Moreover  the  idea  is  not  a  new  one.  It  belongs  to  the 
most  authentic  tradition.  Is  it  not  indeed  the  classical 
teaching  of  theologians  and  scholars  that  in  supernatural 
matters  the  surest  method  of  investigation  is  the  via  nega- 
tionist Permit  me  to  recall  in  this  connection  a  well-known 
text  of  St.  Thomas :  "But  the  via  remotionis  is  to  be  used 
chiefly  in  considering  divine  substance.  For  divine  sub- 
stance by  its  immensity  exceeds  every  form  which  our  mind 
can  touch ;  and  so  we  cannot  grasp  it  by  knowing  what  it 
is,  but  some  sort  of  a  notion  of  it  we  have  by  knowing  what 
it  is  not/'17 

Nevertheless  I  ought  to  point  out  one  objection  which 
might  occur  to  the  mind.  We  will  easily  grant  that  the 
dogmatic  formulation  promulgated  by  the  church  in  the 
course  of  history  has  especially  a  negative  character,  at 
least  when  looked  upon  from  an  intellectual  point  of  view 
as  we  are  doing  at  this  time.  In  fact,  the  church  itself 
declares  that  its  mission  is  not  in  the  least  to  produce  new 
revelations  but  only  to  maintain  the  depositum  revelationis, 
and  the  negative  method  here  adopted  is  entirely  suitable 
for  this  mission.  And  yet,  of  what  does  this  depositum 

17  "Est  autem  via  remotionis  utendum  praecipue  in  consideratione  divinae 
substantiae.  Nam  divina  substantia  omnem  formam,  quam  intellectus  noster 
attingit,  sua  immensitate  excedit;  et  sic  ipsam  apprehendere  non  possumus 
cognoscendo  quid  est,  sed  aliqualem  ejus  habemus  notitiam  cognoscendo  quod 
non  est." — Contra  Gentiles,  I,  xiv. 


5IO  THE   MONIST. 

consist  if  not  of  a  certain  collection  of  original  affirmations? 
Take  the  primary  expression  of  Christian  faith,  the  Credo. 
What  could  be  more  positive?  Now  here  is  the  basis  of 
doctrine,  that  which  characterizes  and  constitutes  it.  More- 
over when  we  say  "revelation"  we  certainly  say  affirmation 
and  not  negation. 

Certainly  we  do.  I  do  not  contradict  it  in  the  least, 
but  we  must  make  a  distinction.  The  creed  of  Nicaea  and 
Constantinople  contains  many  traces  of  a  negative  dog- 
matic elaboration:  for  instance,  on  the  divinity  of  Christ 
as  against  the  Arian  heresy;  on  the  procession  of  the  Holy 
Ghost  in  opposition  to  the  Macedonians,  etc.18  Consequently 
there  is  nothing  on  this  head  to  contradict  our  conclusions. 
It  is  only  the  grammatical  form  which  is  affirmative  here ; 
in  reality  we  are  treating  of  errors  to  be  excluded  rather 
than  theories  to  be  formulated.  But  let  us  take  the  Apos- 
tles' Creed.  Here  indeed  we  have  nothing  negative  but 
neither  do  we  have  anything  properly  intellectual  and  theo- 
retical, nothing  which  belongs  properly  to  the  order  of 
speculative  knowledge,  nothing  in  short  which  resembles  the 
statement  of  theorems.  It  is  a  profession  of  faith,  a  declara- 
tion of  attitude.  We  shall  soon  examine  dogmas  from  this 
practical  point  of  view  (which  I  hasten  to  say  is  in  my 
eyes  the  principal  point  of  view),  yet  we  shall  stop  a  mo- 
ment at  the  intellectual  point  of  view.  The  Apostolic  Creed 
in  its  original  form  affirms  the  existence  of  realities  of 
which  it  gives  not  even  a  rudimentary  representative  the- 
ory, hence  its  only  role  with  reference  to  abstract  and 
reflective  knowledge  is  to  state  objects  and  therefore  prob- 
lems. Finally  we  see  that  the  proposed  objection  is  not 
valid  and  we  can  maintain  our  thesis  until  further  notice. 

Thus  in  so  far  as  they  are  statements  of  a  theoretical 
order  dogmas  have  all  a  negative  meaning.  History  proves 

18  It  would  be  easy  to  insist  on  the  example  of  consubstantialem  or  of 
Filioque. 


WHAT  IS  A  DOGMA?  51 1 

this  when  it  procures  our  assistance  at  the  birth  of  one 
after  another  of  them  in  relation  to  the  several  heresies.19 
The  rise  of  all  dogmas  has  always  followed  the  same 
course,  has  always  presented  the  same  phases:  at  the  be- 
ginning purely  human  speculations,  some  explanatory  sys- 
tems very  similar  to  other  philosophical  systems,  in  short, 
attempts  at  theories  relating  to  religious  facts,  to  mysteri- 
ous realities  experienced  by  Christendom  in  its  practical 
faith;  then  only  come  the  dogmas  for  the  purpose  of  con- 
demning certain  of  these  attempts,  of  taxing  certain  of 
these  conceptions  with  error  and  of  excluding  certain  of 
these  intellectual  representations.  Hence  it  follows  that 
dogmatic  formulas  often  borrow  expressions  from  different 
philosophies  without  taking  the  trouble  to  fuse  together 
and  unify  these  heterogeneous  languages. 

This  offers  no  more  disadvantages  than  does  the  use 
of  concepts  derived  from  different  origins,  from  the  mo- 
ment that  dogmas  do  not  tend  to  constitute  by  themselves 
a  rational  theory,  an  intelligible  system  of  positive  affir- 
mations, but  confine  themselves  to  opposing  certain  excep- 
tions to  certain  hypotheses  and  conjectures  of  the  human 
mind.  On  the  other  hand  it  is  natural  that  each  dogma 
should  put  itself  in  the  point  of  view  belonging  to  the  doc- 
trine that  it  lays  under  an  interdict,  in  order  to  attack  it 
directly  without  danger  of  ambiguity.  Hence  it  also  fol- 
lows that  dogmatic  formulas  can  enact  laws  on  the  incom- 
parable and  the  transcendent  and  yet  not  fall  into  the  con- 
tradictions of  anthropomorphism  or  of  agnosticism.  It  is 
man  who  with  his  opinions,  his  theories  and  systems,  gives 
to  dogmas  their  intelligible  substance;20  these  are  confined 
to  pronouncing  a  veto  at  times,  to  declaring  at  times  that 

19  Compare  the  usual  formula  of  the  decrees  of  council :  "5"*  quis  dixerit. .., 
anathema  sit." 

20  From  the  theoretical  point  of  view,  understand.     Dogmas  are  thought 
in  terms  of  the  human  systems  which  they  oppose.     [This  view  is  recently  en- 
dorsed by  Catholic  theologians  of  such  recognized  authority  as  Cardinal  Billot. 
S.J.-Tr.] 


512  THE   MONIST. 

"such  an  opinion,  such  a  theory,  such  a  system,  is  not  al- 
lowed," without  ever  pointing  out  why  they  should  not  be 
accepted,  nor  by  what  they  must  be  replaced.  Thus  nega- 
tive dogmatic  definitions  do  not  limit  knowledge  nor  put  an 
end  to  progress ;  in  short  they  only  close  up  false  paths. 

From  the  strictly  intellectual  point  of  view  it  seeems 
to  me  that  dogmas  have  only  the  negative  and  prohibitive 
sense  of  which  I  speak.  If  they  formulated  absolute  truth 
in  adequate  terms  (to  assume  that  such  a  fiction  has  a 
meaning)  they  would  be  unintelligible  to  us.  If  they  gave 
only  an  imperfect  truth,  relative  and  mutable,  they  would 
not  be  justified  in  obtruding  themselves.  The  only  radical 
way  to  put  an  end  to  all  the  objections  on  principle  against 
dogma  is  to  conceive  of  them,  as  we  have  already  said,  as 
being  undefinable  in  so  far  as  they  are  speculative  propo- 
sitions, except  with  relation  to  previous  doctrines  upon 
which  they  promulgate  an  unwarranted  judgment.  More- 
over is  it  not  the  teaching  of  theologians,  including  the 
most  intellectualist,  that  in  a  dogmatic  statement  the  rea- 
sons which  can  be  incorporated  in  the  text  are  not  in  them- 
selves objects  of  faith  imposed  upon  belief? 

There  is  one  important  consequence  resulting  from  the 
foregoing,  namely,  that  the  true  method  of  studying  dog- 
mas (from  the  intellectual  point  of  view,  understand)  is 
the  historical  method.  The  science  known  as  positive  the- 
ology, or  rather  the  history  of  dogma,  seeks  to  perform 
this  task.  The  method  has  an  effective  apologetic  value 
much  greater  than  purely  dialectic  dissertations.  Because 
in  any  event  it  is  impossible  to  comprehend  dogmatic  state- 
ments, there  is  the  greater  reason  for  justifying  them  if  one 
would  commence  by  plunging  them  once  more  into  their 
natural  historical  environment  without  which  their  authen- 
tic meaning  becomes  more  and  more  vague  and  finally  ends 
by  vanishing  entirely. 

Nevertheless  dogmas  do  not  have  merely  a  negative 


WHAT  IS  A  DOGMA?  513 

meaning,  and  even  the  negative  meaning  that  they  offer 
when  regarded  from  a  certain  direction  does  not  constitute 
their  essential  and  primary  significance.  This  is  true  be- 
cause they  are  not  merely  propositions  of  a  theoretical 
character,  because  they  must  not  be  examined  solely  from 
the  intellectual  point  of  view,  from  the  point  of  view  of 
knowledge.  This  is  what  we  shall  now  elucidate  further. 


Here  more  than  ever  I  insist  that  the  intention  and 
tendency  of  the  pages  to  follow  must  not  be  misunderstood. 
I  repeat  that  the  affirmative  tone  is  used  only  as  a  means 
for  clearness.  At  bottom  the  question  is  always  the  same 
as  I  specified  at  the  beginning.  Here,  if  I  may  say  so,  is 
the  form  in  which  experience  has  shown  me  that  the  notion 
of  dogma  is  most  easily  assimilable  to  the  minds  of  to-day: 

A  dogma  has  above  all  a  practical  meaning.  It  states 
before  all  a  prescription  of  a  practical  kind.  It  is  more 
than  all  the  formula  of  a  rule  of  practical  conduct.  This 
is  its  principal  value,  this  its  positive  significance.  This 
does  not  mean,  however,  that  it  must  be  without  relation  to 
thought,  for  ( i )  there  are  also  certain  duties  concerned 
with  the  act  of  thought;  (2)  it  is  virtually  affirmed  by  the 
dogma  itself  that  under  one  form  or  another  reality  con- 
tains wherewith  to  justify  the  prescribed  conduct  as  rea- 
sonable and  wholesome. 

I  take  pleasure  in  quoting  in  this  connection  the  follow- 
ing passage  from  R.  P.  Laberthonniere:21  "Dogmas  are 
not  simply  enigmatical  and  obscure  formulas  which  God 
has  promulgated  in  the  name  of  his  omnipotence  to  mortify 
the  pride  of  our  spirits.  They  have  a  moral  and  practical 
meaning;  they  have  a  vital  meaning  more  or  less  accessible 
to  us  according  to  the  degree  of  spirituality  we  possess." 

After  all,  when  converts,  in  spite  of  good  intentions, 

21  Essais  de  philosophic  religieuse,  p.  272.    Paris :  Lethielleux. 


514  THE  MONIST. 

themselves  create  part  of  the  theoretical  difficulties  under 
discussion  do  we  not  answer  them  daily:  "Never  mind  all 
that,  it  is  not  important.  Do  not  believe  that  God  requires 
so  many  formalities.  Come  to  him  fairly,  frankly,  simply, 
according  to  the  wise  words  of  Bossuet.  Religion  is  not 
so  much  an  intellectual  adherence  to  a  system  of  speculative 
propositions  as  it  is  a  living  participation  in  mysterious 
realities."  Why  not  then  make  theory  agree  with  practice? 

Let  us  keep  the  same  examples.  They  represent  well 
enough  the  different  types  of  dogmas.  "God  is  a  person" 
means,  "Conduct  yourself  in  your  relations  to  God  as  in 
your  relations  with  a  human  person."  Likewise  "Jesus 
has  risen"  means,  "Be  in  relation  to  him  as  you  would 
have  been  before  his  death,  as  you  are  with  a  contem- 
porary." In  the  same  way  again  the  dogma  of  the  real 
presence  means  that  one  must  have  the  same  attitude 
toward  the  consecrated  host  as  one  would  have  toward 
Jesus  had  he  become  visible,  and  so  on.  It  would  be  easy 
to  multiply  these  examples,  and  also  to  develop  each  of 
them  farther.22 

That  dogmas  can  and  ought  to  be  interpreted  in  this 
way  there  is  no  doubt,  and  the  fact  will  not  be  contested 
by  any  one.  In  fact,  it  cannot  be  repeated  too  often  that 
Christianity  is  not  a  system  of  speculative  philosophy  but 
a  source  and  regimen  of  life,  a  discipline  of  moral  and 
religious  action,  in  short  the  sum  total  of  practical  means 
to  obtain  salvation.  What  then  is  surprising  in  the  fact 
that  its  dogmas  primarily  concern  conduct  rather  than  pure 
reflective  knowledge?23 

I  do  not  think  it  is  necessary  to  insist  farther  upon  this 

22  I  do  not  claim  in  the  least  that  the  foregoing  comments  exhaust  the 
meaning  of  the  dogmas  mentioned :  they  will  suffice  to  point  out  a  line  of 
inquiry. 

28  This  is  why  assent  to  dogmas  is  always  a  free  act  and  not  the  inevitable 
result  of  a  compelling  line  of  argument. 


WHAT  IS  A  DOGMA?  51$ 

point,  but  I  wish  to  indicate  in  a  few  brief  words  the  most 
important  consequences  of  the  principle  here  laid  down. 

First  of  all  it  is  clear  that  the  general  objections 
summed  up  at  the  beginning  of  this  article  do  not  affect 
this  conception  of  dogma  to  the  same  extent  and  in  the 
same  degree  as  they  do  the  usual  intellectualist  conception, 
for  that  provokes  the  conflict  and  renders  the  difficulty 
insurmountable,  whereas  on  the  other  hand  we  may  now 
catch  a  glimpse  of  a  possible  solution.  As  there  is  no  ques- 
tion of  obtaining  a  theoretical  statement  in  conditions  rad- 
ically opposed  to  those  prescribed  by  scientific  method,  we 
no  longer  find  ourselves  face  to  face  with  a  logical  stum- 
bling-block but  only  with  a  problem  referring  to  relations 
between  thought  and  action — a  difficult  problem  certainly, 
but  not  unapproachable  and  one  which  at  any  rate  does  not 
appear  absurd  after  it  is  stated. 

Of  course  there  are  always  important  questions  to  be 
solved.  It  is  necessary  to  supply  the  dogma  in  some  way 
with  a  demonstration  and  justification,  and  this  is  by  no 
means  a  perfectly  easy  matter.  Nevertheless  one  of  the 
greatest  obstacles  has  been  smoothed  away.  Practical 
truths  are  established  differently  from  speculative  truths. 
Recourse  to  authority  which  is  entirely  inadmissible  in  the 
realm  of  pure  thought  seems  a  priori  less  shocking  in  the 
domain  of  action,  because  if  authority  has  legitimate  rights 
anywhere  it  certainly  has  in  the  domain  of  practical  affairs. 

The  Council  of  the  Vatican  tells  us:  "If  any  shall  say 
that  no  true  mysteries  properly  so-called  are  contained  in 
divine  revelation,  but  that  all  the  dogmas  of  faith  can  be 
comprehended  and  demonstrated  through  reason  duly  per- 
fected by  natural  principles,  let  him  be  anathema."24  Now 
if  faith  in  dogmas  were  first  of  all  knowledge,  an  ad- 
herence to  some  statements  of  an  intellectual  kind,  one 

24  "Si  quis  dixerit  in  revelatione  divina  nulla  vera  et  proprie  dicta  mys- 
teria  contineri,  sed  universa  fidei  dogmata  posse  per  rationem  rite  excultam  a 
naturalibus  principiis  intelligi  et  demonstrari,  anathema  sit." 


5l6  THE  MONIST. 

could  not  comprehend  either  that  assent  to  unsolvable  mys- 
teries could  ever  be  legitimate  or  even  simply  possible,  or 
in  what  it  might  consist,  or  what  sort  of  utility  or  value  it 
might  have  for  us,  or  how  it  might  constitute  a  virtue.  On 
the  other  hand  all  this  can  be  understood  if  faith  in  dogmas 
is  a  practical  submission  to  commandments  which  have  to 
do  with  action.  Nothing  is  more  normal  than  activity  plac- 
ing mysteries  before  intelligence.25 

The  Council  of  the  Vatican  tells  us  further:  "If  any 
shall  say  that  assent  to  the  Christian  faith  is  not  free.  . .  . 
let  him  be  anathema."2  This  text  is  generally  explained 
by  recognizing  that  the  reasons  for  believing,  the  motives 
of  credibility,  are  not  of  insuperable  force,  a  mathematical 
evidence,  and  that  in  consequence  a  decisive  act  of  the  will 
or  of  the  heart  is  always  necessary  to  conclude  the  investi- 
gation definitely.  Is  this  not  virtually  admitting  that  one 
cannot  see  in  belief  in  dogmas  an  act  which  should  first  of 
all  be  intellectual  without  making  it  thereby  inferior  to  the 
ordinary  acts  of  thought?  How  would  such  an  act — an 
act  performed  under  conditions  contrary  to  the  nature  of 
thought — be  even  legitimate  or  merely  possible?  But  on 
the  other  hand  it  is  easy  to  believe  that  the  practical  ac- 
ceptance of  commandments  relating  to  action  depends  on 
our  free  will  and  gains  in  perfection  by  not  being  able  to 
manifest  itself  by  necessary  consequence.  Let  us  insist  a 
little  upon  this  point,  for  it  is  of  highest  importance  in  the 
problem  of  the  relations  between  reason  and  faith. 

From  the  beginning  apologetics  is  confronted  with  a 
grave  difficulty  which  perhaps  cannot  always  be  satisfac- 
torily disposed  of.  On  the  one  hand  it  is  clearly  understood 
that  an  act  of  faith  is  a  free  act  and  that  its  object,  as  well 
as  its  supreme  motive,  is  supernatural.  But  on  the  other 

25  Submission  to  dogmas  then  from  one  point  of  view  is  for  the  believer 
what  submission  to  facts  is  for  the  scholar. 

28  "Si  quis  dixerit  assensum  fidei  Christianae  non  essc  liberum ,  ana- 
thema sit." 


WHAT  IS  A  DOGMA?  517 

hand  an  act  of  reason  ought  to  precede  and  prepare  the  act 
of  faith,  for  it  is  reason  alone  by  which  the  obligation  and 
necessity  of  overreaching  reason  can  be  recognized.  And 
an  act  of  reason  must  also  constantly  accompany  the  act  of 
faith,  for  it  is  necessary  that  the  human  mind  shall  have 
some  sort  of  hold  upon  the  dogma  if  it  wishes  to  ac- 
cept it.  St.  Thomas  said  well:  "Those  things  which  are 
under  the  faith ....  no  one  would  believe  unless  he  sees 
they  ought  to  be  believed."27 

Now  how  shall  we  reconcile  these  two  opposite  require- 
ments in  a  system  of  intellectualist  interpretation?  Either 
we  would  maintain  (as  there  are  some  who  do)  that  the 
apologetic  proofs  are  absolutely  positive  and  exact;  and 
then  what  would  become  of  the  liberty  of  the  act  of  faith  ? 
Or  in  order  to  safeguard  that  liberty  we  would  call  them 
insufficient  and  only  more  or  less  probable;  and  then  our 
faith  would  lack  any  basis,  for  after  all  an  insufficient 
proof  is  not  an  acceptable  proof,  especially  in  so  important 
and  difficult  a  matter.  An  intellectualist  attitude  becomes 
disarmed  in  the  face  of  this  dilemma  since  liberty  does 
not  belong  to  the  domain  of  pure  intelligence  and  has  no 
place  or  part  in  the  proceedings  of  discursive  reason.  But 
with  the  other  attitude  the  dilemma  can  be  solved  because 
this  time  the  dialectic  in  the  case  is  action  and  life  not 
simply  argument,  and  liberty  revives  with  life  and  action. 

Likewise  we  have  here  the  objection  relating  to  the 
intelligibility  of  dogmatic  formulas.  Although  these  for- 
mulas are  hopelessly  obscure,  even  inconceivable,  when  we 
want  them  to  furnish  positive  determinations  of  truth  from 
a  speculative  and  theoretical  point  of  view,  they  neverthe- 
less show  themselves  capable  of  clearness  if  we  are  careful 
not  to- ask  of  them  anything  but  instruction  as  to  practical 
conduct.  What  difficulty,  for  instance,  do  we  find  in  under- 
standing the  dogmas  of  the  divine  personality,  of  the  real 

27  "Ea  quae  subsunt  fidei. . . .  nemo  crederet  nisi  videret  ea  esse  credenda .*' 


5l8  THE  MONIST. 

presence,  or  the  resurrection  in  the  practical  system  of 
interpretation  just  outlined?  Although  these  dogmas  are 
mysteries  for  the  intelligence  that  demands  explanatory 
theories  they  are  nevertheless  susceptible  of  perfectly  clear 
statement  as  to  what  they  prescribe  for  our  actions.  Hence 
the  language  of  common  sense  has  its  place  as  well  as  the 
use  of  anthropomorphic  symbols  and  the  employment  of 
analogies  or  metaphors,  and  neither  the  one  nor  the  other 
gives  rise  to  unsolvable  complications  since  this  time  it  is 
a  question  only  of  propositions  relating  to  man  and  his 
attitudes. 

We  also  see  now  what  the  relation  is  between  dogmas 
and  efficient  life.  We  predict  for  them  a  possibility  of  ex- 
perimental study  and  of  gradual  research  which  has  here- 
tofore escaped  us.  Finally  we  understand  how  they  can  be 
common  to  all,  accessible  to  all,  in  spite  of  the  inequality 
between  intellects,  whereas  to  conceive  them  in  the  intel- 
lectualist  way  one  would  be  inevitably  led  to  make  a  dis- 
tinction of  an  intellectual  aristocracy.  I  have  not  room 
here  to  develop  these  different  considerations  as  much  as 
I  should  like,  but  I  imagine  that  a  simple  indication  after  all 
may  be  sufficient  for  the  time  being,  and  that  the  reader 
can  carry  the  process  on  for  himself  without  any  difficulty. 
Nevertheless  it  seems  necessary  to  me  to  prevent  a  possible 
objection  in  order  to  avoid  all  misapprehension. 

I  have  spoken  of  practice.  This  word  must  be  rightly 
understood.  I  take  it  in  the  widest  acceptation  of  the 
term.  Action  and  life  are  here  synonymous.  Hence  the 
word  does  not  in  the  least  mean  a  blind  step,  without  rela- 
tion to  thought  or  consciousness.  In  fact  there  is  an  act 
of  thought  which  accompanies  all  our  actions,  a  life  of 
thought  which  mingles  throughout  our  life ;  in  other  words, 
to  know  is  a  function  of  life,  a  practical  act  in  its  way. 
This  function,  this  act,  is  also  called  experience,  a  name 
which  indicates  at  the  same  time  that  we  are  not  at  all 


WHAT  IS  A  DOGMA? 

dealing  with  actions  performed  without  any  sort  of  light 
but  that  the  light  in  question  is  not  that  of  simple  argumen- 
tative reason. 

I  have  also  spoken  of  the  activity  which  places  mys- 
teries before  intelligence,  and  by  way  of  elucidation  I  have 
cited  the  example  of  scientific  facts.  To  comprehend  what 
I  mean  by  this,  one  must  not  forget  that  a  scientific  fact  is 
not  a  thing  to  be  submitted  to  passively.  If  there  is  any 
semblance  of  a  purely  external  fact,  of  a  mystery  totally 
opaque,  of  a  violent  commandment  from  without,  it  is  so 
with  respect  to  argumentative  understanding.  But  the 
thought-action  of  which  I  was  just  now  speaking  avoids 
this  appearance.  It  infinitely  exceeds  the  purely  intellec- 
tual thought.  I  have  not  heard  anything  to  affirm  other- 


*  9R 

wise.28 


Hence  there  is  a  necessary  relation  between  dogmas 
and  thought.  It  is  at  the  same  time  both  a  right  and  a 
duty  not  to  be  content  with  a  blind  belief  in  dogmas  but  to 
strive  also  in  proportion  to  one's  strength  to  think  them. 
The  system  of  separation,  of  tight  partitions,  of  the  twofold 
accountability  of  conscience,  is  not  desirable  nor,  to  speak 
truthfully,  possible.  It  is  contrary  to  the  demands  of  that 
faith  which  wishes  to  hold  every  man ;  it  is  contrary  to  the 
requirements  of  philosophy  which  desires  a  spiritual  unity; 
and  finally  it  is  contrary  to  the  requirements  of  morality 
which  cannot  approve  an  action  that  is  systematically  un- 
considered. 

But  thought  when  applied  to  dogmas  should  not  mis- 
understand their  primarily  practical  meaning.  The  path 
to  be  followed  is  the  test  of  practical  experience  and  not 
an  intellectual  dialectic.  The  inspiring  principle  is  per- 
fectly expressed  in  the  sacred  word,  qui  facit  veritatem 
venit  ad  lucem. 

28  The  reader  who  desires  to  pursue  this  point  further  may  refer  to  several 
articles  I  have  published  since  1889  in  the  Revue  de  metaphysique  et  de  morale, 
and  in  the  Bulletin  de  la  societe  fran^aise  de  philosophic. 


52O  •  THE   MONIST. 

Thus  translated  into  terms  of  action  the  traditional 
methods  of  analogy  and  eminence  assume  a  very  clear  sig- 
nificance. Under  the  guise  of  metaphors  and  images  they 
affirm  that  supernatural, reality  contains  the  wherewith  to 
make  obligatory  by  law  that  our  attitude  and  our  conduct 
with  regard  to  it  should  have  such  or  such  a  character. 
The  images  and  metaphors — which  are  hopelessly  vague 
and  fallacious  when  one  tries  to  see  in  them  any  approxi- 
mation whatever  of  impossible  concepts — become  on  the 
other  hand  wonderfully  illuminating  and  suggestive  after 
one  looks  to  find  in  them  only  a"  language  of  action  trans- 
lating truth  by  its  practical  echo  within  ourselves. 

It  remains  finally  to  specify  the  relations  of  dogmas, 
understood  in  the  way  we  have  described  them,  to  theo- 
retical and  speculative  thought,  to  pure  knowledge.  In 
what  respect  do  they  govern  our  intellectual  life  ?  How  does 
their  intangible  and  transcendent  character  leave  the  full 
liberty  of  research  intact  as  wfell  as  the  undeniable  right  of 
the  mind  to  repulse  every  conception  which  tries  to  impose 
itself  from  without?  We  shall  easily  see% 

The  Catholic  is  obliged  to  assent  to  the  dogmas  without 
reservation.  But  what  is  thereby  imposed  upon  him  is  not 
in  the  least  a  theory,  an  intellectual  representation.  Such 
a  constraint  indeed  would  inevitably  lead  to  undesirable 
consequences :  ( I )  The  dogmas  would  in  that  case  .  be 
reduced  to  purely  verbal  formulas,  to  simple  words  whose 
repetition  would  constitute  a  sort  of  unintelligible  com- 
mand; (2)  Moreover  these  dogmas  could  not  be  Common 
to  all  times  nor  to  all  intelligences.29 

No,  dogmas  are  not  at  all  like  that.  As  we  have  seen, 
their  meaning  is  above  all  practical  and  moral.  The  Cath- 
olic, obliged  to  accept  them,  is  not  restrained  by  them  ex- 
cept as  regards  rules  of  conduct,  not  as  regards  any  par- 

29  In  the  two  words  "esotericism"  and  "Pharisaism"  would  be  the  inevi- 
table double  rock  upon  which  they  would  split. 


WHAT  IS  A  DOGMA?  521 

ticular  conceptions.  Nor  is  he  condemned  to  accept  them 
as  simple  literal  formulas.  On  the  contrary,  they  offer 
him  a  very  positive  content,  explicitly  intelligible  and  com- 
prehensible. I  will  add  that  this  content,  having  to  do 
solely  with  the  practical,  is  not  relative  to  the  variable 
degree  of  intelligence  and  knowledge;  it  remains  exactly 
the  same  for  the  scholar  and  the  ignorant  man,  for  the 
exalted  and  the  lowly,  for  the  ages  of  advanced  civilization 
and  for  the  races  still  in  barbarism.  In  short  it  is  inde- 
pendent of  the  successive  states  through  which  human 
thought  passes  in  its  effort  toward  knowledge,  and  thus 
there  is  only  one  faith  for  everybody. 

This  being  granted,  the  Catholic  after  having  accepted 
the  dogmas  retains  full  liberty  to  make  for  himself  what- 
ever theory/  or  whatever  intellectual  representation  he 
wishes  of  the  corresponding  objects — the  divine  personal- 
ity, the  re'a)  presence,  or  the  resurrection,  for  instance.  It 
remains  wtfti  him  to  grant  his  preference  to  the  theory 
which  best  agrees  with  his  own  views,  to  the  intellectual 
representation  which  lie  deems  the  best.  His  position  in 
this  respect  rs  the  same  as  that  toward  any  scientific  or 
philosophical  speculation,  and  he  is  free  to  adopt  the  same 
attitude  in  both  cases.  Only  one  thing  is  imposed  upon 
him,  only  one  obligation  is  incumbent  upon  him ;  his  theory 
must  justify  the  practical  rules  expressed  by  the  dogma, 
his  intellectual  representation  must  take  into  account  the 
practical  edicts  prescribed  by  the  dogma.  Thus  in  a  word 
it  appears  almost  like  the  statement  of  a  fact  with  regard 
to  which  it  is  possible  to  construct  many  different  theories 
but  which  every  theory  must  take  into  account,  like  the 
expression  of  a  truth  many  of  whose  intellectual  represen- 
tations are  legitimate  but  of  which  no  explanatory  system 
can  well  be  independent.80 

30  It  is  at  this  point  that  we  must  distinguish  between  intellectual  formula 
and  the  underlying  reality  in  the  dogma. 


522  THE  MONIST. 

From  this  naturally  follows  the  step  that  we  have  rec- 
ognized as  usual  with  religious  thought  in  its  effort  at 
elaboration.  Let  us  take  any  dogma  whatever,  Divine  Per- 
sonality, the  real  presence  or  the  resurrection  of  Jesus.  By 
itself  and  in  itself  it  has  only  a  practical  meaning.  But 
there  is  a  mysterious  reality  corresponding  to  it  and  there- 
fore it  presents  to  the  intelligence  a  theoretical  problem. 
The  human  intellect  at  once  takes  possession  of  this  prob- 
lem; and  obeying  simply  and  solely  the  laws  of  its  own 
nature  it  imagines  the  explanations,  the  answers,  the 
systems  codified  in  the  precepts  of  scientific  method  and 
the  principles  of  reason.31  As  long  as  the  theory  con- 
structed in  this  way  respects  the  practical  significance  of 
the  dogma  it  is  given  carte  blanche.  Hence  to  pass  judg- 
ment on  the  theories  remains  the  task  of  pure  human  specu- 
lation, and  any  authority  exterior  to  the  thought  itself  has 
neither  the  right  nor  the  power  to  interfere.82  But  once 
let  a  theory  arise  which  makes  an  attack  on  dogma  in  its 
own  domain  by  altering  its  practical  significance,  and  the 
dogma  would  immediately  array  itself  against  it  and  con- 
demn it,  thus  becoming  a  negative  intellectual  statement 
superimposed  upon  the  rule  of  conduct  which  at  first  it 
was,  purely  and  simply. 

Hence  one  sees  positively  how  the  two  meanings  of  a 
dogma,  the  practical  meaning  and  the  negative  meaning, 
are  reunited,  the  latter  being  subordinated  to  the  former. 
Moreover  we  see  how  dogmas  are  immutable  and  yet  how 
there  is  an  evolution  of  dogmas.  What  remains  constant 
in  the  dogma  is  the  orientation  that  it  gives  to  our  practical 
activity,  the  direction  in  which  it  inflects  our  conduct.  But 

31  In  this  respect  the  Middle  Ages  had  an  independence  and  a  boldness 
which  we  have  forgotten. 

32  Religious  authority  which  has  souls  in  its  charge  can  indicate  certain 
theories  as  dangerous,  as  long  as  they  run  the  risk  of  being  wrongly  under- 
stood and  thus  of  reacting  injuriously  upon  conduct.     Hence  arise  censures 
of  an  inferior  note  to  those  of  heresy.    But  these  condemnations  are  not  prop- 
erly dogmatic. 


WHAT  IS  A  DOGMA?  523 

the  explanatory  theories,  the  intellectual  representations, 
change  constantly  in  the  course  of  the  ages  according  to 
individuals  and  epochs,  freed  from  all  the  fluctuations  and 
all  the  aspects  of  relativity -manifested  by  the  history  of 
the  human  mind.  The  Christians  of  the  first  centuries  did 
not  profess  the  same  opinions  on  the  nature  and  personality 
of  Jesus  as  we,  and  they  did  not  have  the  same  problems. 
The  ignorant  man  to-day  does  not  have  at  all  the  same 
ideas  on  these  lofty  and  difficult  subjects  as  the  philosopher 
does,  nor  the  same  mental  preoccupations.  But  whether 
ignorant  men  or  philosophers,  men  of  the  first  or  the  twen- 
tieth century,  every  Catholic  has  always  had  and  always 
will  have  the  same  practical  attitude  with  regard  to  Jesus. 

It  is  time  to  conclude  and  I  will  do  so  in  as  few  and 
brief  words  as  possible. 

Two  main  results  seem  to  me  to  have  been  attained  by 
the  foregoing  discussion: 

1.  The  intellectualist  conception  which  is  current  to-day 
renders  the  greater  number  of  objections  raised  by  the 
idea  of  dogma  unsolvable. 

2.  On  the  other  hand,  a  doctrine  of  primacy  of  action 
permits   a   solution  of  the  problem  without  abandoning 
either  the  rights  of  thought  or  the  requirements  of  dogma. 

If  these  conclusions  were  admitted,  the  apologetics  of 
our  days  would  be  under  the  irresistible  necessity  of  modi- 
fying many  of  its  arguments  and  methods. 

Now,  can  these  conclusions  be  admitted  without  loss  to 
faith  ?  It  is  for  the  theologians  to  tell  us,  and  in  case  their 
response  is  negative  to  teach  us  how  they  expect  otherwise 
to  prepare  to  surmount  the  obstacles  which  perplex  us. 

EDOUARD  LE  ROY. 


LEIBNIZ  IN  LONDON.1 

KIBNIZ  paid  two  visits  to  London  from  Paris,  where 
he  was  staying  from  March,  1672,  to  October,  1676: 
the  first  visit,  which  was  in  connection  with  the  embassy 
from  the  Elector  of  Mainz,  was  from  January  n  to  the 
beginning  of  March,  1673;  the  second  was  made  on  his 
way  home  to  Germany,  when  he  stopped  in  London  for 
about  a  week  in  October,  1676. 

Leibniz  had  a  habit  of  writing  out  all  the  important 
scientific  points  in  the  correspondence  that  he  kept  up  with 
noted  people,  so  that  he  might  thus  impress  them  the  more 
deeply  upon  his  memory.  I  have  discovered  among  his 
manuscripts  three  folio  sheets  on  which  he  has  written 
down  the  things  worth  noting  in  connection  with  these 
two  visits  to  London.2  The  sheets  which  relate  to  his  second 
visit  have  been  known  to  me  for  some  time;  but  the  other 
ones,  referring  to  the  first  visit,  I  came  across  only  during 
my  last  stay  in  Hanover  in  the  summer  vacation  of  the  year 
1890. 

In  what  follows,  1  have  only  paid  attention  to  the  con- 
tents of  these  sheets  which  refer  to  mathematics.3 

1  Translated  by  J.  M.  Child,  from  an  article  by  Dr.  Gerhardt  in  the  Sitzungs- 
berichtc  der  Koniglich  Preussischen  Akademie  der  Wissenschaften  zu,  Berlin, 
1891,  pp.  157-165.    The  notes  are  by  the  translator. 

2  These  highly  important  documents  ought  to  be  photographed  and  pub- 
lished in  facsimile. 

3  It  seems  a  pity  that  Gerhardt  has  not  given  the  contents  of  the  section 
labeled  "Mechanica,"  unless  indeed  this  is  all  non-mathematical;  there  may 


LEIBNIZ  IN  LONDON.  525 

The  sheet  relating  to  Leibniz's  first  visit  to  London,  of 
which  I  have  added  a  partial  transcript  under  the  heading 
I,  is  divided  on  both  pages  into  sections  [the  word  used  in 
the  original  is  Felder  —  columns,  but  it  will  be  seen  that, 
according  to  the  transcript  given  later,  the  sections  are 
horizontal  and  not  vertical],  in  which  Leibniz  has  en- 
tered all  that  he  considered  to  be  worth  noting.  While 
the  sections  labeled  "Chymica,"  "Mechanica,"  "Mag- 
netica,"  "Botanica,"  "Anatomica,"  "Medica,"  and  "Mis- 
cellanea" are  filled  up  with  an  extraordinary  number  of 
memoranda,  the  first  sections,  which  are  allotted  to  mathe- 
matical subjects,  are  very  poorly  filled.  That  labeled 
"Geometrica"  contains  a  note  that  is  especially  worth  re- 
marking :  "Tangents  to  figures  of  all  kinds.  Development 
of  geometrical  figures  by  the  motion  of  a  point  in  a  moving 
straight  line."4  In  all  probability  it  may  be  supposed  that 
this  refers  to  the  lectures  of  Barrow,  delivered  on  his 
method  of  tangents  at  the  University  of  Cambridge  down 
to  the  year  1669.  As  is  well  known,  the  method  of  Bar- 
row is  only  applicable  to  such  curves  as  can  be  expressed 
by  rational  functions.5  Newton's  name  was  mentioned  in 

be  in  it  some  intimation  that  would  lead  to  a  clue  as  to  the  origin  of  Leibniz's 
use  of  the  word  moment,  meaning  thereby,  not  Newton's  use  of  the  word,  but 
the  idea  now  familiar  to  us  in  the  determination  of  the  center  of  gravity  of 
an  area,  expressed  by  the  equation 

x  =  "ZaxfZa, 

where  a  is  the  element  of  the  area  distant  x  from  the  axis,  x  the  distance  of 
the  center  of  gravity  from  that  axis,  and  "Sax  is  the  sum  of  the  'first  moments 
of  the  elements'  or  'the  first  moment  of  the  whole  area.'  See  note  16,  later. 

4  "Tangentcs    omnium    figurarum.      Figurarum   geometricarum    explicatio 
per  motum  puncli  in  moto  lati." 

5  In  a  footnote,  Gerhardt  asserts  that  "Barrow's  Lectiones  Geometricae 
appeared  in  1672."    This  is  incorrect ;  for  they  were  published,  combined  with 
the  second  edition  of  the  Lectiones  Optics,  in  1670 ;  nor  can  Gerhardt  be  referring 
to  the  second  edition,  for  that  appeared  in  1674  and  then  as  a  separate  volume. 
Also,  I  have,  in  the  little  book  on  The  Geometrical  Lectures  of  Isaac  Barrow, 
published  by  the  Open  Court  Publishing  Co.,  given  reasons  for  supposing  that 
these  lectures  were  never  delivered  as  Lucasian  Lectures,  though  they  may 
have  formed  the  subject-matter  for  college  lectures  at  Gresham  and  Trinity. 
Again,  it  is  not  true,  although  "well  known,"  that  "the  method  of  Barrow  was 
only  applicable  to  such  curves  as  can  be  expressed  by  rational  functions" ; 
this  remark  is  even  only  partially  true  about  the  differential  triangle  method ; 
for,  as  I  have  shown  in  the  above-mentioned  book,  Barrow  had  a  complete 
calculus,  which  included,  among  other  things,  the  important  idea  of  substitu- 


526  THE  MONIST. 

the  "Optica."  Leibniz  has  the  remark:  "They  told  me 
about  a  certain  phenomenon  that  Barrow  confessed  he 
was  unable  to  solve.  Newton's  difficulty  has  so  far  not 
been  solved,  Father  Pardies  having  given  it  up."6  Ob- 
viously this  remark  applies  to  Newton's  experiment  on  the 
refraction  of  light  by  a  prism  and  to  the  decomposition  of 
white  sunlight,  and  especially  to  the  fact  that  a  circular 
solar  image  becomes  after  refraction  a  long  spectrum. 
Father  Pardies  of  Clermont  had  published  in  opposition 
to  Newton  his  "Two  Letters  containing  Animadversions 
upon  I.  Newton's  Theory  of  Light,"  in  the  Philosophical 
Transactions  of  1672,  together  with  a  letter  from  Newton. 
It  cannot  be  said  for  certain  that  Leibniz,  during  his 
first  stay  in  London,  met  with  any  of  the  great  English 
mathematicians ;  Wallis  lived  at  Oxford,  while  Barrow  and 
Newton  resided  at  Cambridge.7  Indeed,  it  is  made  a  matter 
of  plaint  by  Brewster,  the  biographer  of  Newton,  that  the 
Royal  Society  of  London  at  that  time  numbered  few  men 
of  distinguished  talents  who  were  in  a  position  to  perceive 
the  truth  of  the  optical  discoveries  of  Newton.  In  the 
letter  which  Leibniz  addressed  to  Oldenburg,  the  Secretary 
of  the  Royal  Society,  during  his  visit  to  London,  he  men- 

tion,  which  is  all  that  is  necessary  to  complete  the  "a-and-e"  method  and  make 
it  applicable  to  surds  and  fractions,  and  probably  was  thus  applied  by  Barrow 
in  working  out  his  constructions ;  but  the  whole  thing  was  geometrical,  which 
apparently  hid  the  inner  meaning  until  recently. 

To  my  mind,  the  mention  of  but  "tangents  and  local  motion"  points  out 
that,  on  Leibniz's  first  reading  of  Barrow,  he  only  perused  at  all  carefully  the 
first  five  lectures,  which  are  relatively  unimportant;  or  rather  it  confirms  an 
opinion  I  had  already  expressed  to  Mr.  P.  E.  B.  Jourdain. 

6  "Locuti  sunt  mihi  de  phaenomeno  quodam  quod  Barrovius  fatetur  se  sol- 
vere  non  posse.    Newtoni  difficultas  soluta  hactenus  non  est,  P.  Pardies  manus 
dante." 

7  It  seems  however  that  Leibniz  attended  the  meetings  of  the  Royal  So- 
ciety ;  at  any  rate  once,  when  he  exhibited  the  model  of  his  calculating  machine. 
It  would  be  interesting  if  the  roll  of  members  present  on  all  occasions  during 
this  period  could  be  obtained,  as  doubtless  they  were  kept.     For  such  men  as 
Ward  were  members  at  the  time  and  attended  the  meetings,  and  Ward  was, 
if  not  in  the  same  class  as  the  three  whose  names  are  given,  an  excellent  math- 
ematician ;  and,  Leibniz,  being  somewhat  of  a  notable,  on  account  of  his  con- 
nection with  the  Embassy  from  Mainz,  would  surely  be  introduced  to  all  emi- 
nent members  present. 


LEIBNIZ  IN  LONDON.  527 

tioned  that  he  had  met  by  accident  the  mathematician  Pell 
at  the  house  of  Boyle,  the  chemist.  The  conversation  fell 
upon  those  number-series  which  in  elementary  mathematics 
were  called  the  higher  arithmetical  series  and  whose  sums 
and  terms  were  found  by  the  help  of  differences.  Leibniz 
showed  that  he  had  gone  deeply  into  the  study  of  such 
series  and  had  partly  found  out  some  new  methods  for 
calculating  the  terms.8  Leibniz's  letter  to  Oldenburg  was 
dated  Feb.  3,  1673  (1672  O.  S.).9 

From  the  preceding  it  appears  that  what  Leibniz  learned 
with  reference  to  mathematics  from  his  first  visit  to  Lon- 
don was  quite  unimportant.10  The  chief  aim  of  his  stay  in 
London  was  to  be  elected  as  a  Fellow  of  the  Royal  Society ; 
and  this  came  to  pass,  owing  in  part  to  an  exhibition  of  a 
model  of  his  calculating  machine,  and  in  part  to  the  friendly 
offices  of  Oldenburg. 

After  his  return  to  Paris  at  the  beginning  of  March, 
1673, 'Leibniz  was  able  to  find  more  leisure  to  follow  up 
his  studies  without  hindrance;  the  political  mission  which 
was  the  cause  of  his  being  sent  to  Paris,  was  now  at  an 
end. 

It  may  be  regarded  as  certain  that,  before  his  first  visit 
to  London,  Leibniz  made  the  personal  acquaintance  of  the 
men  with  whom  he  corresponded  before  he  came  to  Paris, 
and  especially  Antoine  Arnauld  and  de  Carcavy.  The 

8  The  account  given  by  Leibniz  himself  in  the  Historic  (see  The  Monist 
for  October,  1916)   reads  thus:  "He"  [for  Leibniz  wrote  in  the  third  person, 
under  the  guise  of  "a  friend  who  knew  all  about  the  matter"]    "also  came 
across  Pell  accidentally,  and  described  to  him  certain  of  his  own  observations 
on  numbers,  and  the  latter  stated  that  they  were  not  new,  but  it  had  been 
recently  made  known  by  Nikolaus  Mercator. . . .   This  made  Leibniz  get  the 
work  of  Nikolaus  Mercator."     As  a  matter  of  fact  the  suggested  plagiarism, 
or  what  Leibniz  took  for  such  a  suggestion,  was  from  Mouton  and  not  from 
Mercator.     This  is  an  instance  of  the  lack  of  memory  from  which  Leibniz 
suffered ;  such  lack  as  caused  him  to  make  notes  of  all  important  points. 

9  See  Note  32,  on  the  introduction  of  the  Gregorian  calendar. 

10  I  cannot  see  what  reason  Gerhardt  has  for  this  statement,  considering 
the  contents  of  Barrow's  book,  which  we  know  that  Leibniz  had  purchased ; 
that  is,  unless  we  assume  either  that  Leibniz,  as  I  have  suggested,  did  not  at 
that  time  read  the  whole  of  Barrow,  or  failed  to  grasp  what  Barrow  had  given 
owing  to  his  (Leibniz's)  incomplete  knowledge  of  geometry. 


528  THE  MONIST. 

latter  belonged  to  the  circle  in  which  Pascal  moved 
Whether  at  that  time  Leibniz  had  made  the  acquaintance 
of  Huygens  is  not  quite  so  certain ;  at  any  rate  he  did  not 
come  into  close  relations  with  him  until  after  his  return 
from  London.  Huygens  presented  him  with  a  copy  of  his 
great  work,  Horologium  Oscillatorium,  which  had  just 
(1673)  been  published.  The  recognition  that  his  mathe- 
matical knowledge  at  that  time  was  insufficient  to  enable 
him  to  understand  the  contents  of  this  book,  combined  with 
a  reawakening  of  his  former  love  for  mathematics,  had  the 
effect  of  making  Leibniz  devote  himself  with  the  greatest 
fervor  to  the  study  of  mathematcial  subjects.  Cavalieri's 
method  of  indivisible  magnitudes,  the  writings  of  Gregory 
St.  Vincent,  the  letters  of  Pascal  (which  were  especially 
recommended  to  him  by  Huygens),  were  used  by  him  as 
guides  in  his  studies.  As  the  first-fruits  of  these  studies, 
he  obtained  the  theorem  that,  when  the  square  on  the 
diameter  of  a  circle  was  taken  as  unity,  the  area  of  the 
circle  was  expressed  by  the  infinite  series 

1-i  +  i- i+ ad  inf.     ™ 

O         O         I 

He  obtained  it  thus:  Instead  of  dividing  the  circle,  as  in 
the  method  of  Cavalieri,  into  trapezia  by  means  of  parallels, 
he  divided  it  into  triangles  by  lines  radiating  from  a  point ; 
the  areas  of  these  triangles  being  proportional  to  certain 
lines.  With  these  lines  as  perpendicular  ordinates  a  curve 
could  be  constructed  that  was  divided  by  these  ordinates 
into  trapezia,  each  of  \vhich  is  double  the  corresponding 
triangle.  In  this  way  Leibniz  obtained  a  curvilinear  fig- 
ure12 whose  area  was  double  that  of  the  circle,  but  which 
was  expressed  by  a  rational  function,  x  =  y2/(i  -f-  j2),18 

11  Leibniz's  own  date  for  the  discovery  of  this  result,  usually  alluded  to 
by  him  as  the  "Arithmetical  Tetragonism,"  is  1674;  '"But  in  the  year  1674  (so 
much  it  is  possible  to  state  definitely)  he  came  upon  the  well-known  Arith- 
metical Tetragonism; "  (See  Historia,  in  The  Monist,  Oct.,  1916. 

12  See  the  first  critical  note,  page  536. 

13  See  the  first  critical  note,  page  536. 


LEIBNIZ  IN  LONDON.  $2$ 

of  its  coordinates;  and,  using  a  method  that  was  similar 
to  that  employed  by  Mercator  for  the  equilateral  hyperbola, 
this  area  could  be  found  (Ouadratrix).14 

For  the  rest  of  Leibniz's  treatment,  see  the  hitherto 
unpublished  manuscript,  given  under  II  in  the  appendix 
that  follows. 

As  was  often  the  case  in  the  first  scientific  studies  of 
Leibniz,  intimations  of  the  great  problems  that  occupied 
his  attention  his  whole  life  through  are  found  here  in  his 
first  efforts  in  the  domain  of  higher  mathematics.  First 
is  it  to  be  remarked  that  Leibniz  abandoned  the  division 
of  curvilinear  figures  into  trapezia,  as  was  done  by  Cava- 
lieri,  and  instead  divided  them  into  triangles;  from  this 
he  was  led  to  the  "characteristic  triangle,"15  which  formed 
the  foundation  in  the  application  of  the  differential  calculus. 
Further,  Leibniz  constructed,  instead  of  the  proposed  curve, 
another  of  which  the  area  could  be  found  (the  "quadratrix" 
as  he  called  it) ;  this  method  of  procedure  frequently  oc- 
curs in  the  later  works  of  Leibniz  on  the  integral  calculus. 
Closely  connected  also  with  this  is  the  solution  of  the  in- 
verse method  of  'tangents,  that  is,  given  the  tangent,  to 
find  the  curve. 

In  these  first  efforts  of  Leibniz  in  the  domain  of  higher 
mathematics  is  clearly  to  be  seen  the  influence  of  his  study 
of  the  writings  of  Pascal.18  The  French  mathematicians 
Roberval  and  Pascal  did  not  consider  that  Cavalieri's 

14  Observe  that  Leibniz   (or  Gerhardt)  employs  this  word  in  a  different 
sense  from  that  of  Barrow,  with  whom  it  means  the  special  curve  whose  equa- 
tion is  v  =  (r  —  ;r)tan"Mr/2r,  a  curve  that  is  particularly  connected  with  the 
circle. 

15  This  contradicts  both  Gerhardt  and  Leibniz  himself,  who  said  that  he 
got  it  from  a  consideration  of  a  figure  used  by  Pascal  in  finding  the  content 
of  the  sphere.    See  also  the  critical  note  referred  to  in  12,  13  above. 

16  I  hope  to  consider  this  influence  in  a  later  number  of  The  Monist,  in 
connection  with  an  essay  by  Gerhardt  on  this  very  point ;  when  I  shall  en- 
deavor to  substantiate  an  opinion  I  have  formed  with  regard  to  the  earlier 
manuscripts  of  Leibniz,  which  were  discovered  by  Gerhardt,  and  of  which 
translations  appear  in   The  Monist   (April,  1917).     I  suggest  that  these  do 
not  represent  so  much  the  record  of  his  original  investigations  as  notes  made 
while  using  the  works  of  his  predecessors  as  text-books. 


530  THE   MONIST. 

method  was  consistent  with  the  rigorous  requirements  of 
mathematics;17  they  reverted  to  the  study  of  the  Greek 
mathematicians,  and  especially  to  the  writings  of  Archi- 
medes, combining  with  their  method  the  developments 
which  Kepler,  in  particular,  had  brought  about  by  the  in- 
troduction of  infinitely  small  magnitudes  into  geometry. 
Moreover,  in  connection  with  Pascal,  it  is  to  be  observed 
that  he  generalized  into  a  "barycentric  calculus"  the  proce- 
dure used  by  Archimedes  for  the  quadrature  of  the  parab- 
ola by  means  of  the  equilibrium  of  the  lever.18  This 
"calculus"  enabled  him  to  solve  problems  on  the  cycloid 
which  his  contemporaries  had  vainly  attempted.19  It  was 
not  unknown  to  Leibniz  that,  since  the  time  of  Pappus  of 
Alexandria,  quadratures  and  cubatures  had  been  calcu- 
lated by  the  aid  of  the  center  of  gravity  (Guldin's  rule, 
"Centrobaryca") ;  certainly  he  was  now  led,  by  the  works 
of  Pascal,  again  to  notice  the  methods  for  the  determina- 
tion of  the  center  of  gravity,  and  was  also  induced  to  at- 
tempt to  extend  and  perfect  them.  The  manuscript  of 

17  I  fail  to  see  how  this  statement  can  be  completely  reconciled  with  the 
following  well-known  quotation  from  the  "Lettre  de  A.  Dettonville  a  Carcavy" 
(1658): 

"J'ay  voulu  faire  cet  advertissement  pour  monstrer  que  tout  ce  qui  est 
demonstre  par  les  veritables  regies  des  indivisibles  se  demonstrera  aussi  a  la 
rigueur  et  a  la  maniere  des  anciens;  et  qu'ainsi  I'une  de  ces  Methodes  ne  differe 
de  I'autre  qu'en  la  maniere  de  parler;  ce  qui  ne  peut  blesser  les  personnes 
raissonnables  quand  on  les  a  une  fois  avertyes  de  ce  qu'on  entend  par  la"  (Vol. 
VIII,  p.  352). 

Pascal  also  says  on  p.  350:  "....  la  doctrine  des  indivisibles,  laquelle  ne 
peut  estre  rejettee  par  ceux  qui  pretendent  avoir  rang  entre  les  Geometres." 

That  is,  the  method  of  indivisibles  does  not  differ  from  the  method  of 
exhaustions,  except  in  the  way  the  argument  is  put;  and  that  the  former  must 
be  accepted  by  any  mathematician  with  pretensions  to  rank  among  geometers. 

The  page  reference  is  to  the  edition  of  Pascal's  Works  in  14  volumes,  in 
the  series,  Les  Grands  Ecrivains  de  la  France  (pub.  Hachette  et  Cie.,  Paris, 
1914). 

18  Pascal  calls  it  "la  balance."     It  is  worth  noting  in  this  connection  that 
Pascal  uses  the  word  "force"  and  not  "moment"  for  the  product  of  one  of  his 
weights  and  its  lever-arm ;  so  that  we  must  look  elsewhere  for  the  clue  to  the 
use  of  the  word  "moment"  in  this  sense  by  Leibniz. 

19  Several  of  the  problems  proposed  were  solved  by  Huygens,  de  Sluze, 
and  Wren;  but  by  special  methods,  which  did  not  satisfy  Pascal,  who  called 
for  a  general  method.    Later  (1670)  Barrow  gives  the  rectification  of  the  arc, 
as  a  special  case  of  a  general  theorem  (Lect  XII,  App.  3,  Ex.  2,  see  my  Bar- 
row, p.  177). 


LEIBNIZ  IN  LONDON.  531 

Leibniz  which  is  dated  October  25,  October  26,  October  29, 
November  I,  1675,  and  which  contains  the  investigation 
on  the  center  of  gravity,  is  headed,  "Analysis  Tetragonis- 
tica  ex  Centrobarycis."2 

It  is  worth  remarking  that  in  this  Leibniz  continues 
the  method  by  which  he  had  found  the  series  for  the  area 
of  the  circle.  Incidentally  these  studies  were  the  first  occa- 
sion for  the  introduction  of  the  symbol  for  a  sum,  i.  e.,  the 
integral  sign  (October  29,  1675);  from  this  as  the  an- 
tithesis, the  sign  for  the  difference,  i.  e.,  the  symbol  for 
differentiation,  resulted.21  The  equation  in  which  Leibniz 
first  introduced  the  sign  of  integration  was,  in  the  notation 
of  that  time: 


omn.  /LU    n  / 

n  omn.  omn.  - 

2  a 

that  is, 


(omn./)2  /  . 

v '   =  omn.  omn.  -  ; 

2  a 

for  which  Leibniz  writes 

•>>> 

2 
that  is,  when  /  =  ofy, 


After  his  return  to  Paris  in  March,  1673,  Leibniz  was 
in  constant  communication  with  Oldenburg,  the  Secretary 
of  the  Royal  Society;  the  subjects  being  almost  entirely 
mathematical.  In  this  way  he  obtained  his  knowledge 
of  the  work  of  the  English  mathematicians.  Oldenburg's 
mentor  on  all  mathematical  questions  was  John  Collins, 
who  possessed  a  very  wide  acquaintance  among  English 

20  A  translation  is  given  in  The  Monist,  April,  1917. 

21  See  the  second  critical  note,  page  543. 


532  THE   MONIST. 

mathematicians;  and  it  was  through  him  that  what  they 
had  done  was  communicated.  In  this  respect  special  men- 
tion is  to  be  made  of  the  letter  from  Oldenburg  to  Leibniz, 
dated  July  26,  1676,  in  which  Collins  informed  him  of  a 
collection  of  letters  from  English  mathematicians  that  he 
had  in  his  possession.  Collins  mentions  in  it  particularly 
that  script  of  Newton,  of  December  10,  1672,  in  which  the 
latter  makes  a  communication  about  his  method  for  tan- 
gents to  curves,  which  are  given  by  an  explicit  algebraical 
equation;  he  remarks  that  the  method  is  only  a  corollary 
to  a  general  procedure  for  solving  other  problems,  such 
as  those  relating  to  rectification,  determination  of  centers 
of  gravity  and  so  on.22  Collins  stated  in  addition  that,  be- 
sides what  this  letter  showed,  nothing  further  was  known 
at  that  time  about  Newton's  method.  It  was  on  account  of 
these  communications,  and  probably  also  on  account  of  a 
letter  from  Newton  to  Oldenburg,  of  which  Oldenburg  sent 
a  copy  to  Leibniz  at  Paris,  that  Leibniz  was  moved  to  make 
his  return  journey  to  Germany  in  October,  1676,  by  way  of 
London.  Leibniz  stayed  there  about  a  week;  he  made  the 
acquaintance  of  Collins,  who  willingly  let  him  have  access 
to /his  collection  of  treatises  and  letters.23  What  Leibniz 
found  in  them  that  he  thought  worth  noting  he  set  down 

22  Leibniz,  in  the  Ada  Eruditorum  for  the  year  1700,  says,  "I  can  affirm 
that,  when  in  1684  I  published  the  elements  of  my  Calculus,  I  did  not  know 
any  thing  more  of  Mr.  Newton's  inventions  in  this  kind,  than  what  he  formerly 
signified  to  me  by  his  letters,  viz.,  that  he  could  find  tangents  without  taking 
away  surds;.  ..."   As  Newton  says  in  the  article  in  Phil.  Trans.,  Vol.  XXIX, 
No.  342,  Anno  1714  (usually  called  the  "Recensio")  this  "is  very  extraordinary, 
and  wants  an  explanation." 

23  This  is  feasible,  but  there  is  another  alternative  given  by  Dr.  H.  Sloman 
(The  Claim  of  Leibniz  to  the  Invention  of  the  Differential  Calculus,  English 
edition,  pub.  Macmillan,  1860),  which  strikes  me  as  even  more  probable.    Slo- 
man's  points  are  as  follows:   (1)  It  is  highly  probable  that  Leibniz's  week  in 
London  was  the  last  week  of  that  month.     (2)    Oldenburg  had  then  in  his 
possession  two  letters  from  Newton  for  Leibniz,  dated  Oct.  24  and  26 ;  these 
he  showed  to  Leibniz.     (3)  As  Newton  himself  mentions,  these  were  blotted 
and  hastily  written ;  and  thus  Leibniz  asks,  on  this  account,  that  Oldenburg 
should  let  him  see  the  tract  of  Newton  to  which  they  refer ;  which  tract  Leibniz 
knew  was  in  the  possession  of  Oldenburg,  that  is,  a  copy  of  it.    For  the  details 
of  the  argument,  occupying  ten  quarto  pages,  see  the  above-mentioned  book 
by  Sloman,  pp.  97-106. 


LEIBNIZ  IN  LONDON.  533 

on  two  folios ;  the  one  has  the  heading,  "Excerpta  ex  trac- 
tatu  Newtoni  de  Analysi  per  aequationes  numero  termi- 
norum  infinitas."  This  is  the  paper  which  Newton  sent  in 
June,  1699,  to  Barrow,  from  whom  Collins  received  it  on 
July  30,  1699.  Collins  made  a  copy  of  it,  and  sent  the 
original  back;  and  the  original  was  printed  in  the  year 
1711.  The  other  sheet  has  the  heading,  "Excerpta  ex  Com- 
mercio  Epistolico  inter  Collinium  at  Gregorium."  A  partial 
transcript  of  both  these  sheets  follows  under  the  head- 
ing III. 

With  regard  to  the  extracts  from  Newton's  paper,  it 
is  to  be  remarked  that  Leibniz  was  interested  in  the  treat- 
ment of  algebraical  expressions  of  powers  and  in  the  turn- 
ing of  irrational  expressions  into  the  form  of  series  by 
means  of  division  and  root-extraction.  He  noted  indeed 
many  examples  in  their  entirety.  How  to  get  to  quadra- 
tures was  known  to  him;  he  merely  indicated  the  process 
by  the  sign  of  a  sum,  i.  e.,  by  the  symbol  of  integration. 
On  the  other  hand,  the  part  on  the  numerical  solution  of 
adfected  equations  was  new  to  him,  and  this  he  copied  out 
well-nigh  \vord  for  word;  this  is  the  well-known  New- 
tonian method  of  solution  of  equations  by  approximations. 
Leibniz  passes  over  as  well  known  to  him  the  remark,  made 
by  Newton  at  the  close  of  the  quadratures,  that  the  prob- 
lems of  rectification,  determination  of  the  content  of  solids, 
determination  of  the  centers  of  gravity,  can  be  solved  in 
the  same  way,  and  also  the  general  indication  of  the  process 
to  be  followed  in  such  cases.  Then  follows  the  solution  of 
inverse  problems,  for  instance,  to  find  from  the  area  the 
base,  that  is  the  axis  of  the  curve.  This  Leibniz  copied 
out  word  for  word.  In  the  same  way  Leibniz  has  extracted 
the  conclusion  of  Newton's  paper,  "Demonstratio  resolu- 
tionis  aequationum  affectarum."  At  the  end  of  his  manu- 
script Leibniz  adds:  "I  extracted  this  from  the  letter  of 


534  THE  MONIST. 

Newton,  August  20,  1672,  addressed  to  Newton."84  Prob- 
ably this  means  that  from  the  letters  referring  to  Newton, 
Leibniz  picked  out  the  letter  dated  August  20,  1672,  ad- 
dressed to  Newton.25  So  far  as  the  script  can  be  de- 
ciphered,26 its  contents  were  a  graphic  representation  of 
Newton's  method  of  solution  of  equations  by  approxima- 
tions by  means  of  Gunter's  scale.  Gunter's  line  had  been 
noted  by  Leibniz  on  his  first  visit  to  London. 

Of  quite  special  interest  to  Leibniz  were  the  letters  of 
mathematicians  which  Collins  had  collected;  on  a  second 
folio  he  made  excerpts  from  letters  from  James  Gregory. 
In  two  letters  from  Gregory  (1670)  was  Isaac  Barrow 
extolled  as  the  greatest,  not  only  among  living  writers, 
but  also  among  all  those  that  had  written  before  him 
(Barrow).  Further  Leibniz  found  among  these  letters  the 
letter  mentioned  above  of  Newton  to  Collins  of  December 
10,  1672  ;27  he  extracted  what  Newton  had  mentioned  with 
regard  to  his  method  of  finding  the  expression  for  the  tan- 
gent to  a  curve.  Leibniz  added  at  the  end  of  this  extract, 
"This  method  differs  from  that  of  Hudde  as  well  as  from 
that  of  Sluse,  in  that  irrationals  need  not  be  eliminated."28 

24  The  Latin,  "Excerpsi  ex  Epist.  Neutoni  20  Aug.  1672  ad  Neuton,"  as 
given  by  Gerhardt,  seems  somewhat  unintelligible ;  especially  the  word  Neuton. 
What  Collins  had  (or  what  Oldenburg,  as  suggested  by  Sloman,  had)  was  a 
copy  of  a  manuscript  that  Newton  had  sent  to  Barrow.     Gerhardt  says,  "so 
far  as  the  script  can  be  deciphered" ;  perhaps  the  word  Neuton  is  an  error 
of  transcription,  or  maybe  an  error  on  the  part  of  Liebniz,  due  to  the  juxta- 
position of  the  Neutoni  which  comes  just  before.    In  any  case,  note  25  applies. 

25  I  do  not  think  Gerhardt's  translation  of  the  word  excerpsi  is  correct. 

26  Gerhardt  does  not  state  whether  the  extract  is  badly  written  (this  would 
show  that  it  had  been  done  in  a  very  great  hurry,  for  Sloman  says  that  Leibniz, 
in  his  matter  for  publication,  wrote  a  beautiful  hand),  or  whether  spoilt  by 
age;  in  the  latter  case,  as  old-time  inks  contained  salts  of  iron,  the  manuscript 
might  be  restored  by  photography,  by  means  of  a  special  plate,  that  I  under- 
stand is  sometimes  used  for  detecting  forgeries  in  deeds  and  notes. 

27  The  letter  was  sent  to  Barrow  to  be  sent  on  to  Collins,  probably  with 
the  object  of  being  communicated  through  the  latter  to  others;  Collins  seems 
to  have  been  the  regular  channel  of  communication  at  this  period,  in  a  similar 
way  to  Mersenne. 

28  So  we  find  in  a  manuscript,  dated  July  11,  1677,  first  of  all  an  allusion 
to  Sluse's  method  of  tangents,  "in  which  the  equation  is  purged  of  irrational 
or  fractional  quantities" ;  then  the  remark,  "I  have  no  doubt  that  the  gentlemen 


LEIBNIZ  IN  LONDON.  535 

From  these  extracts  it  follows  that  the  contents  of  New- 
ton's letter  were  unknown  to  him  at  that  time  (Oct.,  1676)." 
Regarding  the  verbal  communications  that  Leibniz 
had  from  Collins  during  the  second  stay  in  London,  Collins 
wrote  to  Newton  from  London  on  March  5,  1677  (1676 
O.  S.),  that  the  representation  of  the  roots  of  an  equation 
by  a  series  was  discussed  between  them. 

It  is  clear  that  Leibniz  during  his  second  stay  in  London 
had  made  himself  more  familiar  with  the  results  obtained 
by  English  mathematicians  than  he  was  before.  The  ques- 
tion now  arises:  What  specially  occupied  his  attention? 
What  had  particular  influence  upon  his  studies  ?  It  is  seen 
that  what  Leibniz  found  in  Collins's  collection  relating  to 
algebraical  analysis  was  new  to  him  and  excited  his  in- 
terest; also  the  verbal  exchange  of  ideas  between  himself 
and  Collins  was  upon  the  same  subjects. 

On  the  other  hand,  as  regards  the  infinitesimal  calculus, 
Leibniz  obtained  nothing  during  his  second  visit  to  Lon- 
don; he  had  made  a  progress,  by  the  introduction  of  his 
algorithm  into  the  higher  analysis,  beyond  anything  that 
came  to  his  knowledge  in  London.30  Also  these  algebraical 
results,  at  least  for  the  next  period,  left  behind  no  lasting 
impression;  for  among  Leibniz's  papers  is  to  be  found  an 
extensive  treatise,  written  on  board  the  ship  that  carried 
him  from  London  to  Holland,  wherein  he  considered  the 

I  have  just  mentioned  know  the  remedy  that  is  necessary  to  apply";  then  fol- 
lows the  rule  for  a  quotient,  and  the  remark  that  this  will  be  sufficient  for 
fractions ;  lastly  the  rule  for  powers,  with  the  remark  that  this  will  be  sufficient 
for  irrationals.  Later,  he  says,  "This  method  has  more  advantage  over  all 
others  that  have  been  published  than  that  of  Slusius  over  all  the  rest,  because 
it  is  one  thing  to  give  a  simple  abridgment  of  the  calculation,  and  quite  another 
thing  to  get  rid  of  reductions  and  depressions." 

Thus,  after  the  sight  of  Newton's  paper,  his  whole  business  has  been  to 
improve  the  method  of  Sluse. 

29  I  read  it  quite  otherwise ;  he  has  had  information  of  some  kind,  whether 
from  Oldenburg  direct  or  from  Tschirnhaus,  while  in"  Paris,  and  visits  London 
with  the  express  intent  of  seeing  the  original  papers. 

30  See  the  third  critical  note,  page  546. 


536  THE  MONIST. 

fundamental  principles  of  motion,  in  the  form  of  a  dia- 
logue.31 

It  was  in  the  letter  to  Oldenburg  written  from  Amster- 
dam on  November  18/28,  1676™  which  Collins  spoke  of 
in  the  letter  to  Newton  mentioned  above,  that  Leibniz  first 
refers  to  the  subject  of  the  problem  of  tangents,  and  re- 
marked that  the  method  of  Slusius  was  not  yet  very  per- 
fect.33 

KARL  IM MANUEL  GERHARDT. 

CRITICAL  NOTES  ON  GERHARDT'S  ESSAY. 

BY  THE  TRANSLATOR. 

NOTE  I.    The  origin  of  Leibniz's  "transmutation  of  figures" 
(Referred  to  in  footnotes  12,  13,  IS.) 

In  the  manuscript,  which  follows  under  heading  II,  Leibniz 
appears  to  attach  very  considerable  importance  to  the  method  of 
transmutation  of  figures,  and  to  claim  that  he  had  originated  it. 
This  claim  is  not  incontestible ;  indeed  I  am  almost  inclined  to  think 
it  is  a  deliberate  plagiarism  to  start  with ;  but  Leibniz  has  perceived 

31  Could  this  possibly  have  had  its  rise  in  an  effort  on  the  part  of  Leibniz 
to  understand  fluxions,  or  rather  the  idea  of  fluxions  as  he  had  found  it  in 
Newton's  paper? 

32  In  1582,  Gregory  XIII  had  directed  10  days  to  be  suppressed  from 
the  calendar,  then  in  accordance  with  the  Julian  system  of  intercalation,  in 
order  to  allow  the  error  which  had  crept  into  the  time  of  the  vernal  equinox, 
by  which  Easter-day  was  settled,  to  be  put  right.     The  Gregorian  calendar 
was  introduced  into  all  Catholic  countries  the  same  year,  in  Scotland  in  1600, 
in  the  protestant  states  of  Germany  in  1700,  but  not  in  England  until  1752.    At 
the  same  time  the  commencement  of  the  legal  year  in  England  was  altered 
from  May  25  to  January  1 ;  thus  we  frequently  find  two  years  given  for  dates 
between  January  1  and  May  25 ;  while  there  are  two  days  of  the  month  given 
for  all  months  of  the  year.    For  instance,  February  1673  in  the  new  Gregorian 
calendar  would  be  only  February  1672  in  the  Julian,  distinguished  by  the  letters 
O.  S.  (Old  Style)  ;  and  this  date  was  written  February  1672A-     Similarly  the 
date  November  "/2»,  1676,  was  the  28th  of  November  in  the  New  Style,  and 
the  18th  in  the  Old  Style,  the  number  of  the  year  being  the  same,  since  the  day 
did  not  lie  between  the  1st  of  January  and  the  25th  of  May. 

83  "Methodus  Tangentium  a  Slusio  publicata  nondum  rei  fastigium  tenet." 
These  are  Leibniz's  words ;  Gerhardt  omits  to  translate  the  word  publicata, 
which  probably  refers  to  the  publication  in  the  Phil.  Trans,  of  1672,  by  Slusius, 
of  the  rules  of  his  method,  illustrated  by  examples.  Sluse  had  probably  im- 
proved upon  this  before  1676,  but  there  is  no  evidence  on  this  point.  It  would 
seem  as  if  the  subsequent  work  by  Leibniz,  culminating  in  the  manuscript  of 
July  11,  1677,  was  largely  an  attempt  to  perfect  the  rule  of  Sluse  as  a  rule, 
and  that  Leibniz,  if  ever,  did  not  appreciate  the  idea  fundamental  in  the  cal- 
culus, namely  that  of  rates,  until  very  much  later. 


LEIBNIZ  IN  LONDON.  537 

in  it  something  which  the  original  author  did  not.  Can  it  by  any 
chance  be  the  case  that,  in  conformity  with  several  other  instances 
of  Leibniz's  bad  memory  for  details,  he  is  confusing  author  and 
subject,  when  he  speaks  of  "the  great  light  that  suddenly  dawned 
on  him,  which  the  author  had  missed,"  the  reference  being  to  Pascal 
and  the  discovery  of  the  differential  triangle?  Can  it  be  that  the 
true  connection  is  that  in  considering  the  original  work  of  the  author 
of  such  transmutations  of  figures,  he  perceived  the  method  for  the 
arithmetical  quadrature  ?  For  here  he  really  has  found  a  thing  that 
the  author  missed  though  it  was  almost  staring  him  in  the  face, 
his  discovery  being  due  to  a  habit  that  Leibniz  had  of  writing  down 
everything  that  he  could  get  out  of  any  particular  figure  or  bit  of 
work  that  he  had  in  hand,  whether  it  was  relevant  or  irrelevant. 

Wallis  and  Pascal  had  both  hinted  at  the  method,  i.  e.,  had  used 
it  in  special  cases,  namely  for  proving  the  equivalence  of  the  parab- 
ola and  the  spiral ;  and  Leibniz  was  familiar  with  both  these  authors. 
Again,  James  Gregory  had,  in  the  words  of  Barrow  (Led.  Geom., 
Lect.  XII,  App.  3,  foreword  to  Prob.  IX),  "set  on  foot  a  beautiful 
investigation  about  involute  and  evolute  figures,"  i.  e.,  polar  and 
rectangular  figures  equal  in  area  to  one  another.  Of  course,  Leib- 
niz may  not  have  seen  this  work  of  Gregory  until  later ;  probably 
not,  although  in  one  of  his  manuscripts  he  gives  a  theorem  of 
Gregory ;  this  however  does  not  count  for  much,  for  the  very  same 
theorem  is  given  by  Barrow  (see  my  Barrow,  p.  130)  and  we  know 
that  Leibniz  had  a  Barrow  in  his  possession.  This  book,  judging 
by  his  words,  "as  in  Barrow,  when  his  Lectures  appeared,  in  which 
I  found  the  greater  part  of  my  theorems  anticipated,"  Leibniz  wishes 
to  make  his  friends  believe  was  the  1674  edition,  and  not  the  edition 
of  1670,  which  he  bought  on  his  first  visit  to  London.  Why  did 
Leibniz  wish  to  conceal  this  fact  ?  I.  assert  that  the  reason  for  doing 
so  was  the  fear  that  seemed  always  to  overshadow  him,  the  fear  of 
being  accused  of  plagiarism,  whether  such  was  a  true  or  a  false 
charge.  I  am  firmly  convinced  that  Leibniz  got  his  transmutation 
of  figures  from  Barrow ;  to  this  conclusion  I  have  only  just  come,  it 
never  having  entered  my  head  to  look  for  it  at  the  time  that  I  wrote 
my  articles  for  The  Monist  of  October,  1916,  April  and  July,  1917. 

Before  I  bring  forward  my  arguments,  it  is  right  to  state  as  a 
preliminary  that,  just  as  in  calculus  nowadays  we  usually  draw  a 
curve  with  its  convexity  downward,  and  draw  the  tangent  to  meet 
the  horizontal  axis  beneath  the  curve,  so  Barrow  drew  his  curves 
with  the  concavity  downward  in  many  cases,  mostly,  I  think,  in 
order  to  fit  the  diagrams  conveniently  on  the  old-fashioned  folding 
plates  of  diagrams,  that  in  those  days  were  added  in  batches  at 
the  end  of  a  book  (see  a  specimen  I  have  given  at  the  end  of  my 
Barrow)  ;  in  other  cases,  he  draws  his  figure  on  the  left-hand  side 
of  the  axis.  Whichever  figure  he  draws,  he  always  did  one  thing, 
namely,  he  drew  any  supplementary  figure  he  had  need  of  on  the 
other  side  of  his  axis  or  base.  Leibniz  almost  invariably  drew 


538 


THE   MONIST. 


his  curve  on  the  right-hand  side  of  a  vertical  axis,  and  supplemen- 
tary figures  on  the  same  side.  Hence,  in  the  extract  from  Barrow 
given  below,  I  am  to  be  excused  for  failing  to  notice  before  what 
is  more  than  a  mere  similarity. 

In  the  following  extract  from  Barrow  (Lect.  XI,  Prop.  24), 
I  have  added  Barrow's  proof,  which  I  thought  unnecessary  to  give 
in  my  book;  the  figures  given  are  Barrow's  own  on  the  left,  which 
has  been  "up-ended"  on  the  right ;  the  latter  is  to  be  compared  with 
the  several  figures  by  Leibniz. 

Barrow's  Lectiones  Geometricae,  Lect.  XI,  Prob.  24. 
If  DOK  is  any  curve,  D  a  given  point  on  it,  and  DK  any 
chord ;  also  if  DZI  is  a  curve  such  that  when  any  point  M  is  taken 
in  the  curve  DOK,  DM  is  joined,  DS  is  drawn  perpendicular  to 
DM,  MS  is  the  tangent  to  the  curve,  DP  is  taken  along  DK  equal 
to  DM,  and  PZ  is  drawn  perpendicular  to  DK,  so  that  PZ  is  equal 
to  DS ;  in  this  case  the  space  DZI  is  equal  to  twice  the  space  DKOD. 


Fig.  1. 


Fig.  2. 


For  let  KP  be  considered  to  be  indefinitely  small,  and  let  DT 
be  perpendicular  to  DK  and  KT  the  tangent  to  the  curve  DOK. 
Then,  drawing  the  arc  MP,  we  have  as  before, 

KP :  PM  =  KD :  DT  =  KD :  KI,  and  hence  KP .  KI  =  PM .  KD. 
Take  another  small  part  PQ  and,  with  center  D,  draw  an  arc  QN 
through  Q  cutting  the  chord  DM  in  R ;  then  as  before, 

MR:RN  =  MD:DS,  PQ:RN  =  MD:PZ,  PQ.PZ  =  RN.MD; 
and  so  on  one  after  the  other.     Therefore,  it  is  evident  that  the 
sum  of  all  the  rectangles  KP.KI,  PQ.PZ,  etc.,  is  equal  to  the  ag- 
gregate of  all  the  spaces  PM .  KD,  RN .  MD,  etc. ;  that  is,  the  space 
DKI  =  2  times  the  space  DKOD. 

The  words  I  have  italicized  refer  to  Prop.  22,  in  which  he  uses 


LEIBNIZ  IN  LONDON. 


539 


a  similar  though  rather  more  complicated  figure  to  reduce  a  polar 
area  to  a  rectangle  of  which  one  side  is  a  given  straight  line,  and 
explains  that  the  reasoning  depends  on  the  fact  that  the  line  DK  is 
divided  into  infinitely  small  parts.  Compare  the  words  I  have  ital- 
icized with  the  description  of  Leibniz's  method :  "the  areas  of  these 
triangles  being  proportional  to  lines. 

Further,  Barrow  proceeds  in  Prop.  25  to  prove  the  equivalence 
of  the  spaces  formed  (i)  by  applying  each  MS  to  the  base  and  (ii) 
by  applying  each  chord  to  the  arc,  previously  rectified.  And  he  winds 
up  with  the  words:  "Should  any  one  explore  and  investigate  this 
mine,  he  will  find  very  many  things  of  this  kind.  Let  him  do  so 
who  must,  or  if  it  pleases  him." 

This  all  suggests  that  Leibniz  did  explore  this  mine,  that  he 
did  not  invent  the  method  of  transmutation  of  figures  for  himself, 


Fig.  3. 


Fig.  4. 


that  he  did  find  very  many  things  of  this  kind,  and  that  Barrow  had 
missed  the  arithmetical  quadrature  construction ;  this  Leibniz  ob- 
tained through  his  regular  practice  of  working  every  mine  right 
out,  to  keep  up  Barrow's  simile.  Further  comment  is  needless,  I 
think,  after  a  comparison  of  Barrow's  figure  (the  up-ended  version) 
with  the  figures  of  Leibniz  given  above. 

Fig.  3  occurs  in  a  manuscript  November  21,  1675,  which  ac- 
cording to  Leibniz  is  at  least  a  year  after  he  had  discovered  the 
arithn::tical  quadrature;  and  yet  it  has  a  heading,  "A  new  kind  of 
Trigonometry  of  indivisibles,  etc."  In  this  figure  it  is  to  be  noticed 
that  he  has  the  perpendicular  to  the  chord  BC,  agreeing  with  Bar- 
row's DS  and  DT,  but  has  not  the  tangent  at  the  vertex  that  was 
necessary  for  the  demonstration  of  the  arithmetical  quadrature.  In 
the  working  in  connection,  he  considers  the  similarity  of  all  the  tri- 


54O  THE   MONIST. 

angles  possible,  and  notes  as  one  point  that  "the  sum  of  all  the  tri- 
angles or  the  area  of  the  figure  is  equal  to  the  products  of  the  AB's 
into  the  CE's,  which  is  Barrow's  proof  of  Prop.  24  above. 

Fig.  4  is  the  figure  given  in  the  Historia  (see  Monist  for  Oct., 
1916),  in  connection  with  the  explanation  of  how  he  found  the  area 
of  the  circle.  Notice  the  difference  between  this  figure  and  the 
one  given  in  the  manuscript  that  follows  under  the  heading  II,  also 
that  the  description  there  given  of  the  way  in  which  he  was  led  to 
it  is  much  more  natural.  This  is  probably  the  true  version,  for  the 
use  of  the  notation  B,  (B),  ((B)),  points  out  that  it  was  written 
at  a  comparatively  early  period,  before  Leibniz  had  adopted  the  pre- 
fix notation,  1B,  2B,  3B.  In  the  account  in  the  Historia,  to  which 
Fig.  4  applies,  Leibniz  says,  "he  once  happened  to  have  occasion  to 
break  up  an  area  into  triangles  formed  by  a  number  of  straight  lines 
meeting  at  a  point,  and  he  perceived  that  something  new  could 
be  readily  obtained  from  it."  I  suggest  that  the  occasion  was  most 
probably  while  he  was  digging  in  Barrow's  mine !  This  is  the  reason 
why  he  has  in  the  Historia  given  the  figure  more  according  to  his 
usual  practice,  and  different  from  the  figure  in  the  earlier  manu- 
script, which  is  too  much  like  a  copy  of  Barrow's  (query,  where  did 
Barrow  get  it  from?).  With  regard  to  the  figure  and  proof  in  the 
manuscript  which  follows,  we  find  that  the  reasoning  there  given  is 
unsound,  unless  Gerhardt  has  given  us  a  slightly  erroneous  diagram ; 
for  Leibniz  apparently  does  not  perceive  that  the  ordinates  BA, 
which  are  equal  to  the  corresponding  CE,  must  pass  through  the 
respective  points  D,  before  he  can  say  that  one  figure  is  double  the 
other.  Hence  I  conclude  that  at  the  date  of  this  manuscript,  the 
demonstration  was  imperfect  and  that  he  had  no  proof  until  he  dug 
in  Barrow's  mine ;  in  support  of  which  conclusion  I  will  quote  from 
the  Recensio,  mentioned  in  footnote  22.  "This  quadrature,  com- 
posed in  the  common  manner,  he  began  to  communicate  at  Paris  in 
the  year  1675.  The  next  year  he  was  polishing  the  demonstration 
of  it,  to  send  it  to  Mr.  Oldenburg,  in  recompense  for  Mr.  Newton's 
Method,  as  he  wrote  to  him  May  12,  1676;  and  accordingly  in  his 
letter  of  August  27,  1676,  he  sent  it,  composed  and  polished  in  the 
common  manner."  This  polishing,  I  take  it,  consisted  in  making 
the  slight  but  important  alterations  in  the  demonstration  and  figure, 
from  those  given  in  the  manuscript  II  that  follows,  to  those  given  in 
the  Historia. 

What  had  he  then  got  in  July  1674,  when  he  wrote  to  Olden- 
burg saying  that  he  had  got  a  wonderful  Theorem,  which  gave  the 
area  of  a  circle,  or  any  sector  of  it  exactly,  in  a  series  of  rational 
numbers?  Or,  when  in  the  October  following,  October  26,  1674, 
he  wrote  to  say  that  he  had  found  the  circumference  of  a  circle  in  a 
series  of  very  simple  numbers;  and  also  by  the  same  "method"  (a 
favorite  expression  of  Leibniz)  any  arc  whose  sine  was  given? 
It  was  impossible  that  Leibniz  could  have  had  the  two  things  that 
I  have  italicized;  or  at  least,  the  latter  was  impossible  to  him,  be- 


LEIBNIZ  IN  LONDON.  541 

cause  the  only  way  for  him  to  obtain  it  exactly,  i.  e.,  to  know  the 
law  of  his  series,  was  as  yet  unknown  to  him ;  unless  we  are  to  as- 
sume, contrary  to  his  assertion,  that  the'  binomial  theorem  was 
known  to  him,  which  would  involve  his  also  having  seen  or  been 
told  about  other  parts  of  Newton's  work.  The  only  way  open  to 
Leibniz  was  to  find  the  square  root  of  I-*2,  and  then  its  reciprocal 
by  division ;  and  this  would  not  give  him  the  law  of  the  series,  even 
if  we  assume  that  his  knowledge  of  integration  was  sufficient  to 
enable  him  to  proceed  any  further.  From  his  manuscripts  it  does 
not  seem  that  even  up  to  Nov.  1675  he  had  any  further  knowledge 
of  integrations  than  that  omn.x  =  x2/2,  and  omn.jir2  =  x3/3 ;  but  as 
he  says  that  he  knows  the  latter  from  the  quadrature  of  the  parabola, 
there  is  some  possibility  that  he  might  have  been  able  to  integrate 
every  integral  power  of  the  variable  from  his  reading  of  Wallis  and 
Mercator. 

However,  there  is  the  strongest  probability  that  he  had  not  got 
any  proof  for  the  two  things  italicized,  and  that  the  quadrature  was 
in  the  same  category.  Where  then  had  he  obtained  it  ?  We  find  that 
in  December,  1670,  Gregory  had  found  out  for  himself  Newton's 
method  of  series;  and  two  months  later,  February  15,  1671,  sent 
several  theorems  to  Collins,  one  of  which  was  that  now  known  as 
"Gregory's  series."  "And  Mr.  Collins  was  very  free  in  communi- 
cating what  he  had  received  both  from  Mr.  Newton  and  Mr.  Greg- 
ory, as  appears  by  his  letters  printed  in  the  Commercium"  ( from  the 
Recensio).  One  can  imagine  that  Oldenburg  would  be  one  of  the 
first  to  receive  the  information,  and  that  for  a  certainty  it  would  be 
passed  on  to  Leibniz.  I  think  then  that  Leibniz  perceived  that  by 
putting  x=  1  in  Gregory's  series,  and  making  the  radius  of  the  circle 
equal  to  unity,  he  could  get  an  arithmetical  quadrature;  from  that 
time  onward  he  looked  for  a  proof  by  pure  geometry,  and  found  it 
after  reading  Barrow's  proposition  referred  to  above;  if  we  assume 
the  possibility  of  integration  of  integral  powers,  it  was  an  easy  step 
to  find  that  the  series  he  had  to  integrate  was  y2/(l+y2),  and  all 
he  had  to  look  for  on  his  figure  was  a  line  of  this  length.  This  very 
well  accords  with  the  description  of  the  way  in  which  he  found  his 
demonstration,  as  given  in  the  manuscript  which  follows  under  the 
heading  II. 

Lastly,  in  connection  with  the  suggestion  that  I  have  made 
above,  namely,  that  Leibniz  had  another  method  for  his  arithmetical 
quadrature  than  those  he  has  given,  there  is  one  method  that  is 
bound  up  with  the  change  that  he  made  from  the  Pascalian  char- 
acteristic triangle  which  he  used  at  first,  to  the  Barrovian  differential 
triangle  (see  my  note  in  The  Monist,  Oct.,  1916,  p.  615).  In  Example 
5  of  the  method  of  the  differential  triangle  (see  my  Barrow,  p.  123), 
Barrow  has  found  the  subtangent  for  the  curve  y  =  tznx,  from  a 
consideration  of  the  figures  on  next  page,  and  finds  that 

>=  — rL—  ™.  =  ^.BG 
rr+mm          CG2 


542 


THE   MONIST. 


where  r  is  the  radius  of  the  circle,  m  is  the  ordinate  MP,  which  is 
equal  to  BG,  and  t  is  the  subtangent  TP. 


Fig.  5. 


Now  if  we  put  the  radius  equal  to  unity,  and  for  the  ratio  t/m 
substitute  what  was  known  by  Leibniz  to  be  equal  to  it,  namely, 
QP/RM  or  EF/GH  (by  construction),  we  have  the  sum  of  all  the 
EF's  is  equal  to  the  sum  of  ordinates  equal  to  CK2  ( radius  =1) 
applied  to  G  at  right  angles  to  BG.  Analytically,  calling  BG  z,  we 
have 


arc  BE=sum.omn. 


1  + 


applied  to  the  line  z  \ 


hence  by  division 

arc  BE=sum.omn.(l—  z*  +  z*—  z*  +  etc.  ) 


etc. 

I  can  hardly  see  how  Leibniz  could  have  missed  this  with  his 
analytical  mind,  even  although  Barrow  has  missed  it;  but  there  is 
a  strong  probability  that  at  the  time  of  writing,  Barrow  had  not  seen 
the  quadrature  of  the  hyperbola  by  Mercator,  and,  if  he  had,  such 
algebraical  work  would  not  have  appealed  to  him  at  all. 

As  far  as  I  can  make  out,  there  is  only  one  other  alternative, 
which  involves  a  direct  contradiction  of  Leibniz's  own  statement  ; 
that  is  that  his  proof  was  not  by  the  transmutation  of  figures  in  the 
first  instance.  Color  is  lent  to  this  view  by  a  letter  of  Leibniz  and 
other  papers,  quoted  by  Sloman  (pp.  131ff,  in  the  English  edition 
of  the  work  referred  to  in  footnote  23)  ;  also  even  by  a  passage  in 
the  Historia  (see  Monist,  Oct.,  1916,  p.  599),  where,  while  giving 
the  story  of  the  discovery  of  the  arithmetical  tetragonism,  Leibniz 
distinctly  hints  at  an  algebraical  method;  for  he  says  immediately 
afterwards,  "The  author  obtained  the  same  result  by  the  method  of 
transmutations,  of  which  he  sent  an  account  to  England."  This 
reads  as  if  he  had  another  method  in  addition  to  the  method  by 
transmutations. 

Let  us  consider  this  algebraical  method.  To  square  the  circle, 
Leibniz  has  to  integrate  ^/(l-x2)=y,  say;  let  y=\-xz,  then 
y=  (l-22)/(l+22),  which  is  rational;  moreover,  he  would  also  have 
been  able  to  have  substantiated  his  statement  that  at  this  time  he 


LEIBNIZ  IN  LONDON.  543 

also  had  a  proof  of  the  series  for  the  arc  whose  sine  was  given,  for 
which  he  would  only  have  had  to  integrate  \/^(\-xz).  But  one 
cannot  conceive  that  Leibniz  had  any  means  of  expressing  the  ele- 
ment of  z  in  terms  of  the  element  of  x.  Geometrically,  he  was  in- 
capable of  it,  without  using  Barrow's  infinitesimal  method ;  and  of 
this  we  find  the  first  instance  in  a  manuscript  dated  November  1, 
1675.  Algebraically,  he  could  not,  for  at  this  same  date  he  could 
not  differentiate  a  product.  How  then  are  we  to  account  for  the 
fact  that  he  says  he  has  a  method  for  demonstrating  both  series 
for  the  arc,  given  the  sine  or  the  tangent  ?  I  think  I  can  answer  this. 
Many  times  we  find  assertions  made,  not  only  by  Leibniz  in  those 
times,  but  by  others  in  other  times,  of  the  possession  of  discoveries, 
when  all  that  the  assertor  has  is  the  idea  of  how  they  may  be  ob- 
tained. Thus,  in  the  passage  quoted,  the  concluding  statement  is, 
"and  thus  again  all  that  remains  to  be  done  is  the  summation  of 
rationals."  So  that  if  we  accept  this  alternative  we  are  bound  to 
come  to  the  conclusion  that  Leibniz  did  not  yet  recognize,  what  he 
ought  to  have  done  from  the  work  of  Pascal,  that  an  area  was  not 
a  mere  summation  of  lines,  but  of  rectangles  formed  by  these  lines 
ordinated  at  certain  definite  points  along  a  straight  line.  That  is  to 
say,  he  did  not  recognize  the  fundamental  principle  of  integration, 
namely,  the  importance  of  the  factor  dx  or  ds.  When  he  had  to 
write  out  his  proof  he  found  that  the  summation  of  (l-2*)/(l+.sr2) 
or  its  reciprocal  was  beyond  him ;  or  rather  that  the  series  he  found 
by  Mercator's  method  was  not  correct ;  he  had  to  resort  to  the  geo- 
metrical proof,  of  which  he  got  the  idea  by  digging  in  Barrow's 
mine,  as  above;  he  found  that  this  would  not  work  for  the  other 
series ;  and  consequently  he  dropped  all  claim  to  the  second  series. 
In  his  letters  of  1676,  therefore,  we  find  him  offering  to  send  New- 
ton the  proof  of  his  quadrature  in  return  for  the  method  of  proof 
of  the  series  for  the  arc  when  the  sine  is  given. 

Thus  I  come  to  the  conclusion  that  Leibniz  obtained  these  series 
in  some  way  by  correspondence,  thought  he  had  got  a  proof  of  his 
own,  (which  turned  out  to  be  incorrect),  and  much  later  did  obtain 
a  proof  of  his  arithmetical  quadrature  by  the  transmutation  of 
figures,  after  obtaining  the  idea  from  Barrow.  As  the  special  case, 
when  x  =  l\\z  radius,  had  not  been  specifically  mentioned  by  Gregory, 
Leibniz  considered  that  he  had  a  right  to  claim  it,  more  particularly 
as  he  thought  he  had  devised  a  proof  for  it,  if  it  was  necessary  to 
produce  one ;  for  of  course,  Gregory  had  given  no  proof  according 
to  the  usual  custom  of  the  time.  Then,  when  he  did  find  a  proof, 
after  having  found  that  his  original  idea  was  hopeless,  one  can 
hardly  blame  him  for  sticking  to  his  claim. 

NOTE  2.     On  the  introduction  of  the  Leibnizian  algorithm. 

(Referred  to  in  footnote  21.) 
The  two  passages  in  which  the  signs  for  integration  and  dif- 


544  THE  MONIST. 

ferentiation  are  respectively  introduced  occur  in  the  manuscript  of 
October  26,  1675. 

i.  "It  will  be  useful  to  write  /  for  omn.,  so  that  //  =  omn.  /,  or  the 
sum  of  the  I's." 

ii.  Not  for  some  time  is  the  sign  for  differentiation  introduced, 
and  then  in  these  words :  "I  propose  to  return  to  former  considera- 
tions. Given  /  and  its  relation  to  x,  to  find  //.  Now  this  comes 
from  the  contrary  calculus,  that  is  to  say  if  /  l  =  ya.  Let  us  assume 
that  l  =  ya/d,  or  as  /  increases,  so  d  will  diminish  the  dimensions. 
But  /  means  a  sum,  and  d  a  difference.  From  the  given  y,  we  can 
always  find  ya/d  or  /,  or  the  difference  of  the  y's.  Hence  one 
equation  may  be  changed  into  the  other, " 

Now  of  these  the  introduction  of  the  symbol  for  integration 
can  no  more  be  called  an  invention  than  the  use  of  2  to  stand  for 
"the  sum  of  alj  such  terms  as."  It  was  simply,  as  Leibniz  himself 
says,  a  convenient  and  useful  abbreviation  for  sum.omn.  or  omn. 
It  is  nothing  more  or  less  than  the  long  s  then  in  general  use ;  indeed 
it  was  so  thought  of  by  contemporary  mathematicians,  Newton  for 
one  at  any  rate,  for  we  find  in  the  Recensio  the  passage,  "Mr.  Leib- 
niz has  used  the  symbols  sx,  sy,  ss  for  the  sums  of  ordinates  ever 
since  the  year  1686."  This  may  have  been  an  instance  of  prejudice, 
or  perhaps  the  printers  of  the  Phil.  Trans,  may  not  have  had  an 
integral  sign  in  their  fonts  of  type;  but  it  shows  up  the  fact  that 
the  English  accepted  it  as  the  initial  letter  of  the  \yord  "summa." 

Now  let  us  consider  the  introduction  of  the  letter  d.  Gerhardt 
says  that  it  resulted  as  antithesis  to  the  sign  /.  How  he  can  possibly 
derive  this  from  the  context  I  cannot  surmise.  I  am  well  aware 
that  in  another  passage  he  was  unable  to  assign  a  meaning  to  the 
introduction  of  a  letter,  which  was,  to  me,  clearly  used  for  the  simple 
purpose  of  keeping  the  dimensions  correct.  We  have  this  use  again 
in  the  present  passage.  Leibniz  knows  that  the  sum  of  the  lengths, 
/  /.  is  an  area :  hence  taking  y  to  represent  a  length,  given  in  terms 
of  x,  he  introduces  the  length  denoted  by  a  to  give  with  y  the  area 
of  a  rectangle.  Therefore  he  argues  that  /  must  be  an  area  divided 
by  a  length,  and  he  writes  I  =  ya/d,  where  d  is  another  length,  intro-. 
duced  to  keep  the  dimensions  correct.  This  is  clear  from  the  sen- 
tence that  follows  next :  "so  will  d  diminish  the  dimensions." 

So  far  the  sequence  of  ideas  is  easy  to  follow,  and  there  is  not 
the  slightest  trace  of  any  concept  of  differentiation,  nor,  if  the  /'s 
are  ordinated  to  any  axis,  any  trace  of  a  connection  between  d  and 
an  element  of  that  axis.  The  difficulty  begins  with  the  next  sentence : 
"But  /  means  a  sum,  and  d  a  difference."  The  first  idea  that  strikes 
one  is  that  this  was  added  later,  after  that  he  had  found  out  the 
connection  between  the  inverse-tangent  problem  and  quadratures. 
Gerhardt  gives  no  suggestion  on  the  point,  so  until  the  paper  can  be 
reexamined  for  small  details  like  differences  in  the  ink  or  character 
of  the  writing  this  idea  will  be  disregarded.  The  next  is  that  about 
this  time  he  was  reading  Barrow,  and  then  one  is  at  once  reminded 


LEIBNIZ  IN  LONDON. 


545 


of  Lect.  X,  Prop.  1 1 ;  this  is  the  proposition  in  which  Barrow  proves 
that  differentiation  is  the  inverse  of  integration.  If  we  consider 
this  in  the  manner  of  Leibniz,  we  get  the  equivalent  that  is  set  down 
on  the  right-hand  side  below: 


BARROW 

Let  ZGE  be  any  curve  of 
which  the  axis  is  VD;  and  let 
ordinates  applied  to  this  axis, 
VZ,  PG,  DE,  continually  in- 
crease from  the  initial  ordinate 
VZ ;  also  let  VIF  be  a  line  such 
that  if  any  straight  line  EDF  is 
drawn  perpendicular  to  VD,  cut- 
ting the  curves  in  the  points  E. 
F,  and  VD  in  D,  the  rectangle 
contained  by  DF  and  a  given 
length  R  is  equal  to  the  inter- 
cepted space  VDEZ;  also  let 
DE:DF  =  R:DT,  ^nd  join  DT. 
Then  TF  will  touch  the  curve 
VIF. 

Cor.   It  should  be  observed  that 
VDEZ. 


LEIBNIZ 

Let  AC  be  a  curve,  whose  axis 
is  AB,  and  let  the  ordinate  AB 
be/; 


let  AD  be  another  curve,  having 
the  same  axis,  and  let  its  ordinate 
DB  be  called  y ; 

let  this  curve  AD  be  such  that 
the  area  ABC,  i.  e.,  all  the  /'s  or 
//.  is  equal  to  the  product  of 
BD  and  a  fixed  line,  i.  e.,  equal 
to  ay; 

then,  taking  B(B)  equal  to  unity, 
we  have  /  =  aw,  where  w :  B  ( B ) 
=  DB:BT,  orw  =  y/d,  i.  e., 
l  =  ay/d. 


We  thus  see  that  the  d  that  results  as  the  "antithesis  to  the 

integral  sign"   (als  Gegensatz sich  ergab),  is  not  a  difference 

at  all,  but  the  subtangent ;  it  is  y/d  or  w  (on  account  of  B(B)  being 
taken  as  unity)  that  is  the  difference  between  the  ordinates  y.  But 
there  is  not  the  slightest  trace  of  the  idea  of  differentiation :  this 
is  made  more  manifest  by  the  work  which  follows,  which  is  based 
on  his  idea  of  obtaining  independent  equations,  and  eliminating  all 
variables  but  one  and  thus  reducing  the  problem  to  a  quadrature. 
And  yet  he  seems  to  perceive  from  the  equation  that  gives  the  dif- 
ference of  the  y's  as  a  quotient,  that  in  some  unintelligible  way  a 
division  means  a  difference.  Later  therefore  we  find  him  trying 
to  find  an  interpretation  of  d  as  an  operator,  whether  he  writes  it 


546  THE  MONIST. 

in  front  of  his  y,  or  as  a  denominator ;  namely,  when  he  considers 
what  value  he  is  to  assign  to  d(xy}.  I  venture  to  assert,  unless  we 
assume  that  Leibniz  is  considering  this  proposition  of  Barrow's, 
that  there  is  no  possible  connection  to  be  made  out  between  the 
several  sentences  of  this  passage.  Also  that  in  no  sense  can  this 
introduction  of  the  letter  d  be  looked  on  as  an  algorithm  with  any 
idea  in  it  of  differentiation. 

I  am  well  aware  that  in  the  above  I  have  adduced  no  positive 
proof  that  my  idea  is  correct;  I  have  not  had  the  advantage  of 
Gerhardt  in  seeing  these  manuscripts.  But  I  have  honestly  tried  to 
find  other  ways  of  explaining  the  circumstances  that  lead  from  y/d 
as  a  quotient  to  dy  as  a  difference,  and  I  can  find  none  other  that 
is  feasible  than  that  given  above,  namely,  that,  perhaps  by  accident, 
Leibniz  uses  d  for  the  subtangent  (instead  of  the  usual  t),  and  per- 
ceives from  such  a  figure  as  the  above  (which  of  course  I  do  not 
intend  to  say  he  has  given)  that  y/d  (where  d  is  the  subtangent) 
works  out  the  same  as  dy  (when  d.v  is  taken  to  be  unity)  ;  in  other 
words  the  subtangent  d  is  equal  to  y/(dy/dx). 

NOTE  3.  On  the  progress  made  by  Leibniz  before  November,  1676. 
(Referred  to  in  footnote  30.) 

The  remark  made  by  Gerhardt  that  Leibniz  "had  made  a 
progress,  by  the  introduction  of  his  algorithm  into  the  higher  anal- 
ysis, beyond  anything  that  came  to  his  knowledge  in  London,"  is, 
to  say  the  least  of  it,  a  matter  of  opinion.  From  a  study  of  the  six 
manuscripts,  that  Gerhardt  has  given  us,  that  bear  dates  between 
that  of  the  introduction  of  the  integral  and  differential  symbols 
(Oct.  26,  1675)  and  that  of  his  return  to  Germany,  via  Amsterdam 
(after  Nov.,  1676),  I  fail  to  see  that  there  is  very  much  occasion 
for  the  main  part  of  the  above  statement,  namely,  that  the  progress 
made  by  Leibniz  was  at  all  greater  than  anything  that  came  to  his 
knowledge  in  London ;  as  for  this  progress,  if  for  a  moment  we 
assume  its  superiority,  being  due  to  the  reason  set  in  ftalics,  I  fail 
to  see  that  Gerhardt  has  any  grounds  whatever  for  such  a  state- 
ment. 

The  six  manuscripts  in  question  have  been  given,  translated 
into  English  and  annotated  in  The  Monist,  April,  1917;  but  for  con- 
venience I  here  add  a  precis  of  them. 

i.  Nov.  1,  1675.  A  continuation  of  the  work  on  moments  about 
axes ;  the  new  symbols  do  not  occur,  omn.  being  still  used. 
He  has  now  read  Wallis,  Gregory  and  Barrow,  in  addition 
to  Cavalieri  and  St.  Vincent;  he  speaks  of  his  theorem  of 
breaking  up  a  figure  into  triangles  as  bringing  out  something 
new ;  the  whole  tone  of  this  manuscript  is  in  the  main  Pas- 
calian. 

ii.  Nov.  11,  1675.  He  successfully  obtains  a  solution  of  the  prob- 
lem of  finding  a  curve  such  that  the  rectangle  contained  by 


LEIBNIZ  IN  LONDON.  547 

the  subnormal  and  ordinate  is  constant.  This  he  considers 
to  be  "one  of  the  most  difficult  things  in  the  whole  of  geom- 
etry." He  uses  the  integral  sign,  and  the  denominator  d; 
but  neither  integration  nor  differentiation,  the  fact  that 
yz/2d  =  y,  being  taken  from  the  "quadrature  of  the  triangle." 
In  verifying  his  result  he  quotes  Slusius's  Rule  of  Tangents. 
Further  on,  he  has  the  note  that  x/d  and  dx  are  the  same 
thing,  though  there  is  nothing  to  show  why  he  comes  to  this 
conclusion ;  see  the  last  critical  note.  He  also  comes  to  the 
conclusion  that  d(xy)  is  not  the  same  as  dx.dy;  but  in  the 
last  bit  of  work  in  this  manuscript  he  uses  special  letters 
for  the  infinitesimals,  showing  that  he  has  been  trying  to 
find  the  effect  of  d  as  an  operator,  or  perhaps  trying  to  find 
the  reason  of  the  equality  x/d  and  dx.  He  has  failed  to 
solve  a  problem,  which  results  in  the  differential  equation,  as 
we  should  now  write  it,  x  +  y.dy/dx  =  az/y,  or  as  Leibniz 
has  it  x  +  w=az/y;  although  he  gives  an  incorrect  solution, 
which  he  asserts  to  be  true.  This  time  he  does  not  attempt 
to  verify  his  solution,  the  reason  being  obviously  that  he  is 
unable  to  do  so,  because  one  side  of  his  equation  is  a  product. 
As  a  matter  of  fact,  I  have  it  on  the  authority  of  Professor 
Forsyth  that  there  is  no  solution  of  this  equation  in  elemen- 
tary functions ;  or  at  least  he  says  that  he  has  been  unable 
to  find  one,  which  I  take  it  comes  to  the  same  thing.  The  one 
advance  that  can  be  found  here  is  the  appreciation  that  squares 
and  products  of  infinitesimals  can  be  neglected,  as  he  has 
doubtless  found  in  reading  Barrow.  It  is  worth  noting  that 
he  now  uses  the  differential  triangle  in  Barrow's  form  instead 
of  the  form  he  says  he  got  from  Pascal. 

iii.  Nov.  21,  1675.  In  this  manuscript  he  sets  himself  another 
problem,  which  he  fails  to  solve;  the  curve  required  is  log- 
arithmic, and  this  fact  even  he  fails  to  bring  out.  In  gen- 
eralizations that  arise  from  the  consideration  of  his  problem 
he  obtains  dxy-xy  —  xdy,  in  a  more  or  less  analytical  man- 
ner; but  immediately  afterward  states  that  nothing  new  can 
be  obtained  from  it ;  he  has  already  obtained  this  formula  by 
his  consideration  of  moments,  geometrically;  and  he  does 
not  appreciate  the  advance  there  is  in  obtaining  it  algebra- 
ically. The  manuscript  concludes  with  a  consideration  of  the 
figure  by  means  of  which  it  is  generally  supposed  that  he 
affected  his  arithmetical  quadrature.  This  is  very  remarkable 
on  account  of  the  heading,  which  reads,  "A  new  kind  of 
Trigonometry  of  indivisibles,  by  the  help  of  ordinates  that 
are  not  parallel  but  converge."  What  I  refer  to  is  the  use  of 
the  word  new,  which  I  have  here  italicized.  It  is  to  be  ob- 
served that  the  diagram  and  the  results  are  almost  identical 
with  those  of  Barrow,  Lect.  XI,  Prop.  22-24  (see  the  first 
critical  note).  He  concludes  by  a  reference  to  the  trochoids, 


548  THE  MONIST. 

which  shows  that  he  is  still  under  the  influence  of  Pascal,  if 
indeed  he  is  not  still  studying  his  works. 

iv.  Nov.  22,  1675.  He  returns  to  the  subjects  of  the  previous  day. 
But  there  is  here  no  mention  of  the  signs  of  integration  or 
differentiation. 

v.  June  28,  1676.  Here  we  have  a  certain  advance,  for  there 
occurs  the  statement:  "The  true  general  method  of  tangents 
is  by  means  of  differences."  While  he  uses  dy  and  dz  for  the 
elements  of  3;  and  z,  he  uses  ft  for  the  element  of  x ;  the  rest 
of  the  work  is  merely  Harrovian  in  principle.  This  mere 
substitution  of  dy  and  dz  for  the  special  letters  used  by  Bar- 
row for  the  same  things  can  hardly  be  called  progress.  What 
progress  there  might  be  is  barred  by  the  use  of  equations 
with  three  or  more  variables  in  them. 

vi.  July,  1676.  The  remark  on  the  last  manuscript  is  corroborated 
by  the  contents  of  this  manuscript.  Leibniz  asserts  that 
he  has  solved  two  problems,  of  which  Descartes  had  alone 
solved  one,  and  owned  that  he  could  not  solve  the  other.  The 
truth  is  that  he  has  not  solved  the  former,  which  was  fairly 
easy,  only  given  an  alternative  construction  which  is,  if  any- 
thing, more  difficult  to  carry  out  than  a  construction  from  the 
original  data  for  the  curve.  The  latter  he  gets  out  in  a  hazy 
fashion  (". . .  .which  belongs  to  a  logarithmic  curve").  This 
conclusion  he  comes  to  after  several  erroneous  steps  of  rea- 
soning; whereas  the  solution  stared  him  in  the  face  about  a 
quarter  of  the  way  through  the  work,  where  he  has  the 
equation  cdy  =  ydx,  if  he  could  have  integrated  dy/y  with 
certainty.  The  failure  I  think  arises  from  the  study  of  Pas- 
cal, who  lays  it  down  that  only  one  of  the  variables  can  in- 
crease arithmetically,  and  Mercator's  work  has  been  with  y 
increasing  arithmetically,  and  Leibniz  has  already  considered 
that  the  x  is  increasing  arithmetically  (See  my  notes  on  this 
manuscript  in  The  Monist  for  July,  1917). 

Throughout  the  whole  of  these  manuscripts,  he  makes  no  prog- 
ress, because  he  is  hampered  by  the  idea  of  keeping  one  of  his 
variables  increasing  uniformly;  he  seldom  uses  his  algorithm  for 
differentiation ;  and  when  he  does  do  so,  it  is  merely  a  substitution 
of  dx,  etc.  for  the  special  letters  used  by  Barrow.  In  fact  these 
manuscripts  appear  to  me  to  be  the  records  of  his  work  on  the  text- 
books of  his  study,  Pascal,  Wallis,  Gregory,  and  Barrow ;  and  we 
see  him  trying  to  fit  the  matter  and  methods  found  in  them  into  his 
own  ideas  and  notations.  It  is  not  until  November,  1676,  when  he 
has  arrived  on  the  Continent,  after  having  seen  Newton's  paper, 
that  we  have  any  Differential  Calculus ;  even  then  some  of  the 
standard  forms  that  he  gives  are  not  quite  correct ;  on  the  other  hand, 
he  gives  the  method  of  substitution  to  differentiate  an  irrational, 
though  he  uses  the  Barrovian  method  to  differentiate  the  general 
equation  of  the  second  degree,  merely  using  dy  and  dx  instead  of 


LEIBNIZ  IN  LONDON.  549 

Barrow's  special  letters.  It  is  not  until  July,  1677,  that  he  is  able 
to  give  anything  like  an  intelligible  account  of  the  differentiation  of 
products,  powers,  quotients  and  roots.  Lastly  I  doubt  if  Leibniz 
ever  did  really  appreciate  the  Newtonian  idea  that  dy/dx  was  a  rate, 
or  else  the  example  he  gives  of  the  use  of  the  second  and  third 
differentials  in  his  answer  to  Nieuwentiit  would  not  have  contained 
so  many  ridiculous  errors. 


TRANSLATIONS  OF  THE  MANUSCRIPTS 
Alluded  to  by  Dr.  Gerhardt. 

I. 

Scientific  memoranda  of  the  visit  to  England  at  the  beginning  of 

the  year  1673. 

When  at  the  beginning  of  the  year  1673,  I  accompanied  his 
Excellency  the  Ambassador  of  Mainz,  Baron  Schornborn,  a  nephew 
(on  his  father's  side)  of  the  Elector,  from  Paris  to  London,  although 
I  stayed  in  England  scarcely  a  month,  among  various  distractions, 
I  still  gave  attention  to  increasing  my  knowledge  of  philosophy; 
for  at  that  time  the  English  held  a  high  reputation  in  this  subject. 

To  set  out  a  long  minute  record  of  daily  happenings  is  useless 
on  account  of  its  inequality;  for  the  fortune  of  all  the  days  was 
not  the  same ;  indeed  the  points  worth  remarking  heaped  themselves 
up  one  day,  and  the  next  gaped  with  emptiness.  For  this  reason 
perhaps  it  will  be  more  satisfactory  to  go  by  heading  of  subjects, 
one  remark  recalling  another  as  it  were. 

The  principal  heads  for  the  subjects  noted  may  be  taken  as 
Arithmetic,  Geometry,  Music,  Optics,  Astronomy,  Mechanics,  Bot- 
any, Anatomy,  Chemistry,  Medicine,  and  Miscellaneous. 

ARITHMETIC.  The  line  of  proportions  or  Gunter's  lines  or  the 
double  scale.  Logarithmotechnia  or  compendium  for  calculating 
logarithms.  To  recognize  square  numbers  from  non-squares  by  their 
end  figures.  Morland's  machine. 

ALGEBRA.  Substance  of  English  algebraical  work  of  27  years.  Al- 
gebra of  Pell.  At  first  few  rules,  but  lots  of  selected  examples. 
Renaldinus  not  thought  much  of  in  England. 

GEOMETRY.  Tangents  to  all  curves.  Development  of  geometrical 
figures  by  the  motion  of  a  point  in  a  moving  line. 


550 


THE  MONIST. 


Music.  Its  universal  character.  System  of  Birthincha.  Vossius 
will  publish  Music. 

OPTICS.  They  told  me  of  a  certain  phenomenon  that  Barrow  con- 
fessed that  he  was  unable  to  solve.  The  difficulty  of  Newton  hitherto 
unsolved,  Father  Pardies  giving  it  up.  Hook  adheres  to  a  cata- 
dioptric  instrument  of  9  feet,  because  for  another  of  50  feet  move- 
ment inconveniences  them.  The  secret  of  the  largest  aperture  which 
can  be  given  to  microscopes  is  primarily  as  great  as  the  distance  of 
the  object. 


ASTRONOMY.  Arrangement  of  Hook  for 
observing  whether  the  earth  at  any  time 
sensibly  approaches  or  recedes  from  the 
fixed  stars,  from  which  it  can  be  judged 
that  it  is  not  in  the  center  of  the  uni- 
verse; he  erected  it  in  a  fine  tube  set 
perpendicularly,  and  observed  the  stars 
that  are  vertically  overhead.  He,  lying 
flat  on  his  back,  observed  their  dimen- 
sions most  exactly. 


CHEMISTRY. 


MECHANICS. 


PNEUMATICS. 


METEOROLOGY. 


HYDROSTATICS. 


NAVIGATION. 


MAGNETISM. 


PHYSICS. 


BOTANY. 


ANATOMY. 


MEDICINE. 


MISCELLANEOUS. 


ii. 


[This  manuscript  is  very  lengthy,  the  translation  running  to 


LEIBNIZ  IN  LONDON.  551 

about  6000  words,  of  which  the  first  5000  are  written  as  a  concise 
history  of  all  the  great  geometers  and  their  works,  that  are  antece- 
dent to  Leibniz  himself.  This  part  is  quite  unimportant  for  the 
purpose  of  estimating  the  part  that  was  played  by  Leibniz,  and  it 
passes  my  comprehension  why  Gerhardt  should  give  it  at  length, 
while  he  has  condensed  the  other  two,  which  are  really  important. 
Hence,  in  what  follows,  I  have  given  a  precis  of  the  first  5000  words, 
with  here  and  there  quotations,  in  which  Leibniz  has  something  to 
say  that  is  either  critical  of  the  work  of  others,  or  a  claim  to  superior 
knowledge  or  better  method  of  his  own.  The  last  part,  which  pur- 
ports to  be  the  history  of  his  arithmetical  quadrature,  together  with 
his  claim  to  the  surpassing  value  of  his  achievement,  I  have  given 
in  full.] 

(Precis).  Geometry  is  a  modern  thing,  probably  due  to  the  Greeks. 
The  great  name  among  the  Ancients  is  that  of  Archimedes,  who 
first  used  indivisibles;  this  use  was  more  profound  than  that  of 
Cavalieri,  but  the  method  became  lost.  The  name  of  Apollonius 
must  not  be  altogether  omitted. 

The  learning  of  the  Greeks  passed  on  to  the  Arabs,  who  con- 
quered them ;  among  these  we  have  Alhazen,  and  a  certain  Mahomet, 
who  gave  the  formula  for  the  general  quadratic. 

This  brings  us  to  the  cubic  and  biquadratic  equations,  which 
were  solved  in  the  sixteenth  century.  The  cubic  is  due  to  one  Scipio 
Ferreus  of  Bologna ;  one  of  his  pupils  set  the  solution  as  a  challenge 
after  Scipio's  death ;  Tartalea  took  up  the  challenge,  found  a  solu- 
tion and  told  his  friend  Cardan ;  the  latter  extended  it  and  published 
it  without  the  consent  of  Tartalea.  Vieta,  Descartes,  and  Ferrarius 
gave  the  solution  of  the  biquadratic.  But  even  Descartes  and  Vieta 
failed  at  equations  of  higher  degrees.  With  regard  to  the  work 
of  Descartes,  Leibniz  remarks  that  "its  origin  [that  is,  of  the  method 
of  solution]  was  a  widely  different  and  more  fertile  spring;  and  if 
Descartes  had  only  recognized  this,  he  would  have  rendered  the 
discovery  of  Scipio  more  general  and  carried  it  to  further  heights. 
But  what  has  befallen  me  in  this  connection  I  will  say  in  another 
place."  Leibniz  further  remarks  that  the  method  of  Descartes 
fails  to  give  the  roots  of  equations  of  higher  degree,  although  the 
quality  of  the  roots  may  be  learned  through  it.  "I  will  show  in 
another  place  that  the  reason  for  this  is  clearly  known  to  me  from 
the  most  fundamental  principles  of  the  art,  and  that  I  have  estab- 
lished an  extremely  easy  method,  and  one  that  is  adapted  too  for 
enlarging  science,  by  the  many  things  that  follow  from  it." 

In  the  seventeenth  century,  Leibniz  goes  on  to  say,  after  Archi- 


552  THE   MONIST. 

mtdes  and  Galileo's  several  times  and  influence  are  gone  by,  there 
is  no  writer  from  whom  more  is  to  be  learned  than  from  Descartes ; 
and  yet  he  is  "unable  to  pass  over  certain  boastful  remarks  that  he 
makes,  by  which  the  less  experienced  among  us  may  be  led  into 
error."  Descartes  had  said  that  by  his  method  every  geometrical 
problem  could  be  reduced  to  the  finding  of  the  roots  of  equations. 
Leibniz  remarks  that  this  shows  Dzscartes's  ignorance  of  the  matter. 
"For  when  the  magnitude  of  curved  lines  or  the  space  enclosed  by 
such  is  required  (which  happen  more  frequently  than  perhaps  Des- 
cartes thought,  since  he  had  not  applied  himself  sufficiently  to  the 
'mechanics'  of  Galileo),  neither  equations  nor  Cartesian  curves  can 
help  us,  and  there  is  need  of  equations  of  a  totally  new  kind,  of 
constructions  and  new  curves,  and  finally  of  a  new  calculus,  given 
so  far  by  nobody,  of  ivhich,  if  nothing  else,  I  can  now  give  certain 
examples  at  least,  which  are  remarkable  enough." . ...  7  have  men- 
tioned these  things  so  that  men  may  understand  that  there  are  cer- 
tain methods  in  Geometry,  for  which  they  may  look  in  vain  in  the 
works  of  Descartes." 

Returning  to  geometry  purely,  Leibniz  next  mentions  the  work 
of  Galileo,  Cavalieri  (whose  method  he  considers  is  rough  and  lim- 
ited in  extent),  Torricelli,  Roberval,  Pascal,  Wallis,  Huygens,  and 
Slusius,  as  contributors  to  the  new  geometry.  He  considers  that 
a  new  epoch  opens  with  the  work  of  Neil  and  van  Huraet  (on  recti- 
fication of  curves),  James  Gregory,  and  Brouncker.  "Finally  Mer- 
cator  gave  a  general  formula  for  the  area  under  a  hyperbola."  He 
claims  Mercator  as  "an  eminent  German  geometer" ;  but  rather 
decries  his  discovery  as  being  an  easy  one,  on  account  of  the 
ordinates  working  out  as  rational  in  terms  of  the  abscissa.  "But  it 
was  not  so  easy  to  give  the  magnitude  of  the  circle,  and  its  parts, 
expressed  as  an  infinite  series  of  rational  numbers ; . . . .  for  the 
circle,  however  you  treat  it,  has  ordinates  that  are  irrational.  How- 
ever I,  as  soon  as  I  had  found  a  certain  very  general  theorem,  by 
means  of  which  any  figure  whatever  could  be  converted  into  an- 
other that  is  quite  different  from  it,  but  yet  of  equivalent  area,  set 
to  work  to  try  whether  the  circle  could  not  be  converted  in  some  way 
into  a  rational  figure;  and  the  thing  came  out  beautifully;....  it 
will  be  worth  while  here  to  give  a  short  account  of  the  matter." 

(In  full).  Nearly  everybody  who  has  up  to  now  treated  of  the 
geometry  of  indivisibles  has  been  accustomed  to  break  up  their 
figures  into  rectangles  or  parallelograms  only  by  means  of  ordinates 


LEIBNIZ  IN  LONDON. 


553 


parallel  to  one  another.  But  the  reasoning  of  Desargues  and  Pascal 
always  pleased  me  very  much ;  these  in  Conies,  as  we  can  call  them 
in  general,  include  under  the  name  of  ordinates  not  only  parallels, 
but  also  straight  lines  meeting  in  or  converging  to  a  point,  especially 
when  parallels  are  included  under  the  name  of  converging,  by 
saying  that  the  point  of  convergence  goes  off  to  an  infinite  distance. 
Thus  while  others  only  consider  parallel  ordinates,  and  have  broken 
up  their  figures  into  parallelograms  AB(B)(A),  (A)(B)((B)) 
((A)),  in  the  way  that  Cavalieri  does,  I  employ  converging  lines 
n.nd  resolve  the  given  figure  into  triangles  CD(D),  C(D)((D)),  and 
at  once  draw  another  figure  of  which  the  ordinates  AB,  (A)(B), 
etc.,  are  proportional  to  these  triangles. 


(El  ((£)) 


Now  this  is  the  case  if  the  AB's  are  equal  to  the  CE's  where 
it  is  supposed  that  the  straight  lines  DE  are  tangents  to  the  given 
curve ;  for  in  that  case,  as  I  will  show  below,  it  will  come  out  that 
the  space  B(B)(A)A  will  be  double  of  the  segment  C(D)DC,  and 
for  any  figure  such  as  C(D)DC  another  that  is  equivalent  to  it  can 
be  drawn.  Now,  supposing  that  the  curve  D(D)((D))  is  circular 
and  that  CA  is  a  part  of  the  diameter,  then,  calling  CA  or  FB  x, 
and  CF  or  AB  y,  and  the  radius  of  the  circle  unity,  calculation 
will  show  that  the  value  of  x  is  2y-/(\±yz).  Thus  the  ordinate 
FB  or  x  can  be  expressed  rationally  in  terms  of  the  given  abscissa 
CF  or  3'.  Such  figures  as  these,  in  which  the  ordinates  can  be  ex- 
pressed rationally  in  terms  of  the  abscissae,  I  call  rational.  Thus 
we  have  drawn  a  rational  figure  equivalent  to  the  circle,  and  this 
will  be  soon  seen  to  be  sufficient  to  give  the  arithmetical  quadrature 
of  the  latter.  For,  from  the  sum  of  a  geometric  series  of  an  infinite 
number  of  decreasing  terms  that  is  well  known  to  all  geometers,  it 
follows  that  y*-y*  +  y*  -  y*  +  y10  -  y12  +  etc.  to  infinity  is  the  same  as 
3'V(1+3|2)»  i-  e-»  tne  same  as  \x,  if  only  we  understand  that  y  is  a 
quantity  that  is  less  than  the  radius,  or  unity.  Now,  since  we  have 
to  collect  together  the  infinite  number  of  $x's  into  one  sum,  in  order 


554  THE  MONIST. 

to  obtain  the  quadrature  of  half  the  figure  C(F)(B)BC  and  what 
it  comes  to,  namely,  that  of  the  circle ;  so  also  have  we  to  collect 
together  the  infinite  number  of  series  y*-y*  +  ys-y*  +  ylo-yl2  +  etc., 
into  one  sum,  and  this  by  the  method  of  indivisibles  and  infinites 
can  be  done  without  difficulty.  For,  suppose  that  the  last  y,  which 
in  general  is  taken  as  C(F),  to  be  b,  then  the  sum  of  every  y2  will 
be  &3/3,  and  of  every  y*  will  be  b5/S,  and  of  every  y*  will  be  b1/?, 
and  so  on ;  hence,  the  sum  of  the  infinite  number  of  £-*"'s,  or  of  the 
series  yz-y*  +  ys-y6  +  yl°-yl2+  etc.,  i.  e.,  the  area  of  half  the  space 
C(F)(B)BC,  will  be  b*/3-b«/5  +  b7/7-b»/9  etc.  From  which, 
by  the  help  of  ordinary  geometry,  it  can  be  easily  deduced  that  the 
square  on  the  diameter  is  to  the  area  of  the  circle  as  1  is  to  1/1  - 
1/3 +1/5 -1/7  + etc. ;  also  speaking  in  general,  supposing  b  to  be 
the  tangent,  then  the  arc  is  b/l-b3/3  +  b*/S-b'/7  +  &9/9 -&"/!!  + 
etc.  Hence  it  now  follows  that  any  one  without  the  help  of  tables 
and  continual  bisections  of  angles  and  extractions  of  roots  can  ap- 
proximate to  the  magnitude  of  the  arc  to  any  degree  of  accuracy 
desired,  so  long  as  the  tangent  &  is  a  little  less  than  the  radius ; 
so  that  if  we  take  the  tangent  to  be  a  little  less  than  the  tenth  part 
of  the  radius,  the  arc  may  be  obtained  with  sufficient  accuracy. 
Let  us  take  the  tangent  to  be  a  tenth  part  of  the  radius,  then  if  we 
want  the  arc,  it  will  be 

11  1  1 ,  1 

1 1 =  f»fr>     • 

10      3000     500000     70000000     9000000000 

and  reducing  all  to  a  common  denominator,  and  adding  the  numbers 
into  one  sum  (  for  it  is  not  worth  while  going  any  further) ,  then  the 
arc  will  be  a  little  greater  than  518027821302775/5197500000000000, 
and  the  defect  of  this  value  from  the  true  value  will  be  less  than  the 
1/1000000000000  part  of  the  radius.  For  if  we  do  not  subtract  the 
last  term,  1/1100000000000,  the  value  would  be  too  great,  and  if 
we  do  subtract  it,  the  value  is  less  than  the  true  value,  there- 
fore the  error  is  less  than  1/1100000000000,  and  thus  is  less  than 
1/1000000000000. 

It  is  seen  how  exactly  it  comes  out  with  such  easy  calculation 
involving  only  additions,  subtractions  and  multiplications,  to  an  ex- 
tent that  is  not  obtainable  with  tables.  Also  if  the  ratio  of  the  tan- 
gent to  the  radius  is  anything  else,  the  arc  can  similarly  be  found, 
and  this  is  especially  easy  when  it  can  be  expressed  in  decimal  parts. 
Again,  since  now  the  ratio  of  the  circumference  to  the  radius  is 
given  in  numbers  of  any  required  degree  of  accuracy,  by  this  also 


LEIBNIZ  IN  LONDON.  555 

the  ratio  of  a  given  arc  to  the  circumference  is  given,  and  thus 
also  the  quantity  of  angle  for  a  given  tangent  will  appear  with  any 
required  degree  of  accuracy.  In  this  way  tables  may  be  corrected, 
supplemented,  or,  if  need  be,  enlarged,  with  no  great  trouble.  Any 
one  who  will  just  remember  this  fairly  easy  rule  will  be  able  without 
tables  to  attain  to  any  required  degree  of  accuracy  with  very  little 
labor.  How  great  an  acquisition  this  is  to  geometry,  I  leave  it  to 
those  who  understand  to  estimate. 

CRITICAL  NOTE. 

It  is  difficult  to  see  the  object  that  Leibniz  had  in  writing  this 
long  historical  prelude  to  an  imperfect  proof  of  his  arithmetical 
quadrature,  unless  it  can  be  ascribed  to  a  motive  of  self-praise. 
This  suggestion  would  seem  to  be  corroborated  by  the  claims  that 
Leibniz  makes  in  the  parts  where  I  have  quoted  his  own  words  in 
italics  in  the  precis,  and  by  the  concluding  sentence  of  the  trans- 
lation given  in  full.  Even  if  this  is  so,  there  may  be  some  plea  of 
justification  put  forward ;  for  Leibniz  appears  to  have  been  a  man 
impelled  by  many  contradictory  motives,  but  these  I  think  can  all 
be  traced  back  to  one  origin.  The  time  in  which  he  lived  was  a  time 
of  great  discoveries  in  geometry;  Leibniz  knew  in  his  soul  that  he 
had  it  in  him  to  be  one  of  the  great  men  in  this  branch  of  learning, 
but  as  truly  recognized  his  great  disability  due  to  his  lateness  in 
starting,  and  felt  that  his  only  chance  was  to  belong  to  the  very 
exclusive  set  who  corresponded  with  one  another;  he  saw  that  the 
only  way  of  entering  this  set  was  to  do  something  brilliant.  This 
may  be  taken  as  some  excuse  for  any  self-praise  that  we  find,  and  to 
a  less  extent  for  his,  to  my  mind,  undoubted  plagiarisms.  With 
regard  to  the  behavior  of  Leibniz,  when  charged  with  these  plagiar- 
isms, Sloman  is  not  beyond  calling  Leibniz  a  liar  point-blank:  I 
prefer  to  call  his  statements  perversions  of  the  truth,  made  under 
stress  of  circumstances,  so  that  his  reputation  as  a  great  and  original 
thinker  should  not  suffer.  For  instance,  to  explain  what  I  mean,  I 
will  take  the  statement  of  Leibniz  to  de  1'Hospital  that  he  owed 
nothing  to  Barrow.  As  I  have  said  in  another  place,  from  one  point 
of  view,  the  point  of  view  that  Leibniz  would  take  for  the  purposes 
of  this  letter,  Barrow  would  be  a  hindrance  rather  than  a  help  to 
Leibniz,  in  the  formulation  of  his  algebraical  calculus,  after  he  had 
once  absorbed  all  the  fundamental  ideas.  That  is,  it  would  seem 
that  Leibniz  always  tries  to  tell  the  truth,  but  to  put  it  in  a  form 
that  to  the  uninformed  reader  will  convey  quite  a  wrong  impression. 
Another  example  of  this  juggling  with  words  and  phrases  is  given 
by  Sloman,  in  the  shape  of  a  letter  from  Leibniz,  dated  August  27, 
1676,  and  the  first  draft  of  the  same ;  these  two  read  together  are 
very  much  the  same,  but  read  apart  convey  a  totally  different  im- 
pression. 

A  second  characteristic  of  Leibniz  may  also  be  traced  back  to 


556  THE  MONIST. 

his  desire  to  make  up  for  his  lateness  in  starting ;  that  is,  the  some- 
times ridiculous  claims  that  Leibniz  makes  to  discoveries,  or  rather 
hints  at  having  made  them.  An  instance  is  given  in  the  Historia 
(see  Monist,  Oct.,  1916,  p.  599).  "It  is  required  to  form  the  sum 
of  all  the  ordinates  V (\-xx}  =y;  suppose  y  =  ±\+.-xz,  from  which 
x -=-2z  /  (\  +  zz) ,  and  3;=  (±sz+  \~}/(zz-\- 1)  ;  and  thus  again  all  that 
remains  to  be  done  is  the  summation  of  rationals."  Unless  we 
assume  that  Leibniz  never  understood  in  all  his  life  what  we  now 
call  the  change  of  the  variable  in  integration,  which  to  me  seems 
rather  far-fetched,  the  only  reason  why  this  should  have  been 
allowed  to  appear  in  a  tract  that  was  certainly  written  after  1712, 
is  that  Leibniz  had  never  attempted  this  summation ;  he  had  set  this 
down  in  1674  and  1675  as  a  method  of  quadrature  for  the  circle, 
not  at  that  time  having  perceived  the  importance  of  the  factor  dz, 
or,  in  other  words,  the  way  in  which  the  ordinates  should  be  ordi- 
nated ;  for  as  I  have  already  pointed  out,  at  that  time  Leibniz  could 
not  have  found  dz,  since  he  could  not  differentiate  a  product.  This 

?oes  to  prove  that  his  reading  of  Pascal  was  not  of  the  profoundest ; 
or  Pascal  is  very  careful  over  this  point,  going  to  the  trouble  of 
calling  the  y's  ordinates  when  drawn  through  the  points  of  equal 
division  of  the  base,  and  sines  when  they  are  drawn  through  the 
points  of  equal  division  of  the  arc.  Probably  to  this  characteristic 
is  due  the  claim,  xset  in  italics  in  the  manuscript  above,  with  respect 
to  equations  of  higher  degrees.  He  thought  he  had  a  general 
rrrthod,  which  he  had  not  time  to  verify  by  particular  examples, 
and  so  find  that  his  claim  was  erroneous.  For  surely  this  cannot 
be  read  as  a  claim  to  the  Tschirnhausian  transformation  and  the 
expression  of  a  quintic  in  the  canonical  form  x5  +  fix  +  q  =  0. 

The  date  of  the  above  manuscript  is  almost  certainly  antecedent 
to  the  manuscript  that  Leibniz  got  ready  for  the  press,  De  Quadra- 
tura ;  hence  his  claim  to  be  able  to  give  examples  of  the  calculus, 
except  for  integral  powers  which  had  already  been  done  by  Wallis, 
is  without  foundation. 

With  regard  to  the  arithmetical  quadrature  itself,  the  great 
importance  of  it  in  the  estimation  of  Leibniz  is  apparently  in  the 
correction  and  enlargement  of  tables ;  this  claim,  as  Leibniz  puts  it, 
is  ridiculous,  although  it  could  be  so  used  by  first  constructing  tables 
for  angles  whose  tangents  are  given.  But  Leibniz,  after  giving  a 
calculation  true  to  twelve  places  of  decimals,  states  that  "the  ratio 
of  the  circumference  to  the  radius  is  now  known,"  and  proposes 
to  use  that.  Apparently  he  does  not  see  that  to  calculate  this  ratio 
from  the  series  he  gives,  it  will  be  necessary  to  take  a  billion  or  so 
of  terms !  For  he  does  not  give  any  hint  of  any  modification  of  the 
series,  or  the  use  of  the  value  obtained  for  some  small  angle. 

Lastly,  with  regard  to  the  calculation,  it  is  strange  that  the 
denominator  chosen  as  a  common  denominator  is  15  times  what  it 
need  have  been ;  also  it  is  a  matter  of  wonder,  considering  that  tables 
of  logarithms  were  known  to  Leibniz,  as  a  reader  of  Mercator  and 


LEIBNIZ  IN  LONDON. 


557 


others,  that  Leibniz  puts  the  matter  in  fractional  form  instead  of 
working  in  decimals ;  thus,  the  arc  whose  tangent  is  0. 1  is  equal  to 

0.1  -   0.00033333333333333 

0.000002  0.00000001428571428 

0 . 000000000 1111 0^000000000000909 

0 . 1000020001 11111 0 . 000333347619956 

=  0.99966865249 

Finally,  note  that  while  Wallis  and  Brouncker  are  mentioned, 
Barrow  is  not.  This  is  all  part  and  parcel  of  his  successful  attempt 
to  conceal,  from  all  but  Oldenburg,  the  fact  that  he  had  a  copy  of 
Barrow  in  his  possession,  right  from  the  commencement  of  his 
studies. 

in. 

Transcribed  from  a  manuscript  tract  of  Newton  on  "Analysis  by 
means  of  equations  zvith  an  infinite  number  of  terms." 

ABnjr,  BDny,  a,  b,  c  given  quantities, 
m,  n  whole  numbers.    If  then 


n  y 


__      _  _ 

x  "    n  [  \  y]  n  area  of  ABD. 


m  +  n 

In  connection  with  this  the  following  ex- 
ample is  to  be  noted  : 

If  -3    (  n^r2)  r\y,  that  is  to  say,  if  o=l, 

X 

n  =  -\,  and  m  =  -2,  then  we  shall  have 

r/T  n  ]  -x-1  (or  —  )   naBD, 
1  /  \        x   I 

produced  indefinitely  in  the  direction  of  a; 
the  calculation  makes  this  negative  because 
it  lies  on  the  other  side  of  BD. 

Again,  if  ^  (or  x~l)  n  y  ,  then  ^x\  n  ^x°  n  |*1  (*  this 

ought  to  be  written  —  1*)  n  —  n  infinity,  which  is  the  area  of  the 
hyperbola  on  either  side. 

If  -  —  ~2  ny,  on  division  we  obtain 


ABCD 


1  +  x* 

ynl-  .v- 

x 


.,  and  then 


., 
lf 


_        £        i 

3    '*"  5    "  "  7     "  etc-  ; 

the  term  xz  is  the  first  in  the  division,  the  value 

of  y  will  be  x~z  -  x~*  +  x~*  -  etc., 


and  hence  BD  a  n   —  - 


3    -5     +etC' 


558  THE   MONIST. 

The  first  method  is  to  be  used  when  x  is  small  enough,  and  the 
second  when  x  is  large  enough. 

Gerhardt  then  remarks  that  Leibniz  has  noted  completely  the 
following  two  cases  of  extractions  of  roots: 

Hy. 

Gerhardt  further  notifies  the  reader  that  he  has  omitted  everything 
that  he  has  found  Leibniz  to  have  copied  out  word  for  word,  on 
comparison  with  Biot's  edition  of  the  Commercium  Epistolicum 
(1856). 

In  the  above,  Leibniz  marks  interpolated  remarks  of  his  own 
with  either  [  ]  or  (*  *). 

In  the  same  manner,  Leibniz  has  written  out  word  for  word 
the  part  of  the  manuscript  dealing  with  the  solution  of  adfected 
equations  (against  this  he  has  put  the  final  observation:  "And  these 
things  that  have  been  given  will  be  sufficient  for  the  investigation 
of  areas  of  curves"),  in  addition  to  the  part  which  follows,  "the 
application  of  what  has  been  given  to  other  problems  of  the  same 
kind,"  which,  as  being  already  known  to  him,  he  has  not  copied  out. 
He  goes  straight  on  to  the  next  section,  "To  find  the  converse  of  the 
foregoing,  that  is,  to  find  the  base  when  given  the  area,  and  to  find 
the  base  when  given  the  length  of  the  curve."  He  has  written  this 
out  word  for  word ;  also  he  has  noted  fully  to  the  end  the  "proof  of 
the  method  of  solution  of  adfected  equations." 

At  the  end  of  these  extracts  from  Newton's  tract  follow  the 
words,  "I  extracted  these  things  from  the  letter  of  Newton  20  Aug. 
to  Newton."  Gerhardt  states  that  he  has  already  said  all  that  is 
necessary  about  the  contents  of  these  extracts. 

SECOND  SHEET. 

Extracts  from  the  correspondence  between  Collins  and  Gregory. 

Among  a  number  of  partly  illegible  and  unintelligible  notes  the  fol- 
lowing were  to  be  noticed. 

Gregory,  January,  1670:  Barrow  shows  himself  to  be  most 
subtle  in  the  geometry  of  optics.  I  think  that  he  is  superior  to  all 
whose  works  I  have  looked  into,  and  I  esteem  this  author  beyond 
anything  that  can  be  imagined. 

Sept.,  1670:  I  think  that  Barrow  has  gone  infinitely  further 
than  all  those  who  have  written  before  him.  From  his  method  of 
drawing  tangents,  combined  with  certain  meditations  of  my  own, 
I  found  a  general  geometrical  method  of  drawing  tangents,  without 
calculation,  to  all  curves,  which  not  only  contain  his  particular 


LEIBNIZ  IN  LONDON.  559 

methods,  but  the  general  method  as  well.  This  is  shown  in  12 
propositions. 

Letter   of    Newton,    1672:   ABC   is   any 

angle,  ABn^r,   BCny.     Take,   for  example, 

the  equation, 

0  10023 

Xs  -2  Jy  +  bx*  -&x  -by*  -/  n  0. 

DAB  3  22100 

Multiply  the  equation  by  an  arithmetical  pro- 
gression, both  for  the  second  dimension  y  and  for  x\  the  first 
product  will  be  the  numerator,  and  the  other  divided  by  x  will  be 
the  denominator  of  a  fraction  which  will  express  BD,  thus. 

-2*»y  +  2fr»  ~3/ 
3    #    -4   xy  +2    bx  -#' 

Moreover  that  this  is  only  a  corollary  or  a  case  of  a  general  method 
for  both  mechanical  and  geometrical  lines,  whether  the  curve  is 
referred  to  a  straight  line,  or  to  another  curve,  without  the  trouble 
of  calculation,  and  other  abstruse  problems  about  curves,  etc.  This 
method  differs  from  that  of  Hudde  and  also  from  that  of  Sluse, 
in  that  it  is  not  necessary  to  eliminate  irrationals. 

NOTE. 

It  is  almost  useless  trying  to  write  a  critical  note  on  the  above 
in  such  an  incomplete  state.  But  I  may  remark  that  Leibniz  appar- 
ently was  at  the  time  quite  ignorant  of  what  we  now  term  "putting 
in  the  limits  for  a  definite  integral." 

Gerhardt  considers  that  the  existence  of  this  extract  proves 
conclusively  that  Leibniz  did  not  see  the  letter  of  Newton  so  often 
referred  to ;  forgetting,  as  Sloman  remarks,  that  Leibniz  ought  not 
to  have  seen  the  tract  at  all ! 

P.  S.  In  allusion  to  footnotes  3  and  18,  with  regard  to  the  use 
of  the  word  "moment"  or  "momentum"  in  the  sense  used  by  Leibniz, 
I  have  found  (since  the  above  was  written)  that  Cavalieri,  in  his 
Exercitationes  Sex,  defines  the  term  in  the  mechanical  sense  and 
gives  much  of  the  matter  of  Pascal  on  Centers  of  Gravity,  as  it 
appears  in  the  "Letters  of  Dettonville."  I  suggest  that  Leibniz  saw 
it  in  Cavalieri,  and  that  its  origin  is  to  be  traced  to  Galileo.  J.  M.  C. 


OUR  MUSICAL  IDIOM. 
WITH  AN  INTRODUCTION  BY  GLENN  DILLARD  GUNN. 

INTRODUCTION. 

The  effort  to  expand  the  means  of  musical  expression  is  as  old 
as  the  art  itself.  It  is  recorded  in  each  chapter  of  musical  history ; 
it  has  been  interrupted  only  during  those  periods  of  the  art's  devel- 
opment wherein  the  composer  has  been  concerned  with  the  com- 
pletion of  art-types  already  defined. 

Every  advance  in  the  art  has  been  prefaced  by  a  period  of  ex- 
perimental effort,  which  has  sought  new  modes  of  expression.  So 
soon  as  these  modes  of  expression  have  been  defined  and  their 
tendencies  and  the  laws  governing  them  have  been  apprehended, 
experiment  has  been  replaced  by  careful  conformance  to  law  and 
tradition,  which  has  operated  to  the  perfection  of  the  new  art  type. 

The  present  is  preeminently  an  epoch  of  experimentation.  The 
old  art  types  have  been  completed.  The  harmonic  vocabulary  based 
upon  the  sequences  of  tonality  established  in  these  completed  art 
forms  also  has  been  exhausted  and  for  the  past  half  century  com- 
posers have  been  concerned  with  the  development  of  new  harmonic 
idioms.  (Viz.,  Liszt,  Wagner,  Strauss,  Franck,  D'Indy,  Debussy, 
Ravel,  Shoenberg,  Busoni.)  As  these  composers  have  discovered 
and  employed  new  harmonies,  new  scales  and  new  sequences  of 
tonality  with  their  resultant  new  harmonic  progressions,  the  theo- 
retician has  endeavored  to  classify  their  discoveries  according  to 
his  established  system  with  results  weirdly  confusing.  The  crying 
need  of  the  moment  seems  to  be  a  new  system  for  the  naming  and 
classifying  of  all  possible  tonal  combinations. 


OUR  MUSICAL  IDIOM.  $l 

"Harmony  is  that  which  sounds  together,"  wrote  Bernard  Ziehn 
twenty-five  years  ago.  But  the  average  theoretician  comprehends 
only  those  simultaneously  produced  sounds  which  may  be  arranged 
in  series  of  superimposed  thirds.  In  the  meantime  the  composer 
has  consciously  employed  many  harmonies  which  are  not  formed  of 
superimposed  thirds.  (Viz.,  Debussy's  major  second  as  the  first 
interval  of  the  tonic  chord,  or  Shoenberg's  combinations  of  super- 
imposed fourths,  to  cite  familiar  examples.)  The  executive  artist, 
upon  whom  the  composer  is  dependent  for  the  delivery  of  his  mes- 
sage, is,  in  turn,  dependent  upon  the  theoretician  for  a  logical 
classification  of  the  new  harmonies.  The  composer  of  the  present 
is  almost  equally  dependent  upon  some  scientific  classification  of  his 
material.  Naturally  the  public  has  first  looked  to  him  for  this  classi- 
fication* But  he  seems  able  to  make  it  only  for  himself,  as  Reger 
and  Shoenberg  have  done.  In  any  event  the  world  has  been  slow 
to  adopt  these  special  classifications  and  is  still  seeking  a  general 
system  that  will  include  all  possible  harmonies  in  logical  order. 

That  system  has  been  evolved  by  Mr.  Ernst  Lecher  Bacon  of 
Chicago,  who  by  applying  the  principles  of  algebraic  permutations 
to  the  problem  has  succeeded  in  formulating  all  harmonies  that  may 
possibly  exist  in  the  present  system  of  twelve  tones  (of  itself  a  most 
important  service)  and  having  formulated  them,  has  found  a  system 
of  nomenclature  which  actually  describes  any  possible  combination 
of  tones  and  makes  a  general  or  special  classification  possible. 

The  value  of  this  new  system  of  nomenclature  to  the  executive 
artist  is  immediately  apparent.  That  puzzled  individual  may  name 
and  classify  the  new  tonal  combinations  which  he  is  required  to 
memorize  and  present  convincingly  to  the  public.  The  composer  is 
even  more  importantly  served  by  Mr.  Bacon's  researches.  For  he 
is  shown  at  a  glance  all  possible  harmonies  (th°re  are  but  350)  and 
all  possible  scales  (of  which  there  are  about  1490).  He  may  select 
from  the  clear  and  concise  tables  placed  at  his  disposal  those  har- 
monies and  scales  which  seem  to  him  useful  and  beautiful  and  having 
familiarized  himself  with  their  color  and  feeling,  in  short,  made 
them  a  part  of  his  own  consciousness,  may  employ  them  subjectively 
to  the  expression  of  feeling  and  sensibility,  to  the  building  up  of 
his  own  especial  harmonic  idiom.  For  though  the  composer's  work 
is  best  when  it  is  most  subjective,  he  is  constantly  obliged  to  concern 
himself  with  the  facts  of  his  art  and  out  of  these  facts  to  fashion 


562  THE  MONIST. 

that  delicate  fabric  of  feeling  and  fantasy  which  is  to  give  freer 
and  fuller  powers  of  expression  to  the  music  of  the  future. 

GLENN  DILLARD  GUNN. 

THE  FORMATION  OF  SCALES. 

The  chromatic  scale  has  become  established  as  the  basis  of 
modern  harmony.  Though  the  major  and  minor  modes  are  still 
accorded  that  recognition  which  the  printed  key  signature  seems  to 
imply,  modern  compositions  so  bristle  with  accidentals  that  even  to 
the  eye,  and  still  more  to  the  listening  ear,  is  it  evident  that  the 
restrictions  of  the  major  and  minor  system  have  been  destroyed. 

However,  few  of  our  modern  composers  treat  the  chromatic 
scale  purely  in  itself;  the  intrusion  of  other  scales  which  are  syn- 
thetically formed  from  the  chromatic  scale  is  always  felt.  The 
chromatic  scale  is  only  the  analytic  product  of  others,  which  consist 
of  combinations  of  its  semitones,  wherein  intervals  are  found  which 
are  collections  of  semitones.  Of  these  synthetically  formed  scales 
there  are  many  in  number  but  few  in  use;  and  each  may  form  a 
separate  harmonic  basis.  Beethoven  and  Liszt,  the  latter  more 
notably,  occasionally  used  scales  differing  markedly  from  the  major 
and  minor ;  but  their  appearance  was  only  incidental,  and  the  scales 
were  rarely  made  use  of  as  bases  for  harmonic  systems.  Busoni 
created,  as  he  says,  113  different  scales  through  rearrangements  or 
permutations  of  the  intervals  of  the  major  and  minor  scales.1  De- 
bussy has  used  a  few  unfamiliar  scales,  notably  the  whole  tone  scale, 
for  a  thorough  harmonic  basis.  However,  as  will  be  shown,  a  vast 
number  of  scales  that  have  never  before  been  conceived  are  opened 
to  discovery  through  the  application  of  the  principles  of  algebraic 
permutation  to  arrangements  of  tonal  sequences. 

A  scale  is  a  series  of  ascending  or  descending  tones.  Such  a 
series  may  conform  to  a  pattern,  a  regularly  recurring  succession 
of  intervals  in  certain  order,  bounded  by  a  fixed  interval ;  or  it  may 
not  conform  to  pattern.  The  pattern  may  or  may  not  conform  to 
the  duodecimal  system.  If  the  scale  conform  to  pattern  it  must  be 
jf  bounded  by  a  fixed  interval.  Intervals  of  simple  physical  ratio  are 
preferred,  and  these  are  found,  slightly  tempered,  in  the  duodecimal 
system. 

Until  now,  the  octave  only  has  been  consciously  used  as  a  fixed 

1  His  figures  are  incorrect  as,  mathematically  computed,  the  number  of 
permutations  of  the  combination  of  intervals  is :  (a)  of  the  major  scale,  21 ; 
(&)  of  the  minor  scale,  140. 


OUR  MUSICAL  IDIOM.  563 

interval,  but  there  is  no  reason  why,  in  specialized  cases,  other  inter- 
vals could  not  exist  between  corresponding  tones  in  recurrences  of 
the  pattern.  We  may  have  scales  repeating  at  each  fourth,  fifth, 
sixth,  seventh,  ninth,  tenth,  etc.  But  because  we  are  at  present 
engaged  in  a  classification  of  scales  which  incidentally  involves  the 
discovery  of  a  multitude  of  unheard-of  ones,  and  because  a  classi- 
fication of  such  scales  as  these  would  be  of  formidable  length,  we 
must  be  content  to  study  that  most  important  class  of  scales  in 
which  each  succeeding  repetition  begins  an  octave  above  or  below 
the  preceding  one. 

Again  we  must  distinguish  between  two  classes  of  scales  whose 
basis  is  the  octave.  The  first  class  is  that  one  in  which  the  smallest 
scale-units  in  the  octave  number  12  ;  this  is  our  duodecimal  system. 
The  second  class  contains  many  systems,  in  each  of  which  the  num- 
ber of  the  smallest  units  is  either  greater  or  less  than  12.  We  will 
consider  both  of  these  classes,  for  in  the  consideration  of  the  first 
class  we  can  enlarge  considerably  the  present  scope  of  the  duo- 
decimal system,  while  in  the  consideration  of  the  second  class  we 
may  discern  dimly  certain  possibilities  of  the  future.  First  we  will 
discuss  the  scale  possibilities  of  the  duodecimal  system. 

By  a  division  of  the  octave  into  twelve  parts  the  common 
chromatic  scale  is  formed.  Now  by  grouping  together  certain  of 
these  twelfths  of  an  octave,  the  so-called  "semitones,"  we  may  form 
scales  whose  gradation  is  uneven  and  less  refined  than  that  of  the 
chromatic  scale.  If  we  are  given  a  certain  combination  of  intervals 
which,  added  together,  give  the  octave,  we  can  permutate  these  in 
a  number  of  different  ways  ;  that  is,  we  can  rearrange  the  given 
intervals  to  form  different  scales.  We  also  may  have  combinations 
in  which  the  same  intervals  occur  more  than  once.  If  n  is  the  num- 
ber of  intervals  between  octaves  in  the  scale  and  «x  of  them  are 
alike,  and  n2  others  are  alike,  etc.,  the  number  of  scales  (P)  that 
can  be  formed  by  permuting  the  given  combination  of  intervals  is: 


(The  exclamation  point,  read  "factorial,"  denotes  that  the 
number  which  it  follows  is  a  product  of  all  integers  less  than  and 
including  itself,  each  integer  being  a  factor  only  once.) 

For  example,  we  desire  to  find  the  number  of  scales  that  can 
be  formed  with  the  intervals  of  the  major  scale.  The  major  scale 
consists  of  5  whole  tones  and  2  half  tones,  making  a  total  of  7 
intervals. 


5^4 


THE  MONIST. 


Thus      P=(n!/«1!n2!)  =  (7!/5!.2!) 

=  [1.2. 3. 4. 5.6.7/(1.2.3.4. 5.)  (1.2)] 
P  =  42/2=21. 

As  an  example  of  the  way  in  which  all  possible  scales  can  be 
formed  out  of  a  certain  combination  of  intervals,  20  scales  will  be 
formed  out  of  the  combination  (three  minor  thirds  and  three  minor 
seconds)  as  follows: 

The  symbol  ( —  3)  will  be  written  below  respective  minor  thirds. 
123 


-3-3-3 


-4 


-3     -3    -3 


-3 


io&E 


-3-3 


-3        -3     -3-3 
8 


-3      -3       -3 
9 


-3      -3  -3      -3  -3  -3  -3  -3        -3 

10  11  12 


o 


-3 
13 


-3-3  -3-3-3 

14 


-3-3        -3 


' 


15 


^ 


feb 


-3  -3  -3  -3       -3  -3 

16  17 

•  •  •  o 


18 


o 


-3        -3  -3 


«. 


fey 


>o*» 


o^± 


-3  -3-3 


-3  -3   -3 


-3  -3          -3 


19 


20 


"  n       i       Ip  °1T 

torfeo-^1-0^ 


-3  -3 


-3    -3  -3 


OUR  MUSICAL  IDIOM.  565 

It  will  be  observed  that  a  new  series  begins  with  each  double 
or  triple  bar. 

A  triple  bar  is  written  before  each  chromatic  elevation  of  the 
lowest  minor  third. 

A  double  bar  is  written  before  each  chromatic  elevation  of  the 
middle  minor  third. 

The  uppermost  minor  third  always  starts  at  its  lowest  possible 
position  and  is  raised  successively  to  its  highest  possible  one,  after 
which  a  change  is  made  in  the  relative  position  of  the  lower  minor 
thirds. 

This  method  of  forming  all  scales  from  a  certain  combination 
of  intervals  is  purely  arbitrary. 

Now  it  is  possible  to  form  a  great  number  of  combinations  in 
which  the  sum  of  the  intervals  is  an  octave.  Moreover,  as  we  have 
seen,  usually  a  number  of  scales  can  be  formed  out  of  each  combina- 
tion. Each  scale  is  to  be  considered  a  permutation  of  the  combina- 
tion's intervals. 

In  the  following  table  will  be  found  every  combination  possible 
with  intervals  as  small  as  the  minor  second  and  not  greater  than 
the  major  third.  Intervals  larger  than  the  major  third  are  not 
used  because  in  the  formation  of  scales  they  would  make  gradation 
too  abrupt  and  uneven.  The  table  will  also  include  calculations 
of  the  number  of  permutations  (to  be  regarded  as  the  number  of 
scales)  possible  with  each  respective  combination,  according  to  the 
formula.  The  vertical  columns  contain  the  intervals  minor  2d, 
major  2d,  minor  3d,  major  3d,  respectively.  The  horizontal  rows 
of  numbers  are  the  combinations.  A  number  (n)  falling  in  a 
vertical  column  (v)  means  that  the  interval  (v)  is  repeated  n  times 
in  the  combination  in  which  n  lies  (see  Table  I). 

To  make  the  function  and  construction  of  the  table  more  plain 
two  of  the  combinations  may  be  explained.  Combination  1  indi- 
cated by  the  number  in  the  extreme  left-hand  column,  consists  of 
twelve  semitones.  It  is  therefore  a  formula  of  the  chromatic  scale, 
and  has  therefore  only  one  permutation.  Combination  21  contains 
two  minor  seconds,  two  major  seconds  and  two  minor  thirds.  From 
it  may  be  formed  fifteen  scales  or  permutations. 

Means  have  now  been  shown  to  find  all  possible  scales  in  the 
twelve-tone  system,  scales  which  have  intervals  exceeding  the  major 
third  in  size  being  omitted.  Adding  the  number  of  permutations 
formed  with  all  combinations  a  total  of  1490  scales  is  found. 


566 


THE  MONIST. 


A  systematic  study  of  these  1490  new  scales  would  lead  to  the 
discovery  of  many  valuable  scales.  I  have  found  many  that  are 
interesting  by  this  method,  but  will  mention  only  a  certain  class  of 
these  scales,  which  I  will  call  equipartite  for  want  of  a  better  name. 

TABLE  I. 


I 

COMBINATIONS 

PERMUTATIONS 

18 

COMBINATIONS 

PERMUTATIONS 

ii 

O  H 

MINOR  ] 
SRCONDS 

MAJOR 
SECONDS 

8 

ta 

O 

ta 

in 

PERMU- 
TATIONS 

gS 

*£ 

s£ 

MAJOR 
THIRDS 

J 

1% 

<2 

0  H 

PERMU- 
TATIONS 

£8 
a£ 

go 

<« 

Sw 

II 

II 

12 

Pl2!/I2! 

I 

3 

3 

6!/3l3l 

2O 

2 

10 

I 

n!/io! 

II 

19 

2 

5 

7!/2!5! 

21 

3 

9 

I 

101/9! 

IO 

20 

2 

3 

I 

61/213! 

60 

4 

8 

2 

io!/2!8! 

45 

21 

2 

2 

2 

6!/2!  2!  2! 

go 

5 

8 

I 

9!/8! 

9 

22 

2 

I 

2 

51/2!  2! 

30 

6 

7 

I 

I 

9!/7! 

72 

23 

2 

2 

I 

51/2!  2! 

30 

7 

6 

3 

9!/3!6! 

84 

24 

I 

4 

I 

6l/4i 

30 

8 

6 

i 

I 

8!/6! 

56 

25 

I 

2 

I 

I 

Si/2! 

60 

9 

6 

2 

8  !/2!  6! 

28 

26 

I 

I 

3 

573! 

20 

10 

5 

2 

I 

81/215! 

1  68 

27 

I 

i 

2 

4!/2! 

12 

ii 

5 

I 

I 

71/5' 

42 

28 

6 

6!/6! 

I 

12 

4 

4 

8!/4!4! 

70 

29 

4 

I 

§1/41 

5 

13 

4 

2 

I 

7!/2U! 

105 

30 

3 

2 

5!/2!3! 

10 

M 

4 

I 

2 

7!/2!4! 

i°5 

31 

2 

2 

41/2!  2! 

6 

15 

4 

2 

61/214! 

15 

32 

I 

2 

I 

4l/a! 

12 

16 

3 

3 

I 

71/513! 

140 

33 

4 

474! 

I 

J7 

3 

i 

I 

I 

61/3] 

120 

34 

3 

31/3! 

I 

Total     1490 

An  equipartite  scale  is  one  in  which  the  same  pattern  of  inter- 
vals is  repeated  an  integral  number  of  times  within  the  octave. 
If  a  scale  is  bipartite  a  group  of  intervals  will  appear  twice  within 
the  octave  with  no  remainder;  if  the  scale  is  to  begin  on  F  its  two 
parts  begin,  respectively,  on  F  and  B.  As  a  result  in  this  case  it  is 
immaterial  whether  the  tonic  is  B  or  F,  for  the  scale  sounds  alike 
either  way,  except  for  the  transposition. 


OUR  MUSICAL  IDIOM. 


567 


We  may  split  the  sum  of  twelve  semitones  (semitones  being 
regarded  as  intervals)  into  two  parts  or  three.  Dividing  it  into  two 
parts,  each  part  containing  six  semitones,  allows  us  again  to  divide 
this  semi-octave  into  two  or  three  parts.  Dividing  the  octave  into 

TABLE  II  (FOR  BIPARTITE  SCALES). 


COMBINATIONS 

PERMUTATIONS 

No. 

I 

MINOR 
SECOND 

2 

MAJOR 
SECOND 

3 

MINOR 
THIRD 

4 

MAJOR 
THIRD 

CALCULATIONS 

No.  OF 
PERM  . 

I 

6 

P=6!/6! 

I 

2 

4 

I 

P==5!/4' 

5 

3 

3 

I 

P=4'/$l 

4 

4 

2 

2 

P=4!/2!  2! 

6 

5 

2 

I 

P=3!/2! 

3 

6 

I 

I 

I 

P=*i 

6 

7 

3 

P=3'/3> 

i 

8 

I 

I 

P=2! 

2 

9 

2 

P=2!/2! 

I 

Total     29 


TABLE  III  (FOR  TRIPARTITE  SCALES). 


COMBINATIONS 

PERMUTATIONS 

No. 

I 

MINOR 
SECOND 

MAJOR 
SECOND 

3 

MINOR 
THIRD 

4 

MAJOR 
THIRD 

CALCULATIONS 

No.  OF 
PERM. 

I 

^4 

P=4!/4! 

L_J 
I 

2 

2 

I 

P=3!/2! 

3 

3 

I 

I 

P=2! 

2 

4 

2 

P=2!/2! 

I 

5 

I 

P=i! 

I 

Total     8 


three  parts,  each  part  has  four  semitones,  which  may  again  be 
divided  by  two.  Thus  we  may  split  the  octave  into  2,  3,  4,  and  6 
equal  parts.  Scales  formed  by  such  divisions  may  be  called,  respec- 
tively, bipartite,  tripartite,  quadripartite,  and  sexpartite.  As  the 


568 


THE  MONIST. 


last  two  types  may  be  classed  under  the  first  and  second  they 
do  not  require  a  separate  classification.  In  Tables  II  and  III 
the  combinations  in  the  bipartite  and  tripartite  types  are  given;  in 
other  words,  the  possibilities  of  combinations  with  six  and  four 
semitones,  respectively,  are  shown.  Each  arrangement  of  a  com- 
bination is  then  repeated  in  the  remaining  half  or  two-thirds  of  the 
octave. 

A  few  interesting  equipartite  scales  are  herewith  shown: 

1  2 


-o- 


(1,  2  and  3  are  from  Table  2,  combination  No.  4;  4,  5  and  6  are  from 
Table  3,  combination  No.  3.) 

Scales  formed  by  permutating  combination  No.  4  in  Table  II, 
and  combinations  Nos.  2  and  3  in  Table  III  are  especially  interesting. 
No.  1  is  formed  by  alternating  major  and  minor  seconds,  while  No. 
3  is  formed  in  the  same  way,  except  that  in  it  the  order  of  the  inter- 
vals of  No.  1  is  reversed.  Even  such  a  mechanically  formed  scale 
as  this  sounds  beautiful  and  original.  It  is  a  noteworthy  fact  that 
in  scales  1  and  3  the  chords  formed  on  every  degree  are  diminished. 
Scales  Nos.  4  and  5  are  built  similarly;  only  a  minor  third  and  a 
minor  second  alternate.  Chords  formed  on  every  degree  of  these 
scales  are  augmented. 

SCALES  FORMED  FROM  SYSTEMS  OTHER  THAN  THE  DUO- 
DECIMAL. 

Although  to-day  the  importance  of  systems  containing  other 
intervals  than  multiples  of  semitones  is  questionable,  it  is  neverthe- 
less interesting  to  know  that  such  systems  may  be  exploited  for 
scale  and  harmonic  possibilities  in  the  same  manner  as  our  present 
system.  Busoni  has  already  experimented  with  the  tripartite  tone 
scale ;  that  is,  a  scale  in  which  each  whole  tone  is  divided  into  three 
instead  of  two  whole  parts.  The  physicist  may  scorn  the  idea  of  a 


OUR  MUSICAL  IDIOM.  569 

new  system,  knowing  that  the  duodecimal  system  contains  the  sim- 
plest physical  intervals,  yet  it  must  be  remembered  that  the  perfect 
intervals  are  also  not  found  in  the  12-tone  system,  because  of  "tem- 
pering." Moreover  in  the  other  systems  many  of  the  most  impor- 
tant intervals  of  the  duodecimal  system  will  be  duplicated.  Al- 
though probably  no  system  will  ever  be  of  equal  importance  with 
the  duodecimal,  it  is  not  inconceivable  that,  just  as  certain  new 
scales  within  our  present  system  have  been  chosen  by  recent  com- 
posers as  harmonic  and  melodic  idioms  of  expression,  so  certain 
"foreign"  systems  may  once  be  chosen  for  similar  purposes. 

Accordingly,  we  are  to  consider  any  equal  divisions  of  the 
octave.  However,  certain  divisions,  as  for  example  into  11  or  13 
equal  parts,  are  not  of  importance,  since  the  intervals  formed  in  this 
way  would  only  be  confounded  with  poorly  tuned  intervals  of  the 
12-tone  scale.  In  order  to  discriminate  in  the  selection  of  numbers 
with  which  to  divide  the  octave  it  is  well  to  choose  only  those  num- 
bers which  are  multiples  of  the  smallest  prime  numbers,  2,  3,  and 
5.  We  may  call  each  of  these  systems  an  "N-tone  chromatic  sys- 
tem." If  the  system  is  one  in  which  the  number  of  smallest  intervals 
is  9,  we  may  call  it  a  9-tone  chromatic  system.  We  are  not  bound 
to  confine  the  use  of  the  term  "chromatic"  to  our  duodecimal  system, 
since  in  its  musical  application  the  word  is  used  to  describe  a  suc- 
cession of  the  smallest  possible  intervals. 

In  considering  the  N-tone  chromatic  systems  we  may  go  through 
the  same  steps  through  which  we  have  passed  in  considering  the 
duodecimal  system.  In  each  of  these  unfamiliar  systems  there  are 
chromatic  intervals  which  may  be  combined  and  permutated  to  form 
scales  of  more  rapid  and  uneven  gradation.  Just  as  before,  we  have 
to  set  a  certain  limit  to  the  size  of  an  interval  employed  in  one  of 
these  scales.  In  the  five-tone  system  a  coupling  of  only  two  chro- 
matic intervals  produces  an  interval  almost  too  great  to  exist  in  a- 
scale  of  moderately  refined  gradation.  In  the  24-tonal  system  a 
coupling  of  as  many  as  6  intervals  into  1  is  acceptable.  It  will  readily 
be  seen  that  to  construct  tables  for  all  N-tonal  systems  through 
which  an  infinite  number  of  gradations  is  possible,  would  require 
much  space.  It  has  already  been  stated  that  those  systems  having 
numbers  of  chromatic  intervals  equal  to  multiples  of  2,  3  and  5,  are 
most  important.  They  are  systems  of  4,  5,  6,  8,  9,  10,  12,  14,  '15, 
16,  18,  20,  21,  22,  24,  etc.,  chromatic  tones ;  for  demonstration  I  will 
select  only  2  of  these;  namely  8-  and  9-tone  systems. 


570 


THE  MONIST. 


In  the  tables  that  follow  I  will  give  the  number  of  combinations 
and  calculate  the  number  of  permutations  for  each  combination  with 
the  selected  N-tonal  systems.  In  other  words,  scale  possibilities 
with  systems  having  8  and  9  tones  will  be  shown  in  each  respective 
table. 

TABLE  II,  N=8  TABLE  III,  N— 9 


I 

COMBINATIONS 

PERMUTATIONS 

i 

COMBINATIONS 

PERMUTATIONS 

a 

PERMU- 
TATIONS 

H 

-*? 
U 

O 

8 

OCTAVE 

1 

OCTAVE 

< 

h 
<° 

O  H 

PERMU- 
TATIONS 

•AVXDQ 

? 

i 

OCTAVK 

1  OCTAVK 

i 

=  » 

31 

U  H 

81/8! 

I 

9 

9!/9! 

I 

2 

I 

7!/6| 

7 

2 

7 

I 

81/71 

8 

3 

5 

I 

61/51 

6 

3 

6 

I 

71/01 

7 

4 

4 

2 

61/214! 

15 

4 

5 

2 

71/2!  5! 

21 

•5 

3 

I 

I 

5l/3l 

20 

5 

4 

I 

I 

6!/4! 

30 

6 

2 

3 

5!/2!3! 

10 

6 

3 

3 

6!/3!3! 

2O 

7 

2 

2 

41/2!  2! 

6 

7 

3 

2 

5V2!3! 

10 

8 

I 

2 

I 

4'/2! 

12 

8 

2 

2 

I 

51/2!  2! 

30 

9 

4 

4'/4! 

I 

9 

I 

4 

5»/4l 

5 

10 

i 

2 

3!/2! 

3 

10 

ii 

I 

I 
3 

2 

I 

4!/2! 
4!/3! 

12 

4 

Total     81 

Intervals  used  do  not  exceed  |  octave. 

12 

3 

3!/3l 

I 

Duodecimal  System^. 


Total     149 

Intervals  used  do  not  exceed  |  octave 
or  a  major  third,  as  translated. 

Number  of  intervals  corresponding 
to  Duodecimal  System  =  3. 

The  Numbers  of  Tones  and  Intervals  Found  Correspondingly  in 

Any  Two  N -Tonal  Systems. 

If  we  choose  a  common  tonic  for  all  N-tone  chromatic  scales 
we  will  find  certain  other  tones  which  are  common  to  two  or  more 
of  these  scales.  For  example,  if  we  form  both  a  9-tone  chromatic 
and  a  duodecimal  scale  upon  C,  we  will  expect  to  find  two  tones  in 
common  besides  the  C  and  its  octave.  They  will  be  E  and  G  sharp ; 
for  each  of  these  tones  marks  the  partition  of  the  octave  into  three 
equal  parts.  This  means  that  certain  intervals  in  one  system  are 


OUR  MUSICAL  IDIOM.  571 

the  same  as  intervals  of  another.  But  an  interval  common  to  two 
systems  cannot  be  the  same  multiple  of  the  smallest  unit  in  each  sys- 
tem. If  we  desire  to  find  the  number  of  intervals  which  are  found 
correspondingly  in  each  of  the  two  systems,  we  need  merely  to 
find  the  largest  factor  common  to  the  number  of  chromatic  divisions 
of  both  systems.  For  example,  to  find  the  number  of  intervals 
which  are  common  to  the  18-tone  and  the  12-tone  chromatic  scales 
we  find  the  G.  C.  F.  of  18  and  12,  which  is  6.  This  is  the  desired 
number.  Of  course,  intervals  which  are  multiples  of  this  common 
interval  (the  whole-tone,  in  this  case)  are  also  common  to  both 
systems. 

Intervals  of  N-Tone  Chromatic  Scales. 

Throughout  our  entire  treatment  of  scale  possibilities  there  is 
one  interval  which  remains  constant ;  namely,  the  octave.  The  ratio 
of  this  interval,  that  is  the  ratio2  of  the  vibration  frequency  of  the 
higher  tone  to  that  of  the  lower  tone  is  always  2.  If  N  is  the  fre- 
quency of  the  lower  tone,  its  octave  is  2  N.  Now  N  and  2  N  may 
be  written  as  2°  N  and  21  N  respectively,  since  any  quantity  with  an 
exponent  0  equals  unity.  It  is  evident  that  the  frequencies  of  any 
tones  between  NX 2°  and  Nx2*  can  be  expressed  as  N  times  the 
coefficient  2  with  an  exponent  varying  between  0  and  1. 

If  the  octave  contains  r  equal  intervals,  the  difference  between 
0  and  1  of  the  exponent  of  2  will  be  divided  into  r  parts.    This  is 
true  because  (a)  equal  intervals  form  equal  ratios  of  vibration;  and 
(&)  equal  ratios  may  be  expressed  as  the  quotients  of  a  constant 
in  which  the  difference  of  the  constant's  exponents  in  the  numerators 
and  respective  denominators  remains  constant.     To  illustrate: 
21/2°  =  21-°  =  2 
26/25  =  26-6  =  2. 

Hence  21/2°  =  26/2*. 

Thus 

2^  or  v  2 

expresses  the  ratio  of  any  interval  formed  by  two  adjacent  tones 
in  an  equally  tempered  scale  of  r  intervals.  Moreover  the  intervals 
which  any  tonic  (arbitrarily  chosen  in  the  case  of  the  equally  tem- 
pered scales)  forms  with  the  successive  ascending  tones  above  it, 
are,  respectively: 

2iA  22A  2»A  2*A 2<'-1>A  2'/'  or  2. 

2  This  ratio  is  physically  defined  as  the  interval  itself. 


572  THE  MONIST. 

From  these  facts  we  derive  two  general  formulas:  (A)  ex- 
pressing the  physical  interval-  or  vibration  ratio  between  2  tones 
and  (B)  the  vibration  frequency  of  any  tone  lying  above  a  given 
tone,  N. 

(A)  I  (interval }=2c/r, 

J* 

where  r  =  the  number  of  chromatic  intervals  per  octave,  in  the  given 
system;  and  c  -  the  number  of  chromatic  intervals  separating  the 
two  tones  whose  physical  ratio  is  to  be  found. 

With  these  formulas  we  can  express  the  various  intervals  of 
any  equally  tempered  scale. 

NOTATION  OF  SCALES. 

In  considering  the  great  number  of  scales  of  which  we  have 
learned  in  the  previous  section  we  are  confronted  with  the  problem 
of  their  notation.  Our  present  notation  is  really  suited  for  seven 
scales  only ;  namely,  the  major  scale  and  the  scales  formed  by 
cyclically  rotating  the  permutation  of  the  intervals  of  the  major 
scale,  that  is,  the  Dorian,  Phrygian,  Lydian,  Mixolydian,  etc.  We 
cannot  write  even  a  minor  scale  without  the  use  of  an  accidental. 
Then  with  regard  to  the  1483  other  scales,  because  of  this  great 
number  and  variety,  we  cannot  do  more  than  make  general  state- 
ments. 

We  realize,  to  begin  with,  that  the  ideal  notation  of  our  present- 
day  music  should  be  one  which  is  designed  to  eliminate  the  incon- 
veniences of  accidentals.  Such  a  notation  would  be  naturally  one 
designed  from  the  chromatic  scale;  and  because  the  chromatic 
scale  contains  all  of  the  1490  other  scales  of  the  duodecimal  system, 
it  would  be  adaptable,  in  a  perfect  sense,  to  all  of  these  scales.  We 
could  accomplish  the  notation  of  the  chromatic  12-tone  scale  with  a 
six-line  staff  giving  each  degree  a  separate  line  or  space,  as  shown : 

f^_  O  N 


-<3- 


O   *  ' 


The  major  scale  on  this  staff  would  be: 


do     re  mi  fa  sol    la    ti    do 
The  minor  scale  would  be: 


OUR  MUSICAL  IDIOM.  573 


The  mental  picture  we  obtain  of  the  relation  of  the  intervals  of 
these  two  scales  in  this  manner  is  alone  an  advantage.  Furthermore, 
in  the  six-line  staff  notation  we  are  less  bound  to  avoid  deviations 
from  our  chosen  scale ;  we  are  freer  to  escape  from  the  tyranny  of 
sharps  and  flats.  An  abhorrence  of  accidentals  has  always  tied  us 
to  our  chosen  scales.  Other  advantages  of  this  notation  could  be 
cited,  but  the  chief  one  is,  of  course,  that  merely  through  the  addi- 
tion of  another  line  (which  does  not  confuse  us  optically)  we  are 
able  entirely  to  avoid  accidentals. 

However,  the  difficulty  of  introducing  this  system  into  common 
use  would  be  almost  too  great  to  be  overcome.  An  attempt  at  this 
could  be  likened  to  the  recent  attempts  at  introducing  a  universal 
language;  for  were  we  all  to  learn  a  universal  language  we  would 
still  have  to  retain  a  knowledge  of  the  old  for  its  literature.  We 
are  therefore  compelled  to  adjust  our  new  scales  to  the  common 
notation  of  the  five-line  staff. 

We  may  eliminate  from  consideration  not  only  the  major  scale 
and  those  scales  formed  by  a  cyclic  rotation5  of  its  permutation  of 
intervals,  but  also  the  minor  scale  with  its  corresponding  scales 
formed  by  a  similar  cyclic  rotation.  This  suggests  to  us  a  process 
that  will  greatly  simplify  our  whole  problem.  We  see  that  the 
notation  for  one  scale  is  suitable  for  all  other  scales  formed  by  a 
cyclic  rotation  of  the  permutation  of  its  intervals.  The  number  of 
these  scales  will  depend  upon  the  number  of  tones  or  intervals  in 
the  original  one.  The  notation  for  a  scale  of  n  tones  or  n  intervals 
will  serve  for  (n-1)  cyclically  related  scales.  Thus  one  notation 
serves  for  n  scales. 

We  realize  that  out  of  a  certain  combination  of  intervals  we 
may  form  more  than  one  cyclic  group,  for  some  combinations  have 
as  many  as  168  scales  while  in  no  cyclic  group  can  there  be  more 
than  11  different  scales.  A  formula  with  which  we  may  calculate 
the  number  of  cyclic  groups  in  each  combination  is : 

G*  («- !)!/%!  *, in,!.... 

where  n  is  the  number  of  intervals  in  a  combination,  and  nlt  n.,,  n3, 

3  The  term  defines  itself.  A  cyclic  rotation  of  a  permutation  is  one  in 
which  the  terms  are  always  written  in  the  same  order,  but  each  successive 
permutation  begins  with  the  second  term  of  the  preceding  one.  The  following 
is  a  cyclic  group  of  permutations :  ABCD,  BCD  A,  CDAB,  DABC. 


574  THE  MONIST. 

etc.  are  the  numbers  of  times  respectively  which  certain  intervals 
are  repeated  in  the  combination.  There  are  few  exceptions  to  this 
formula,  all  of  which  are  of  one  type.  The  erroneous  type  is  that 
in  which  («1  +  M2  +  w3  +  n4. . . )  exceeds  (w-1).  These  exceptions 
often  cause  fractions  which  cannot  be  integrally  expressed.  In  cases 
of  this  exception  we  must  find  our  number  of  cyclic  groups  by  actual 
trial.  But  if  we  have  found  one  signature  suitable  for  each  whole 

•  cyclic  group  of  scales  we  have,  in  general,  shown  only  one-twelfth 
of  possible  signatures,  for  in  most  cases  a  different  signature  is 
necessary  for  each  chromatic  degree.  Only  in  equipartite  scales 

%  are  fewer  signatures  than  twelve  necessary  to  each  group.  If  a 
scale  is  bipartite  only  six  signatures  are  necessary;  if  tripartite, 
four;  if  quadripartite,  three;  and  if  sexpartite,  two. 

As  we  are  considering  these  227  scales  representing  cyclic 
groups  primarily  for  their  notation,  we  are  confronted  with  the 
question,  what  signature  shall  we  give  to  a  work  based  on  a  scale 
like  the  following? 


oP" 


None  of  our  conventional  signatures  for  major  scales  will  apply 
to  this  scale ;  for  we  see  the  three  essential  signatures  are : 


d  flat  being  unnecessary  as  a  signature  because  it  is  cancelled  im- 
mediately, the  scale  being  an  8-tone  scale,  which  necessitates  the 
repetition  of  one  note. 

We  will  find  that  most  of  the  scales,  like  this  one,  will  require 
signatures  other  than  those  which  we  have  employed  for  our  major 
and  minor  modes.  Consequently  we  will  not  try  to  reconcile  our 
customary  signatures  with  those  natural  to  the  new  scales.  There- 
fore, in  order  to  make  a  signature  for  any  scale  on  any  degree,  write 
down  those  accidentals  which  appear  in  the  notation  of  the  scale, 
omitting  those  accidentals  only  which  are  cancelled  as  the  scale  con- 
tinues. We  may  rightly  call  this  a  natural  system  of  signatures. 

Concerning  the  method  of  finding  each  scale  representative  of 
a  cyclic  group  for  a  given  scale  degree,  the  following  means  are 
perhaps  the  simplest: 


OUR  MUSICAL  IDIOM.  575 

1.  Choose  an  interval  which  occurs  singly  in  the  combination 
and  place  it  in  the  lowest  position  in  the  scale. 

2.  Permutate  the  other  intervals  above  it  in  every  possible  way. 

3.  Each  permutation,  with  the  first  interval  remaining  in  a 
fixed  position,  will  form  a  desired  scale. 

4.  When  no  interval  occurs  singly  in  the  combination  there  is 
no  rule  which  applies  generally;  but  because  of  the  small  number 
of  combinations  of  this  character  the  desired  scales  can  be  easily 
found  by  trial. 

There  is  little  need  for  investigating  the  problems  of  notation 
of  N-tonal  systems  until  such  systems  come  into  use.  Solutions  to 
such  problems  are  really  simple  and  arbitrary.  Suffice  it  to  say, 
there  is  no  need  of  retaining  the  five-line  staff  for  N-tonal  notation. 
It  would  be  unfortunate  if  one  were  compelled  to  read  a  totally 
new  system  of  intervals  from  a  staff  with  which  one  would  con- 
stantly associate  accustomed  intervals. 

Although  it  may  seem  strange  that  so  much  attention  is  paid 
a  subject  like  the  formation  of  scales,  there  is  nevertheless  justi- 
fication in  an  investigation  of  this  sort.  A  scale  has  far  greater  im- 
portance than  the  mere  sequence  of  tones  comprising  it  would  imply. 
Practically  all  of  the  hundreds  of  melodies  we  know  can  be  formed, 
almost  without  accidentals,  from  the  major  and  minor  scales.  Vir- 
tually all  of  the  common  harmonies  can  be  constructed  from  these 
modes.  The  vast  amount  of  musical  thought  and  feeling  has  until 
recently  expressed  itself  in  major  and  minor.  But  the  chromatic 
scale  offers  a  much  wider  field  of  expression ;  for  it  contains  not 
only  the  major  ^nd  minor  scales,  but  over  fourteen  hundred  others. 
Nevertheless,  although  the  chromatic  scale  has  become  the  basis  for 
modern  harmony,  melody  does  not  seem  to  flow  freely  chromatically. 
Our  musical  speech  continually  demands  some  simple  group  of  tones 
and  larger  intervals.  Without  some  limitation  more  binding  than 
the  chromatic  scale,  we  are  helplessly  confused  with  the  wealth  of 
possibilities.  Such  limitations  are  found  among  the  multitude  of 
scales  derived  synthetically  from  the  chromatic. 

Debussy  and  some  of  his  colleagues  have  made  their  idiom  or 
"dialect,"  as  it  were,  the  whole-tone  scale.  This  one  scale,  because 
of  its  uncertain  "tonality,"  and  its  "color,"  has  been  the  outstanding 
characteristic  of  the  French  impressionists. 

A  few  other  scales,  such  as  the  Greek  "modes,"  which  are  all 
cyclically  related  to  the  major  scale,  have  been  the  basis  for  numer- 


576  THE  MONIST. 

cms  works.  On  the  whole,  there  is  no  reason  why  other  scales, 
among  the  vast  number  shown  to  exist,  should  not  become  equally 
important  idioms  of  expression. 

Limitations  of  space  unfortunately  prevent  me  from  tabulating 
completely  the  fourteen  hundred  and  ninety  scales  of  the  duodecimal 
system. 

Concerning  the  N-tone  scales,  it  is  well  to  consider  for  illus- 
trative purposes  the  words  of  Busoni  in  regard  to  his  tripartite  tone 
scale:4  "The  tripartite  tone,"  says  he,  "has  for  some  time  been  de- 
manding admittance,  and  we  have  left  the  call  unheeded."  With 
the  tripartite  tone  he  encounters  a  difficulty  which  will  be  found 
also  in  considering  other  N-tone  scales.  He  says  we  would  lose 
through  the  tripartite  tone  the  minor  third  and  the  perfect  fifth. 
Now  a  chromatic  scale  in  which  the  most  important  intervals  do  not 
occur  (intervals  whose  ratios  are  expressed  as  quotients  of  the 
smaller  integers)  will  never  form  quite  as  valuable  a  system  as  a 
chromatic  scale  that  contains  them.  Realizing  this,  Busoni  has  at- 
tempted to  reconcile  the  12-tone  with  the  18-tone  system;  that  is,  a 
system  of  bipartite  tones  with  one  of  tripartite  tones.  His  solution 
is  naturally  a  36-tone  scale  involving  the  sexpartite  tone.  To  enter- 
tain any  hopes  for  a  system  of  sexpartite  tones  seems  to  me  futile. 
A  system  of  24-tone  chromatics  might  be  better  reconciled  with  our 
duodecimal  system.  This  example  merely  shows  us  that  we  cannot 
attempt  to  reconcile  the  N-tonal  systems  with  each  other  or  with 
the  duodecimal  system.  An  18-tone  chromatic  system  is  perhaps 
next  in  importance  to  the  duodecimal  system,  but  it  is  comprehensive 
and  important  enough  in  itself,  even  though  it  does  not  contain 
minor  thirds  and  perfect  fifths. 

Again  we  must  consider  how  we  are  to  produce  these  tones, 
as  Btisoni  has  mentioned  in  regard  to  his  tripartite  tone  scale.  For 
experiment  and  a  training  of  the  ear  to  the  tripartite  tone  Busoni 
recommends  Dr.  Thaddeus  Cahill's  dynamophone,  an  instrument 
which  would,  however,  be  very  difficult  to  obtain  or  to  construct. 
A  Seebeck's  siren  with  a  special  disk  for  each  system  would  be  a 
good  substitute.  The  number  of  holes  in  each  circular  row  could 
be  mathematically  computed  with  the  help  of  the  formulas: 


4  For  Busoni's  statements  read  his  Sketch  of  a  New  Esthetic  of  Music, 
New  York,  1911. 


OUR  MUSICAL  IDIOM.  577 

which  are  explained  in  previous  pages.    A  motor  to  Devolve  the  disk 
would  furnish  a  constant  speed  of  rotation. 

Such  experiments  would  furnish  means  of  acquiring  a  sense  of 
intervals  other  than  those  to  which  we  are  accustomed ;  but,  in 
Busoni's  words,  "only  a  long  and  careful  series  of  experiments  and 
a  continued  training  of  the  ear  can  render  this  material  approach- 
able and  plastic  for  the  coming  generation,  and  for  art." 

A  NEW  HARMONY. 

Our  present  system  of  harmony,  the  system  of  chords  (har- 
monies formed  of  superimposed  thirds),  is  deficient  in  two  important 
respects.  First,  it  is  often  unwieldy ;  and  second,  it  is  not  fully  com- 
prehensive. This  latter  shortcoming  is  partly  responsible  for  the 
former,  since  it  is  true  that  we  may  represent  certain  harmonies, 
seemingly  not  within  the  scope  of  our  system,  in  complex  ways.  To 
illustrate:  let  us  examine  the  various  unsatisfactory  ways  of  de- 
scribing the  simple  harmony: 


If  the  harmony  is  a  chord,  we  must  be  able  to  build  it  up  by 
superimposing  thirds.  But  no  complete  chord  exists  that  contains 
each  of  these  and  only  four  notes. 

But  there  are  chords  containing  more  than  four  notes  which 
contain  the  notes  of  the  harmony,  such  as  the  following: 

A  B  C^    D 


tl 

From  these  chords  we  may  strike  out  those  notes  foreign  to 
the  harmony  and  derive  what  we  call  an  incomplete  chord.  The 
harmony : 


may  therefore  be  termed  an  incomplete  nth  chord  (as  in  A,  B,  or 
D)  or  an  incomplete  ijth  chord  (as  in  C).  If  we  are  willing  to 
recognize  an  incompleteness  of  this  sort  as  mathematically  rigid, 
we  must  still  admit  that  such  a  naming  of  the  harmony  as  an  in- 
complete llth  does  nothing  more  than  justify  its  existence  among 
chords.  It  does  not  name  the  harmony  for  there  are  innumerable 


578  THE  MONIST. 

incomplete  llths.  To  name  the  llth  chord  in  each  case  is  difficult, 
the  general  method  being  that  of  determining  upon  what  degree  of 
the  major  or  minor  scale  it  is  built.  But,  again,  must  we  consider 
all  harmonies  in  the  light  of  the  major  or  minor  scales  to-day  when 
many  other  scales  are  being  used?  Furthermore  we  must  find  where 
the  incompleteness  lies.  Lastly  one  would  suppose  that  every  har- 
mony has  one  fundamental  position,  but  here  are  four.  One  should 
be  able  to  tell  what  sort  of  inversion  of  the  fundamental  harmony 
the  one  in  question  is.  How  is  this  possible  when  the  harmony  in 
its  position  is  a  different  inversion  with  each  fundamental? 

If  we  allow  ourselves  the  latitude  of  recognizing  diminished 
thirds,  we  may  say  the  harmony  is  composed  of  two  diminished 
thirds  separated  by  a  minor  third.  Taking  this  liberty  we  might 
have  a  specialised  chord  or  so-called  altered  chord,  but  how  shall 
we  describe  any  particular  one  ?  Moreover,  it  is  false  to  assume  that 
diminished  thirds  are  thirds  at  all;  for  they  are  seconds. 

Sometimes,  if  the  harmony  is  preceded  or  followed  by  others, 
we  may  analyze  it  under  our  present  system  by  considering  certain 
tones  as  "passing  tones,"  "suspensions,"  "afterbeats,"  "syncops," 
"organ-point,"  etc.  The  awkward  system  of  figured  bases  sometimes 
affords  a  means  for  expressing  simpler  chords. 

If  such  is  the  fate  that  a  simple  harmony  like  the  above  suffers 
in  analysis,  what  lot  befalls  the  multitiude  of  more  complex  har- 
monies ?  The  best  that  modern  analysis  can  do  for  them  is  to  treat 
them  in  relation  to  surrounding  harmonies.  Even  then,  "unresolved 
suspensions,"  etc.  are  continually  met  with  in  modern  music.  If 
harmony  is  "that  which  sounds  together,"  we  should  be  able  to 
define  any  combination  of  simultaneously  sounding  tones,  whether 
this  combination  is  surrounded  by  others  or  not.  A  note  suspended 
from  a  consonant  to  a  dissonant  chord  is  sounding  in  the  second  as 
well  as  in  the  first  harmony.  Does  not  an  organ-point  form  a 
separate  harmony  with  each  of  a  series  of  chords  "moving  through 
it,"  even  though  these  chords  are  dissonant  with  the  organ-point? 
A  harmony  is  a  harmony  whether  dissonant  or  consonant.  Yet  of 
the  vast  majority  of  dissonant  harmonies  few  can  be  adequately 
named  and  classified  in  themselves. 

The  chord  system  is  adequate  in  analysis  of  older  works  only. 
It  can  give  only  a  superficial  analysis  of  modern  works. 

A  more  important  objection  even  than  that  of  inadequate 
nomenclature  is  that  by  reason  of  our  use  of  the  chord  system  we 


OUR  MUSICAL  IDIOM.  579 

are  hindered  in  enlarging  our  scope  of  harmonies.  The  conception 
of  harmonies  given  us  by  this  system  restrains  us  from  enlarging 
our  harmonic  vocabulary.  Bred  in  the  chord  system,  we  are  prone 
to  regard  any  harmony  which  is  not  chordic  in  construction  as  a 
mere  variance  of  some  "simple"  chordic  form.  Many  a  stereotyped 
theorist  would  shudder  at  the  notion  of  giving  the  above  harmony 
a  prolonged  and  separate  existence.  It  must  be  immediately  re- 
solved into  a  stable  form ;  the  tonic  triad  of  G  major,  etc.,  etc.  Are 
we  blind  to  the  existence  of  harmonies  not  made  up  of  superimposed 
thirds?  Shall  we  refuse  to  recognize  non-chordic  harmonies  merely 
for  the  technical  reason  that  we  employ  a  system  of  superimposed 
thirds,  which  was  an  expedient  solution  to  theoretical  problems  two 
centuries  ago?  Because  of  the  limitations  of  our  present  system,  a 
vast  number  of  harmonies  remain  to-day  virtually  undiscovered. 
Although  many  have  been  employed  passingly  and  subconsciously, 
few  have  been  employed  deliberately,  few  are  spoken  of  as  a  part 
of  the  composer's  vocabulary. 

Fully  realizing  the  importance  of  the  chord  system  in  the  anal- 
ysis of  older  works  (for  these  were  conceived  in  the  spirit  of  the 
chord  system),  I  believe  it  is  important  that  a  new  and  fully  com- 
prehensive system  should  supplement  it,  a  system  that  would  prove 
adequate  for  the  analysis  of  modern  writings.  Whereas  the  old 
system  embraces  chords  only,  the  new  should  embrace  all  harmonies. 
The  chord  scheme  would  then  take  its  place  as  a  sub-system  of  the 
more  general  and  all-inclusive  system. 

Just  as  the  present  method  is  more  than  a  mere  scheme  of 
nomenclature,  so  the  one  which  I  propose  should  be  considered  as 
affecting  more  than  the  mere  naming  of  harmonies.  The  chord 
system  teaches  us  that  all  harmonies  are  chords,  are  built  by  imposing 
major  and  minor  thirds  upon  each  other.  The  proposed  system 
should,  as  will  be  shown,  teach  us  to  recognize  harmonies  which  are 
built  by  superimposing  any  intervals.  It  should  teach  us  a  broader 
conception  of  harmonies  and,  as  I  believe,  a  more  valuable  one, 
since  the  importance  of  a  notion  usually  depends  upon  its  generality. 

To-day  the  modern  composer  habitually  employs  the  twelve- 
tone  scale  as  the  source  of  his  harmonic  invention.  The  abundance 
of  accidentals  in  our  modern  composition  is  superficial  but  none  the 
less  accurate  evidence  of  the  passing  of  the  feeling  for  the  diatonic 
modes.  To-day  there  are  also  a  few  scales  which  are  formed  of 
new  arrangements  in  the  intervals  of  our  duodecimal  system. 


580  THE   MONIST. 

But  the  octave  still  remains  the  common  basis  for  all  scales  now 
used ;  each  scale  repeats  itself  at  successive  octaves. 

It  seems  only  natural  that  we  make  this  interval  which  is  of 
the  greatest  importance  because  it  has  the  simplest  ratio,  the  basis 
for  our  harmony.  We  may  therefore  call  its  interval  unity.5 

Having  established  the  octave  as  the  basic  interval,  and  having 
assigned  to  it  the  number  one,  we  turn  our  attention  to  the  lesser 
intervals.  The  semitone,  since  it  is  one-twelfth  the  gradation  of 
the  octave,  will  be  known  as  the  interval,  one-twelfth.  The  ''whole 
tone"  becomes  two-twelfths.  Tabulating  all  of  our  intervals  in  their 
old  and  new  nomenclature  we  have: 

DIATONIC  NAME  NATURAL  OR  CHROMATIC  NAME 

Minor  Second One  Twelfth 

Major  Second Two  Twelfths 

Minor  Third     . Three       " 

Major  Third Four 

Perfect  Fourth Five 

Augmented  Fourth Six 

Perfect  Fifth Seven 

Minor  Sixth Eight 

Major  Sixth      .     .     .     . Nine 

Minor  Seventh Ten 

Major  Seventh  .     .     .     .'".     .     .     .     .     Eleven     " 

Octave One. 

Although  many  of  these  fractions  expressing  intervals  are  not 
reduced  to  their  simplest  form,  it  is  of  advantage  to  retain  the  com- 
mon denominator,  twelve ;  for  if  all  intervals  can  be  expressed  as 
quotients  of  variable  integers  and  the  constant  twelve,  we  need  con- 
sider only  the  numerators  and  eliminate  the  common  denominator. 
Thus  the  intervals,  one-twelfth,  two-twelfths,  three-twelfths  etc., 
may  be  called  respectively,  one,  two,  three,  etc.  It  is  clear  that  this 
nomenclature  is  founded  entirely  upon  the  chromatic  scale  since 
every  interval  is  measured  as  a  multiple  of  the  intervals  comprising 
the  chromatic  scale. 

In  naming  harmonies  having  more  than  one  interval,  the  ad- 
vantages of  the  chromatic  nomenclature  are  immediately  apparent. 
For  instance,  the  major  triad  is  said  to  be  formed  of  a  minor  third 
placed  above  a  major  third.  In  other  words  the  interval  3  is  placed 
above  the  interval  4.  Thus  the  major  triad  in  fundamental  position 

5  For  simplicity  we  call  the  interval  unity,  although  the  physical  interval  of 
the  octave  is  2. 


OUR  MUSICAL  IDIOM.  581 

is  a  34  or  4-3  harmony.  Likewise,  the  minor  triad  is  a  3-4  harmony ; 
the  dominant  sept  chord  in  fundamental  position,  a  4-3-3  harmony ; 
and  the  dominant  sept  chord  in  its  first  inversion  is  a  3-3-2  harmony. 
The  harmony  under  previous  discussion  is,  in  the  form 


a  2-3-2  harmony. 

This  change  in  nomenclature  means  more  than  is  at  first  ap- 
parent. It  means  an  expansion  of  our  conception  of  harmonies 
which  may,  perhaps,  offset  the  limitations  felt  in  the  minds  of  many 
who  can  think  of  music  only  as  varying  arrangements  of  groups  of 
superimposed  thirds.  We  may  freely  think  of  any  harmonies  as 
being  composed  of  superimposed  intervals  of  any  sort,  instead  of 
being  shackled  by  considering  every  harmony  made  up  of  super- 
imposed thirds,  or  inversions  of  these.  Our  vocabulary  of  har- 
monies, instead  of  embracing  only  chords,  will  embrace  all  har- 
monies. 

It  may  be  well  here  to  forestall  a  possible  objection  that  the 
chord  system  is  the  nearest  to  the  ideal  from  the  physical  point  of 
view.  It  is  true  that  chords  are  physically  the  "cleanest"  harmonies, 
i.  e.,  their  tones  have  the  simplest  vibration  ratios.  It  may  be  said 
from  this  that  the  system  of  building  up  thirds  is  founded  not  upon 
an  arbitrary  choice,  but  upon  an  acoustic  basis.  But  I  answer  that 
from  this  point  of  view  it  is  immaterial  whether  we  think  of  major 
and  minor  thirds  or  of  4s  and  3s  in  building  up  the  most  important 
harmonies.  If  thirds  and  sums  of  thirds  are  the  most  important 
intervals,  then,  after  we  have  learned  the  chromatic  nomenclature, 
3s,  4s,  6s  and  7s  will  come  to  be  recognized  as  the  most  important 
intervals.  There  is  no  ground  for  any  charge  that  the  chromatic 
nomenclature  is  more  empiric  and  less  scientific  than  the  chord 
system. 

Again,  is  not  the  chromatic  nomenclature  a  simple  and  an  ac- 
curate method  for  naming  and  classifying  any  harmony?  Instead 
of  grappling  with  such  harmonies  as  these: 


ft  r:          \  jf — **         i 


considering  them   in   relation   to   surrounding   harmonies,   and   in 
themselves  by  devious  ways,  we  simply  name  them  chromatically 


582  THE  MONIST. 

as  consisting  of  superimposed  intervals.  They  are  respectively: 
6-4-6-4  and  5-5-4-4  harmonies. 

Our  next  task  is  to  study  more  closely  the  nature  of  harmonies 
and  to  discover  suitable  means  for  systematically  finding  all  of 
these  harmonies. 

First  we  will  recognize  a  tone  essentially  the  same  whether  its 
vibration  frequency  is  increased  or  diminished  by  a  power  of  two; 
for  the  ordinary  ears  hear  it  as  essentially  the  same.  For  this  reason, 
we  are  compelled  to  regard  an  inversion  of  a  harmony  only  as  a 
different  position  of  the  harmony  and  not  as  a  different  harmony. 
In  the  chord  system  we  decide  upon  one  position  of  a  harmony 
which  we  call  close  fundamenal,  namely  that  position  in  which  the 
harmony's  root  is  in  the  base.  (There  is,  however,  no  distinction 
made  between  "open"  and  "close"  fundamental  positions.)  Like- 
wise, and  for  purpose  of  classification,  we  should  decide  upon  a 
fundamental  position  of  a  harmony  in  our  chromatic  system  of 
nomenclature.  The  choice  of  a  fundamental  position,  though  arbi- 
trary, will  be  made  later  in  the  discussion. 

The  relations  between  the  tones  comprising  a  harmony  are 
intervals.  A  harmony  may  be  thought  of  as  a  combination  of  inter- 
vals. However,  a  combination  of  intervals  may  form  more  than 
one  harmony.  The  major  triad  is  a  4-3  harmony,  that  is  it  is  made 
up  of  the  intervals  4  and  3 ;  yet  if  we  merely  reverse  the  order  of 
these  intervals  we  have  another  harmony,  the  minor  triad.  It  is 
clear  then  that  a  harmony  is  one  permutation  only  of  a  given  com- 
bination of  intervals. 

Now  it  might  be  thought  that  one  could  form  all  harmonies 
existing  in  our  duodecimal  system  by  permutating  all  possible  com- 
binations of  intervals  in  all  possible  ways.  However,  by  doing  this 
we  would  calculate  an  immense  number  of  harmonies,  very  many  of 
which  would  only  be  inversions  of  each  other.  To  avoid  the  repe- 
titions occasioned  by  inversions,  in  classifying  all  harmonies,  we 
come  to  the  consideration  of  an  extra  interval  with  each  harmony. 
If  we  invert  the  major  triad  or  the  4-3  harmony  we  have  first  a 
3-5  harmony  (commonly  called  the  first  inversion)  and  then  a  5-4 
harmony  (called  the  second  inversion).  The  extra  interval  to  be 
considered  in  this  case  is  the  interval  5,  which  is  bounded  by  the 
highest  tone  of  the  triad  and  the  note  an  octave  above  the  triad's 
lowest  tone,  the  triad  standing  in  fundamental  position — an  interval 
which  will  be  defined  as  the  complement.  If  the  triad  is  in  4-3 
position  of  f,  the  complement  is  the  interval  c-f : 


OUR  MUSICAL  IDIOM.  $83 

^complement  s  5 


» 
it 


If  the  triad  is  in  5-3  position  on  c,  the  complement  becomes  the 
interval  a-c  or  3. 


zn  complement  =  3 


The  complement  will  be  denoted  by  the  letter  R  in  the  following 
paragraphs.  The  term  will  be  employed  in  the  discussion  of  har- 
monies having  any  number  of  tones  or  intervals. 

Now  if  A  represents  the  lowest  interval  of  a  harmony  in  close 

position,  that  is,  such  a  position  in  which  all  tones  are  within  the 

compass  of  an  octave ;  B  the  interval  above  A ;  C  the  interval  above 

B,  etc.,  and  R  the  complement,  a  harmony  could  be  represented  thus : 

R 


or  ABC R 

C 
B 
A 

The  first  inversion  of  this  harmony  would  be  represented  thus : 
A 
R 


or  BC..RA 


C 

B 

the  second  inversion  thus: 
B 
A 
R 


or  C...RAB8 


C 

•  Each  different  letter  does  not  necessarily  denote  a  different  sized  interval 


584  THE   MONIST. 

We  can  think  of  all  the  different  positions  of  the  same  harmony 
as  arranged  clockwise  in  a  circle.  Taking  the  intervals  in  clockwise 
order  we  find  a  different  inversion  taking  each  letter  as  a  starting- 
point.  It  is  clear  that  the  inversions  of  a  harmony  are  cyclic  rota- 
tions of  the  permutation  of  intervals  forming  the  fundamental 
position,  whichever  it  may  be.  Therefore  if  we  desire  to  form  all 
possible  harmonies  from  a  given  combination  of  intervals,  we  employ 
only  those  permutations  of  the  given  combination  which  are  not 
cyclically  related ;  that  is,  out  of  each  group  of  permutations  which 
are  cyclic  rotations  of  each  other  we  select  only  one  permutation 
as  representative  of  a  single  harmony. 

The  Number  of  Harmonies  with  a  Given  Combination. 

Assume  we  are  given  a  harmony  A-B,  in  which  A  and  B  are  of 
different  magnitude.  Furthermore  let  us  make  A  and  B  such  inter- 
vals that  the  complement  (R)  of  A-B,  is  unequal  in  magnitude  to 
either  A  or  B.  Including  R  in  our  letter  representation,  we  denote 
the  harmony  as  A-B-R.  The  minor  (3-4-5)  and  major  (4-3-5) 
triads  are  good  examples  of  such  a  harmony.  We  have  already 
seen  that  the  inversions  of  a  harmony  are  cyclic  rotations  of  its 
arrangement  of  intervals.  Conversely,  if  two  or  more  harmonies 
are  cyclically  related  they  are  only  different  positions  or  inversions 
of  one  and  the  same  harmony. 

Since  we  are  now  engaged  in  finding  how  many  harmonies  can 
be  formed  from  a  given  combination  of  intervals,  our  problem  be- 
comes one  of  finding  the  number  of  cyclically  unrelated  permuta- 
tions of  the  given  combination. 

Let  us  experiment  with  our  harmony  A-B-R.  The  permuta- 
tions of  the  combination  are: 

ABR       BRA       RAB 
ARB       BAR       RBA 

The  arrangements  in  the  upper  row  are  cyclically  related ;  likewise 
those  in  the  lower;  yet  no  one  of  the  permutations  in  the  upper 
row  is  related  cyclically  to  any  one  in  the  lower  row.  Hence,  only 
two  harmonies :  A-B-R  and  A-R-B,  can  be  formed  of  the  combina- 
tion A,  B,  and  R.  We  observe  that  both  harmonies  are  represented 
in  each  vertical  group;  moreover  in  each  vertical  group  one  letter 
occupies  the  same  position  (i.  e.,  first  in  this  case)  in  both  harmonies 
while  the  other  two  letters  are  permutated. 

Let  us  experiment  in  like  manner  with  a  combination  of  4 
intervals  (R  included  as  one),  in  which  all  the  intervals  are  of 


BCRA 

CRAB 

RABC 

ERCA 

CABR 

RCAB 

BRAC 

CBRA 

RACE 

BACR 

CRBA 

RBAC 

BCAR 

CARB 

RBCA 

BARC 

CBAR 

RCBA 

OUR  MUSICAL  IDIOM.  585 

different  magnitudes.     We  represent  its  parts  as  A,  B,  C,  and  R. 
Its  permutations  are : 

ABCR 
ABRC 
ACER 
ACRE 
ARBC 
ARCB 

Here,  too,  the  permutations  have  been  so  arranged  that  those 
in  any  one  horizontal  row  are  cyclically  related,  while  no  permuta- 
tion in  one  horizontal  row  is  cyclically  related  to  any  one  in  any 
other  horizontal  row.  Thus  any  one  of  the  vertical  columns  contains 
all  possible  harmonies :  in  this  case  3 !  or  6.  It  will  be  noticed  here 
as  before,  that  by  retaining  one  letter  in  a  stationary  position 
throughout  while  permutating  the  remaining  letters,  all  possible  har- 
monies are  formed  from  the  given  combination ;  for,  although  many 
of  the  permutations  of  the  remaining  letters  are  cyclically  related, 
the  stationary  letter  will  bear  a  different  relation  to  the  other  letters 
in  each  case.  In  each  vertical  column  one  letter  is  held  in  the  same 
position,  allowing  the  remaining  3  letters  to  be  permutated  in  3! 
different  ways. 

With  a  combination  of  two  intervals  (R  included)  we  form 
1 !  or  1  harmony;  with  three  intervals  (R  included)  we  form  2! 
or  2,  with  four  intervals  we  form  3 !  or  6,  etc.  Thus  with  n  different 
intervals  we  form  (n-1)  !  different  harmonies. 

But  this  formula  does  not  apply  to  combinations  in  which  two 
or  more  intervals  are  alike ;  and  such  combinations  are  by  far  more 
numerous  than  the  others.  In  fact,  no  harmony  can  have  as  many 
as  five  or  more  different  intervals,  provided  of  course  that  all  its 
tones  are  bounded  by  the  octave.  To  illustrate,  we  find  the  sum 
of  the  five  smallest  intervals  (1,  2,  3,  4,  5)  to  be  15,  which  exceeds 
the  octave  12  by  3. 

To  experiment  with  harmonies  in  which  two  or  more  intervals 
are  alike  let  us  take  the  combination :  A,  B,  B,  and  R.  Its  permu- 
tations are: 

ABBR          BBRA          BRAB          RABB 
ABRB          BRBA          BABR          RBAB 
ARBB          BARB          BEAR          RBBA 
Here  again  the  horizontal  rows  contain  cyclically  related  permuta- 
tions while  the  vertical  columns  do  not.    We  find  all  three  of  the 


586  THE  MONIST. 

possible  harmonies  represented  in  any  one  vertical  group  (in  each 
of  which  one  letter  has  a  stationary  position  throughout).  How- 
ever, since  the  combination  contains  two  B's  there  are  two  "B  rows," 
each  of  which  contains  the  three  harmonies.  Thus  if  one  B  is  held 
stationary  while  the  remaining  intervals  are  permutated,  we  obtain 
six  instead  of  three  harmonies ;  showing  that  our  rule  of  stationaries7 
holds  only  for  singly  occurring  intervals.  But  if  we  hold  A  sta- 
tionary, we  permutate  B-B-R  in  3  !/2 !  different  ways  forming  3 
different  harmonies. 

With  the  combination  A,  B,  B,  R  and  R  we  can  form  the  fol- 
lowing harmonies:  ABBRR,  ABRBR,  ABRRB,  ARBRB, 
A  R  B  B  R,  A  R  R  B  B,  by  holding  A  in  the  same  position  and  per- 
mutating  the  remaining  letters  in  4!/2!2!  (=6)  different  ways. 

The  general  formula  then  for  the  number  of  harmonies  to  a 
given  combination  in  which  at  least  one  interval  occurs  singly  is: 

H  (harmonies)  =  (n-1)  !/w1!n2!n3! 

where  n  is  the  total  number  of  intervals  (R  included)  in  the  com- 
bination, and  nlf  n2,  ns,  etc.,  are  the  respective  number  of  times 
which  certain  intervals  are  found.  This  is  the  general  formula  for 
the  number  of  harmonies  to  a  combination,  as  the  large  majority  of 
combinations  contain  singly  occurring  intervals.  To  find  the  num- 
ber of  harmonies  in  a  combination  containing  no  singly  occurring 
intervals  actual  trial  must  be  resorted  to;  any  formula  for  this 
would  be  beyond  the  scope  of  this  work.  As  an  example  of  the 
error  to  which  the  formula  leads  if  it  is  applied  to  combinations 
having  no  singly  occurring  interval,  we  will  apply  it  to  the  com- 
bination 3  A's  and  3  B's.  Here 

(n-l)!/w1!n2!....=5!/3!3!  =  1.2.3.4.5/(1.2.3)  (1.2.3)  =  10/3 
The  result  is  an  impossibility,  since  a  fractional  number  of  harmonies 
cannot  exist.8 

The  Number  of  Possible  Combinations. 

We  will  henceforth  regard  a  harmony  as  in  a  prime  position 
if  its  tones  are  reduced  to  within  the  compass  of  an  octave. 

7  That  is,  the  method  of  holding  one  interval  stationary  and  permutating  the 
remaining  intervals  in  all  possible  ways  to  obtain  the  several  different  har- 
monies. ' 

8  We  remember  that  throughout  the  last  pages  we  have  considered  the 
complement  R  as  one  of  the  n  intervals  of  a  combination.    It  might  be  sup- 
posed that  we  could  now  eliminate  R  from  consideration  and  thereby  make  our 
formula,  n !/MI \nz\. .. ,    This  could  not  be  done  generally,  since  R  might  be  an 
interval  having  a  magnitude  equal  to  that  of  another  interval  (i.  e.,  A,  B,  or 
C,  etc.)   in  the  combination;  in  that  case  it  would  necessarily  figure  in  the 
denominator  of  the  fraction. 


OUR  MUSICAL  IDIOM. 


587 


No  harmonies  in  prime  position  can  consist  of  less  than  two 
or  more  than  twelve  tones,  in  our  duodecimal  system.  Hence  no 
prime  harmony  can  have  less  than  two  (R  included)  or  more  than 
twelve  intervals.  Now  the  number  of  combinations  possible  within 
the  octave  could  be  computed,  but  the  result  would  be  of  no  value, 
since  in  finding  the  number  of  harmonies  we  must  treat  each  com- 
bination separately.  Furthermore,  mere  numbers  interest  us  only 
speculatively  while  a  concrete  method  of  obtaining  all  harmonies 
is  of  real  value. 

To  simplify  the  task  of  finding  the  combinations  possible  with 
the  twelve  units  within  the  octave,  we  will  treat  separately  those 
groups  of  combinations  having  a  different  number  of  tones  or 
intervals.  A  separate  table  will  be  made  for  each  group  of  com- 
binations having  a  certain  fixed  number  of  intervals.  It  is,  of 
course,  evident  that  the  sum  of  the  intervals  of  each  combination 
equals  the  octave  12,  since  R  is  always  one  of  the  combination's 
intervals.  In  the  tables  the  vertical  columns  contain  the  respective 
intervals,  varying  in  magnitude  as  the  number  of  intervals  of  the 
combination  permit.  In  the  horizontal  rows  are  found  the  com- 
binations. If  a  number  greater  than  1  is  found  in  a  combination, 
it  indicates  how  many  times  the  interval  in  whose  vertical  column 
it  lies  is  repeated  in  the  combination.  Thus  a  number  3  lying  in 
a  column  headed  by  the  number  2,  indicates  that  the  combination 
contains  the  interval  2  (or  the  whole  tone)  thrice.  The  tables  follow.9 

TABLE  I;  N  (NUMBER  OF  INTERVALS  INCLUDING  R)=2 


VARIOUS  INTERVALS 

fNo.  OF 

I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

HARMONIES 

H 

I 

I 

I 

H=(«—  1)!=(2_  i)! 

I 

2 

I 

I 

« 

I 

3 

i 

i 

« 

I 

4 

i 

i 

« 

I 

5 

i 

i 

« 

I 

6 

2 

u 

I 

Total     t> 


9  The  combination  numbers  found  to  the  left  of  the  respective  combinations 
are  only  for  future  reference. 


DC 

_„„„„-„„--„- 

IH 

~ 

k. 

04 

Z 

O 

^> 

r-  ' 

«:  « 

Z   X 

^~^            ^~N                                                                                                                                                                                                                                        i- 

s  ; 

-             M                                                             ~                                            ~                                              *- 

P  0 

«      3      >"^7,'-^-~7i>^ 

—   X 
3  < 

^    co                   pi               N              pa 

DM 

J* 

II 

CO 

II 

y 

E 

II 
Z 

s 

— 
t—t 

o 

- 

w 

ON 

M 

s 

oo 

M 

t^ 

~ 

0 

IH                                              M                              IH 

VI 

M                                                               PI                              w 

rh 

«        rr. 

CO                                                       >H                                                               M                                              CS             IH 

n                    M                              <N      «      M      - 

>H 

pj              M              |_              M              M 

IH        N        co^iOvOt^.iO        QiO        IH        c< 

s 

>H         CO       CO       COW         COVOVD         COCOIH         POO)         coin 

5 

CALCULATIONS  OF 
HARMONIES 

_.    _.                 .2    —                 _.           —    —    ."«    —    75 

CO       CO                            .^       CO                            "co                  rO       CO       >,       CO       >, 

||      ||                  «                                                    PQ           W 

"i 

o 
H 

Oi 

M 

00 

~ 

r^ 

M 

o 

IH 

«0 

Ol                                                    IH              IH                                                    IH 

•«*• 

IH                                                                       IH                                 M                                                   P)              IH 

co 

C4         M                     -H                     M         rf- 

d 

IH                                                                      dWIH                                                    POOICSlH 

«       W        fO^-u^vO       t^oo        0>0        «        <N        rOTl-u^ 

OUR  MUSICAL  IDIOM. 


TAUI  E  IV;  N=5 


I 

2 

3 

4 

5 

6 

7 

8 

CALCULATIONS  OF 
HARMONIKS 

H 

I 

4 

i 

H=4!/4! 

i 

1 

3 

I 

i 

473! 

4 

3 

3 

i 

i 

" 

4 

4 

3 

i 

i 

" 

4 

5 

2 

2 

i 

4!/2l2! 

6 

6 

2 

I 

i 

i 

4!/2! 

12 

7 

2 

I 

2 

4!/2!  2! 

6 

8 

2 

2 

I 

" 

6 

9 

I 

3 

i 

4W 

4 

10 

I 

2 

I 

I 

4!/2! 

12 

ii 

I 

I 

3 

473! 

4 

12 

4 

I 

4!/4! 

i 

13 

3 

2 

Bv  trial 

2 

Toial     oo 


TAB.E  V;  N=6 


i 

2 

3 

4 

5 

6 

7 

CALCULATI--NS  OK 
HARMONICS 

H 

I 

5 

i 

H=5!/5! 

i 

2 

4 

I 

i 

5V4! 

5 

3 

4 

i 

i 

n 

5 

4 

4 

2 

By  trial 

3 

5 

3 

2 

i 

5!/2!3! 

10 

6 

3 

I 

i 

I 

5V3! 

20 

7 

3 

3 

By  trial 

4 

8 

2 

3 

I 

SVa'3' 

10 

9 

2 

2 

2 

By  trial 

16 

10 

I 

4 

I 

574! 

5 

ii 

6 

By  trial 

i 

Total     80 


590 


THE  MONIST. 


TABLE  VI;  N= 


i 

2 

3 

4 

5 

6 

CALCULATIONS  OF 
HAKMONIES 

H 

I 

6 

i 

H=6!/6! 

i 

2 

5 

I 

i 

61/51 

6 

3 

5 

i 

i 

" 

6 

4 

4 

2 

i 

6!/2!  4! 

i5 

5 

4 

I 

2 

" 

15 

6 

3 

3 

I 

6!/3!3! 

20 

7 

2 

5 

By  trial 

1 

Total     66 


TABLE  VII;  N= 


i 

2 

3 

4 

5 

CALCULATIONS  OF 
HAKMONIES 

H 

I 

7 

i 

H=7!/7! 

i 

2 

6 

I 

i 

7!/6! 

7 

3 

6 

2 

By  trial 

4 

4 

5 

2 

I 

7!/2!  5! 

21 

5 

4 

4 

By  trial 

IO 

Total     43 


TABLE  VIII;  N= 


i 

2 

3 

4 

CALCULATIONS  OF 
HARMONIES 

H 

I 

8 

i 

H=8!/8! 

i 

2 

7 

I 

i 

81/71 

8 

3 

6 

3 

By  trial 

to 

Total     19 


TABLE  IX;  N=no 


I 

2 

3 

CALCULATIONS  OF 
HARMONIES 

H 

I 

9 

I 

H=9!/9! 

i 

2 

8 

2 

By  trial 

5 

Total     6 


OUR  MUSICAL  IDIOM.  59! 

TABLE  X;N=n  TABLE  XI;  N=i2 


i 

i 

2 

CALCULATIONS  OF 
HARMONIES 

H 

10 

I 

H=IO!/IO! 

i 

I 

CALCULATIONS  or 
HARMONIES 

H 

12 

By  trial 

i 

Let  us  tabulate  the  number  of  harmonies  found  in  each  respective 
table : 


Table  No.  1 


10 
11 


Intervals  2 
3 
4 
5 
6 
7 
8 
9 

"  10 
"  11 
"  12 


Total  No.  H    6 


Total  350 


We  notice  that  as  the  number  of  intervals  increases  to  6  the  number 
of  harmonies  increases,  while  as  the  number  of  intervals  increases 
beyond  6  the  number  of  harmonies  decreases.  Thus,  six-tone 
harmonies  (or  harmonies  of  six 'intervals  including  R)  are  most 
numerous ;  five-  and  seven-tone  harmonies  next  in  number ;  four-  and 
eight-tone  harmonies  next ;  etc.  More  harmonies  can  be  formed 
from  combination  4  Table  VII  than  from  any  other  combination. 
From  it  we  obtain  21  harmonies.  Finally  we  see  that  the  total  num- 
ber of  harmonies  in  the  duodecimal  system  is  350. 

The  harmonies  of  least  dissonance  will  be  those  having  the 
fewest  small  intervals.  There  are  9  harmonies  having  intervals  no 
smaller  than  3  (minor  third)  ;  there  are  28  having  intervals  no 
smaller  than  2  (major  second)  ;  and  there  are  55  harmonies  having 
only  one  interval,  1  (minor  second). 

So  far  we  have  only  shown  the  number  of  harmonies  with 
each  separate  combination.  Now  it  remains  to  show  every  har- 
mony on  the  staff.  Means  have  been  shown  for  finding  the  number 
of  harmonies  from  each  combination.  We  merely  retain  a  singly 
occurring  interval  in  one  position  (preferably  the  lowest)  and  per- 
mutate  the  remaining  intervals.  But  in  notating  harmonies  we 


592 


THE   MONIST. 


should  at  least  represent  them  in  some  standard  form;  and  thus 
we  arrive  at  the  long-delayed  decision  about  fundamental  position. 

Fundamental  Position. 

Among  the  350  complex  harmonies  which  we  have  found,  there 
are  many — nay,  a  large  majority — which  could  not  be  represented 
as  plain  chords.  Furthermore,  since  our  vocabulary  of  intervals 
and  harmonies  has  become  a  chromatic  one,  we  will  no  longer  at- 
tempt to  reconcile  the  limited  number  of  harmonies  known  as  chords, 
with  all  of  350  harmonies ;  hence  no  attempt  to  make  a  chordic  posi- 
tion the  fundamental  form.  Arbitrarily  we  might  choose  as  our 
fundamental  a  form  in  which  the  span,  or  the  interval  between  the 
extreme  tones  of  a  harmony,  is  smallest.  Or,  since  we  have  found 
it  convenient  to  place  any  singly  occurring  interval  in  the  lowest 
position  in  forming  harmonies  from  combinations,  we  might  call 
such  a  position  fundamental.  The  question  is  difficult,  and  although 
my  solution  is  only  arbitrary  I  believe  the  fundamental  position 
should  be  one  which  satisfies  the  following  conditions: 

I.  The  harmony  is  prime. 

II.  The  smallest  interval   (it  may  be  R)    occupies  the  lowest 
position,  and  thereby  becomes  A. 

III.  In  case  there  exist  two  or  more  smallest  intervals,  the  one 
or  more  other  smallest  intervals  are  nearest  A. 

IV.  In  case  the  two  or  more  smallest  intervals  are  spaced  regu- 
larly apart,  the  next  smallest  interval  is  nearest  A. 

Illustrations  follow  in  order  respective  to  the  conditions  of  the 
definition. 

ANY  POSITION  FUNDAMENTAL  POSITION 

4 Q- 


Condhionl]!    -i 


2- 2 -4 -(4) 


OUR  MUSICAL  IDIOM.  593 


Condition  IV 


3-7-6-5 

The  first  example  illustrates  how  a  harmony  in  more  or  less 
spread-out  position  (left-hand  column)  is  reduced  to  prime  position. 

The  second  illustrates  how  another  harmony  (9-8-5)  is  reduced 
to  prime  position,  following  which  it  is  placed  so  that  the  smallest 
interval  (1)  occupies  the  lowest  position. 

The  third  illustrates  a  harmony  which,  in  any  prime  position, 
contains  two  smallest  intervals.  We  are  not  satisfied  with  making 
either  of  these  A ;  the  equivalent  of  A  must  be  nearest  A.  Thus  in 
this  position: 


2-2-4- (4) 
A's  equivalent  is  nearer  A  than  in  the  position: 


2-4-4-12) 

The  fourth  illustrates  a  harmony  containing  two  smallest  inter- 
vals (=1)  which  are  equally  separated  in  any  prime  position  of  the 
harmony.  Thus  the  two  intervals  (1)  are  always  separated  by  the 
interval  5.  But  because  the  next  smallest  interval 


lies  nearer       -/v  —     than 


we  consider  the  interval  e-f  as  A. 

Thus  in  classifying  any  harmony,  only  three  short  steps  are 
necessary.  First  we  reduce  its  tones  to  within  the  compass  of  the 
octave ;  second,  we  select  from  the  prime  positions  the  fundamental 
position ;  third,  we  name  the  harmony  according  to  the  chromatic 
nomenclature. 

Having  determined  the  fundamental  position,  we  are  prepared 
to  write  out  on  the  staff  all  existing  harmonies  with  the  help  of 
the  previous  tables.  Every  harmony  will  appear  in  fundamental 
form;  while  each  will  be  respectively  named.  The  harmonies  fol- 
low: 


594 


THE  MONIST. 


ALL   EXISTING   HARMONIES   OF   THE   DUODECIMAL    SYSTEM. 

LISTED  IN  FUNDAMENTAL  POSITIONS  AND  BY  TABLES. 
(The  names  of  the  harmonies  are  written  respectively  below;  the  number  of 

combination  in  which  each  is  found  appears  above  the  staff.) 
Table  I 
A.      C.I  C.  2  C.  3  C.4  C.5  C.6 


Table  H 


C.2 


3  -(9)  etc.     4 
C.3 


6 
C.4 


1-1         '  1-2-  (9) 
C.5 


1-3 

C.6 


1-8 
C.7 


1-4 


-o- 


1-7 

C.8 


c>y» 


1-5  1-6  2-2  2-3  2-7 

C.9          C.10         C.ll  C.12 


2-4  2-6          2-5 

Table  III 


C.2 


3-3          3-4          3-5          4-4 


C.3 


1-3-6  1-6-2       1-6-3       1-2-4       1-2-5        1-4-2       1-4-5 


OUR  MUSICAL  IDIOM. 
C.9 


595 


C.10 


« 


^t 


s 


o*»     i    -&*. 


-e^t 


C»E* 


1-5-2          1-5-4         1-3-3         1-3-5          1-5-3         1-3-4 
C.ll       C.12 


* 


^ 


Ho 


1-4-3  1-4-4  2-2-2        2-2-3         2-2-5        2-3-2 

C.13  C.14  C.15 


S; 


2-2-4        2-4-2        2-3-3        2-3-4        2-4-3          3-3-3 


Table  IV 
C.I 


C.2 


^S 


k£§£ 


ti: 


1-1-1-1  1-1-1-2  1-1-1-7 

C.3 


1-1-2-1 


1-1-7-1  1-1-1-3 

C.4 


1-1-1-6          1-1-3-1 


1-1-6-1 
C.5 


ft»     M'l'g 

-gT  «» 


1-1-1-4         1-1-1-5.  1-1-4-1         1-1-5-1 


1-1-2-2 


-o- 


s 


tfc 


1-1-2-6        l-l-6-2-(2)   l-2-2-l-(G)    1-2-1-2-16)       1-2-1-6 -(2) 
C.6 


1-1-2-3         1-2-1-3         1-1-3-5          1-1-3-2 


1-3-1-2 


1-1-2-5        1-2-3-1       -1-3-2-1          1-3-1-5  1-1-5-3 


596 


THE  MONIST. 
C  7 


3X 


Btt 


1-2-1-5  1-1-5-3  1-1-2-4           l-4-l-4-(2)      1-1-4-2 

0.8 


* 


1-2-1-4  1-4-1-2- (4)     1-1-4-4- (2)    l-l-3-3-(4)      1-3-1-3 

C.9 


l-4-l-3-(3)   1-1-3-4         1-3-1-4         1-1-4-3          1-2-2- 2- (5) 

C.1(T 


1-2-2-5       1-2-5-2        1-5-2-2         1-2-2-3 -(4)    1-2-2-4     1-2-3-2 


i± 


1-2-3-4        1-2-4-2      1-2-4-3       1-3-2-2      1-3-2-4        1-3-4-2 

C.ll 


-G 


O 


1-4-2-2         1-4-2-3 


1-4-3-2        l-2-3-3-(3)       1-3-2-3 

CA  <)  f*   i\  Q- 

•  i.4t  v.  lu 


1-3-3-2         1-3-3-3         2-2-2-2-(4)    2-2-2-3-(3)     2-2-3-2 


Table  V 


C.2 


[r^^Mmygf-  Y3- 


1-1- 1-1-1- (7)          l-l-l-l-2-(6)       1-1-1-2-1  1-1-2-1-1 

C.3 


\f\  b*w 


1-1-1-6-1  1-1-1-1-6  M-l-l-3-(5)       1-1-1-3-1 


OUR  MUSICAL  IDIOM.  597 

C.4 


>  Ujft^! 


1-1-3-1-1  1-1-1-5-1 


1-1-1-1-5  1-1-1-1 -4 -(4) 

C.5 


1-1-1-4-1  1-1-4-1-1  l-l-l-2-2-(5)      1-1-2-1-2 


1-1-2-2-1  1-1-2-5-1  1-2-1-2-1  1-1-5-1-2 


1-1-1-2-5 
C.6 


1-1-2-1-5  1-1-5-2-1  1-1-1-5-2 


»,-H»ai  Ibi'^au    Ib'^gt    Mq8j 


1-l-l-2-3-(4)          1-1-1-3-2  1-1-2-1-3  1-1-2-3-1 


1-1-3-1-2  1-1-3-2-1  1-1-3-4-1  1-2-1-3-1 

* 


1-1-4-1-2  1-1-2-4-1  1-2-1-4-1  1-1-4-1-3 


1-1-1-3-4  1-1-3-1-4  1-1-4-2-1  1-1-1-4-2 


^%Ul!>!' 


<^ 


1-1-1-2-4  1-1-2-1-4  1-1-4-3-1  1-1-1-4-3 


598 


THE  MONIST. 


C.7 


Fte 


S!^=H 


1-1-1-3-3- (3)         l-l-3-l-3-(3)       1-1-3-3-1  1-3-1-3-1 

C78 


l-l-2-2-2-(4)   l-2-l-2-2-(4)  -1-2-2-1-2     1-4-1-2-2        1-1-2-2-4 


go: 


1-2-1-2-4 
"C.9 


1-2-2-1-4      1-1-2-4-2      1-2-1-4-2       1-1-4-2-2 


-ft 


1-1 -2-2 -3 -(3)       1-1-2-3-2 -(3)       l-l-2-3-3-(2)       1-1-3-2-2 


y  |        'Vf  -T- 

^~1>"g»        £E 


&^ 


-vt® 


1-1-3-2-3       1-1-3-3-2      1-2-1-2-3       1-2-1- 3-2 »     1-2-1-3-3 


4g$n 


-4*— e 


1-3-1-2-2     1-3-1-2-3 


-V  V       '.     ^>i» 


1-3-1-3-2       1-2-2-1-3        1-2-3-1-2 
C.10 


1-2-3-1-3  1-3-2-1-3  1-2-2-2-2-C3)     1-2-2-2-3 


C.ll 


— o»j 

1-2-2-3-2  1-2-3-2-2         1-3-2-2-2  2-2-2-2-2-(2) 


Table  VI 
C.I 


C.2 


1-1-1-1-1-1- (6)       1-1-1-1-1 -2 -(5)      1-1-1-1-2-1 -(5)    l-l-l-2-l-l-(5) 


OUR  MUSICAL  IDIOM. 


599 


C.3 


\r      5^C"%  I      f 


<£| 


111131(4)       111311(4)        111411(3)      111141(3) 
C.4 


111114(3)        111182(4)     111212(4)      112112(4) 


112141(2)  111214(2)        112211(4)       114121(2) 


112114(2)      111412(2)        111421(2) 
C.5 


111142(2) 


111123(3)       111321(3)         113121(3)        112131(3) 


11213(3)       111132(3)          111312(3)       113112(3) 


lAuE; 


111231(3)        111133(2)         111313(2)  113113(2) 

C.6 


111331(2)      113131(2)       112113(3)         111222(3)      • 


6oo 


THE  MONIST. 


fife 


112122(3)  112231(2)        111223(2)       112212(3) 


S 


121212(3)  112123(2)        112312(2)        112321(2) 


111232(2)         112221(3)        121221(3)  112213(2) 


& 


& 


121312  (2) 


121213  (2)  112132  (2) 


113122(2)  113212(2)  113221(2) 

C.7 


111322(2)        112222(2)       121222(2)        122122(2) 
C.2 


Table  VII 
C.I 


1111111(5) 


1 1 1 1 1 1  2  (4)  1  1  1  1  1  2  1  (4) 


gs 


£ 


11211(4)  1112111(4) 


±ka 


1111411(2) 
C.3 


5 


1111141(2)  1111114(2)  1111113(3) 


m^ 


B 


£ 


1111131(3)  1111311(3)  1113111(3) 


OUR  MUSICAL  IDIOM. 


601 


C.4 


}>™      \1\ 


HH- 


111  1122(3) 

. 


,1111212(3)  1112113  (3) 


1112311(2)  1111231(2)  1111123(2) 


s* 


1111221(3)  1112121(3)  1121121(3) 


1112131(2)  1111213(2)  1112211(3) 


& 


1121211(3) 


1121131(2)  1112113(2) 


1113112(2)  1113121(2)  1113211(2) 


1111  31  2  (2) 
C.5 


111^1  321(2)  11111.32  (2) 


ffi 


1111222(2)          1112122(2)  1121122(2) 


fcs 


1112221(2)  1112212(2)  1122112(2) 


v^ai  u 


1122121<2)      1121221(2)    1121212(2)    1212121(2) 


602 


Table  Vffl 
.C.I 


THE  MONIST. 


C.2 


0- 

3 


11111111(4)       11111112(3)     11111121(3)11111211(3) 


f^Mi  A  i  *l>  L&M    AH?,  i  al"  8n      i.  H?i  i  *l*  lAl.&i 
p^lJy  %^{    IJr  N*  &§}{     Ibw  ^  aifS{ 


11112111(3) 


11113111(2) 


11111311(2) 
C.3 


11111131(2)  11111113(2)          11111122(2) 


11111212(2) 
krr^rrStth-^ 


11112112(2)          11121112(2) 


•M«^ 


11112211(2)  11111221(2)  11112121(2) 


11121121(2)          11121211(2)  11211211(2) 


Table  IX 


C.I 


C.2 


111111111(3)  111111112(2)        111111121(2) 


P    B» 


111111211(2)  111112111(2)  111121111(2) 


Table  X 


Table  XI 
C.I 


\\r-j? *\>  41?-^N~I 

U  [J[J    ^l^  >j'-  : —  I 

--     *^  J    -.--r-      .  C» 


1111111111 


111111  11111 


OUR  MUSICAL  IDIOM.  603 

NOTES  ON  THE  NOTATION  OF  THE  ABOVE  HARMONIES. 

1.  R  is  represented  in  parentheses  where  it  is  included  in  the  notation. 

2.  The  degree  most  convenient  for  representation  is  chosen  for  the  base  note. 

3.  The  Number  Names  under  each  respective  harmony  have  their  integers 
separated  at  first  by  dashes.    Later  these  dashes  are  omitted. 

4.  When  the  number  of  sharps  and  flats  becomes  excessively  great,  it  is  written 
n  b  or  n  Jt.    Thus  8  8  $  #  becomes  4  8. 

5.  There  are  other  means  of  notating  some  of  these  denser  harmonies.     For 
example,  it  would  be  possible  to  employ  two  staffs,  or  double  stems.    Our 
present  system  of  notation  allows  of  no  better  methods. 


Inversions  of  the  Harmonies. 

Many  of  these  harmonies,  especially  those  of  many  tones,  may 
sound  unesthetic  in  their  fundamental  form  because  of  their  dis- 
sonance, even  to  an  ear  trained  to  an  appreciation  of  the  most  "ultra- 
modern" music.  A  conglomeration  of  slow  beats  caused  by  adjacent 
tones  will,  indeed,  almost  approach  a  common  noise.  However, 
such  dissonance  can  be  largely  reduced  in  the  same  harmony  when 
the  tones  of  the  fundamental  position  are  scattered  by  octaves. 
Thus  many  harmonies,  seemingly  obscure  in  their  fundamental  posi- 
tion, become  more  appreciable  to  us  by  inverting  them  or  spreading 
them  out.  The  different  forms  and  inversions  of  almost  every  har- 
mony (made  possible  by  the  range  of  modern  instruments)  allow 
of  the  greatest  variety  of  effects.  The  number  of  inversions  and 
positions  of  most  harmonies  is  astounding.  Now,  in  making  our 
rather  superficial  study  of  inversions,  we  will  be  obliged  to  use  a  few 
technical  expressions ;  which  are  enumerated  below. 

A  prime  position  of  a  harmony  has  been  previously  defined. 

Any  prime  inversion  of  a  fundamental10  harmony  will  be  known 
as  primary  inversion. 

An  inversion  not  prime,  but  containing  no  interval  as  great 
as  the  octave  will  be  considered  a  secondary  inversion. 

An  inversion  containing  one  or  more  intervals  exceeding  the 
octave  in  magnitude  will  be  known  as  a  tertiary  inversion. 

An  example  of  each  type  is  given,  respectively,  below ;  the  three 
inversions  are  in  the  same  harmony.  Although  a  form  like  (3) 
would,  according  to  the  chord  system,  be  considered  a  fundamental 
position  since  the  root  (e)  occupies  the  lowest  position,  we  will 

10  That  is,  a  harmony  in  fundaemntal  position,  according  to  definition. 


604 


THE   MONIST. 


find  it  more  convenient  to  regard  any  position  which  is  not  prime, 
even  though  it  have  the  root  in  lowest  position,  as  an  inversion. 

Al  A  2  A3 A  4      JO 


9)  tr  «;  «; 

Fundamental   Pos.        Primary  Inv.        Secondary  Inv.        Tertiary   Inv. 

In  the  following  paragraphs,  a  few  general  principles  are  dis- 
cussed in  the  form  of  propositions. 

Proposition  I.  A  harmony  of  n  tones  has  (w-1)  inversions  of 
the  first  degree. 

Since  a  prime  inversion  can  be  formed  with  each  tone  as  a 
lowest  tone,  and  since  (n-1)  tones  are  available  as  lowest  tones 
(one  tone  being  employed  as  the  lowest  tone  for  the  fundamental) 
it  follows  that  there  are  (»-l)  inversions  possible. 

In  other  words,  an  w-tone  harmony  has  n  prime  positions. 

Proposition  II.  The  number  of  secondary  inversions  of  a  har- 
many  of  n  tones  is  (n\-n). 

Let  us  experiment  with  two-,  three-  and  four-tone  harmonies, 
as  follows,  allowing  no  interval  between  adjacent  notes  to  be  as 
great  as  12. 

With  two-tone  harmonies  we  can  form  only  two  or  2 !  positions 
which  conform  to  our  limitations. 
For  example : 


-o- 


With  three-tone  harmonies  only  6  or  3!  such  positions  are 
possible;  as  for  example: 


-o — &- 


-o- 


o 


With  four-tone  harmonies  only  24  or  4 !  are  possible : 


OUR  MUSICAL  IDIOM.  605 

With  5-tone  harmonies  5!  or  120  such  inversions  are  possible, 
etc.,  etc.  With  n  tones  n!  positions  of  this  type  are  possible.  But 
since  these  positions  in  which  the  intervals  are  less  than  12  include 
n  prime  positions,  the  secondary  inversions  number  (n!-n). 

It  is  evident  that  the  number  of  tertiary  inversions  is  entirely 
dependent  upon  the  range  of  the  instrument  employed  to  represent 
them. 

By  the  span  of  a  harmony  is  meant  the  interval  of  its  extreme 
tones  or  the  sum  of  its  intervals.  Thus  the  span  of 


5-9 
is  (9  +  5)  equals  14.11 

Proposition  III.  In  a  harmony  of  n  tones  the  sum  of  the  spans 
of  its  prime  positions  is  (w-1)  octaves.  Let  us  take  an  example 
from  a  4-tone  harmony.  Adding  its  prime  positions  we  have 

A+   B+  C 

B+  C+  R 

A         +  C+  R 

A+   B         +  R 


=  3(A  +  B  +  C-tR)=3  octaves 

=  (4-1)  octaves. 

Progressions  of  Harmonies, 

This  subject  interests  us  more  from  a  speculative  than  a  prac- 
tical standpoint,  since  the  possibilities  in  this  direction  are  well-nigh 
unlimited,  as  will  be  shown.  However,  if  the  few  suggestions  that 
follow  are  carried  out  in  limited  form  practical  ends  are  attainable. 

Any  harmony  may  of  course  be  preceded  or  followed  by  any 
other  harmonies.  Whether  the  progressions  between  harmonies 
sound  abrupt  or  smooth  depends  partly  upon  the  harmonies  in  ques- 
tion, partly  upon  the  positions  chosen,  and  partly  upon  the  degree 
of  broad  appreciation  to  which  we  have  been  trained.  However, 
smoothness  or  abruptness  of  progression  does  not  concern  us  here, 
for  either  may  be  more  desirable  according  to  the  character  of  a 
composition.  The  inclination  of  a  composer  with  ideas  certainly 

11  In  finding  the  span  of  a  harmony,  R  is  evidently  not  included  to  make 
the  sum  12,  since  the  span  of  all  harmonies  would  consequently  be  multiples 
of  12. 


606  THE   MONIST. 

deserves  more  consideration  than  the  ever  weakening  law  of  the- 
orists. 

Let  us  now  calculate  the  number  of  progressions  of  two  suc- 
cessive harmonies  which  can  be  made  with  a  given  number  of  har- 
monies, let  us  say  n  harmonies.  One  of  these  n  harmonies  placed 
upon  one  degree  of  the  chromatic  scale  can  progress  to  the  same 
harmony  placed  on  11  other  degrees.  The  same  harmony  can  form 
12  progressions  with  any  of  the  other  harmonies  given,  since  any 
other  harmony  can  be  placed  on  12  different  degrees.  And  since 
there  are  (w-1)  such  other  harmonies,  the  first  harmony  can  form 
(n-l)12  progressions  with  the  remaining  harmonies.  Thus  the 
total  number  of  progressions  possible  between  a  single  harmony 
and  the  remaining  harmonies  is:  ll  +  12-(w-l). 

But  as  many  progressions  are  possible  with  each  of  the  (n-  1) 
remaining  harmonies.     Hence  the  total  number  of  progressions  of 
two  successive  harmonies  possible  with  n  harmonies  is: 
n[ll  +  12(w-l)]  =  (12n-l)n  or  12w2-n. 

Thus,  with  only  2  harmonies  we  can  form  (24-1)2,  or  46 
progressions.  With  5  harmonies  (which  is  the  limit  of  vocabulary 
with  many  persons,  and  in  which  may  be  included  the  major  triad, 
minor  triad,  dominant  sept,  supertonic  sept  and  leading-tone  sept), 
(60—1)5  or  295  progressions  of  only  two  successive  ones  are  pos- 
sible. With  the  350  existing  harmonies,  the  possibilities  of  pro- 
gression of  two  at  a  time  are:  (350x12-1)350=1,469,650. 

The  number  of  possibilities  of  progressions  of  three  at  a  time 
will  be  (\2nz-n)\2n,  since  each  progression  of  two  harmonies  may 
be  followed  by  one  of  n  harmonies  placed  on  any  one  of  12  different 
degrees  of  the  scale.  Thus  with  350  harmonies  6,172,530,000  pro- 
gressions of  three  are  possible. 

The  general  formula  expressing  the  number  of  progressions 
possible  is: 


where  n  is  the  number  of  harmonies  among  which  the  progressions 
are  to  be  made,  while  s  is  the  number  of  harmonies  to  be  used  at 
a  time  in  a  progression. 

As  mentioned  before,  the  enormous  figures  just  given  mean 
little  practically,  yet  they  serve  to  emphasize  the  fact  that  variety 
is  not  only  desirable  but  possible;  and  this  is  only  variety  of  one 
kind,  harmonic  variety. 

Melody,  rhythm  and  form  are  quite  as  variable  as  harmony, 


OUR  MUSICAL  IDIOM.  607 

and  the  variability  of  music  is  measured  as  the  product  of  these 
respective  elements  of  it  and  is  therefore  quite  beyond  the  bounds 
of  comprehension.  Formerly  I  shared  the  foolish  and  common  fear 
that  as  more  music  is  written  the  possibilities  of  future  invention 
narrow.  I  actually  felt  that  the  field  for  contemporary  composition 
is  narrower  than  it  was  a  century  ago.  To-day  it  seems  to  me  that 
every  great  musical  work  enlarges  the  field  of  the  future. 

"And  myriad  strains  are  there  since  the  beginning  still  waiting 
for  manifestation."12 

ERNST  LECHER  BACON. 
CHICAGO,  ILLINOIS. 

12  Busoni,  A  New  Esthetic  of  Music. 


CRITICISMS  AND  DISCUSSIONS. 


THE  PRIMITIVE  AND  THE  MODERN  CONCEPTIONS  OF 
PERSONAL  IMMORTALITY. 

In  an  interesting  review  of  my  recent  book  (The  Beliefs  in  God  and  Im- 
mortality: an  Anthropological,  Psychological,  and  Statistical  Study)  in  the 
April  number  of  this  journal,  it  is  inadvertently  written  that  the  book  is 
"simply  a  statistical  investigation."  This  statement  is  true  of  Part  II  only. 
It  is  not  applicable  to  Part  I,  for  it  treats  exclusively  of  "The  two  conceptions 
of  immortality:  their  origins,  their  different  characteristics,  and  the  attempted 
demonstration  of  the  truth  of  the  modern  conception."  No  more  does  the 
statement  apply  to  Part  III,  which  discusses  "The  present  utility  of  the  beliefs 
in  personal  immortality  and  in  a  personal  God." 

In  the  first  half  of  the  present  paper,  I  set  forth  very  briefly  that  which  I 
consider  the  main  contribution  contained  in  Part  I  of  my  book.  In  the  second 
half,  I  give  some  information  concerning  the  statistics  (Part  II).  J.  H.  L. 

A  curious  contradiction  seems  to  exist  with  regard  to  the 
origin  of  the  belief  in  survival  after  death.  It  is  authoritatively 
affirmed  by  anthropologists  that  that  belief  is  to  be  found  in  every 
tribe  now  extant.  Frazer,  less  dogmatic,  writes  that  "it  might  be 
hard  to  point  to  a  single  tribe  of  men,  however  savage,  of  whom 
one  could  say  with  certainty  that  the  faith  is  totally  wanting  among 
them."1  And  yet  historians  no  less  competent  in  their  field  than 
the  anthropologists  to  whom  we  refer  state  with  disconcerting  una- 
nimity that  the  belief  in  immortality  appeared  late  among  the  people 
from  whom  Europe  drew  its  civilization.  We  are  told,  for  instance, 
that  the  Israelites'  belief  in  immortality  cannot  be  traced  much 
further  back  than  the  beginning  of  the  Christian  era.  The  covenant 
Yahveh  made  with  his  people  does  not  allude  to  a  future  life.  The 
nation  alone  was  an  object  of  his  care.  The  great  prophets  them- 

*J.  G.  Frazer,  The  Belief  in  Immortality,  pp.  25,  33. 


CRITICISMS  AND  DISCUSSIONS.  609 

selves,  when  they  inveigh  against  sin,  care  only  for  the  danger  there- 
from to  the  existence  of  the  nation.  Among  the  Greeks  also  the 
belief  in  immortality  is  said  to  have  app  ared  late.  Pythagoras, 
the  Mysteries,  and  Plato  are  named  as  marking  the  rise  of  the  faith. 
Of  the  Romans,  Carter  says  that  they  did  not  have  the  idea  of  a 
personal  soul:  "It  was  not  present  at  the  time  of  the  Punic  wars. 
We  see  only  scanty  traces  of  it  in  the  literature  of  the  Ciceronean 
age."2 

These  apparently  contradictory  affirmations  may  be  explained 
in  two  ways:  either  the  particular  survival-idea  expressed  in  the 
belief  in  ghosts,  universal  among  primitive  p  ople,  had  at  the  be- 
ginning of  the  historical  period  disappeared  from  among  the  nations 
just  mentioned ;  or  the  immortality  which  the  historians  of  these 
nations  have  in  mind  is  so  different  from  the  primary  conception 
of  continuation  after  death  that  they  disregard  that  belief. 

The  first  of  these  two  suppositions  is  not  tenable.  When  the 
historical  period  opens,  a  belief  in  survival  was  incontrovertibly 
present  among  the  peoples  of  whom  the  historians  we  have  quoted 
speak.  In  the  Old  Testament  traces  of  polydemonistic  belief  are 
definite  enough  to  preclude  divergence  of  opinion.  The  evidence  is 
just  as  clear  in  the  case  of  the  Greeks  and  of  the  Jews.  The 
Homeric  conception  of  man  is  of  a  dual  personality  composed  of  a 
visible  earthly  being  and  of  its  shadow  or  copy,  which  manifests 
its  presence  in  dreams  and  continues  to  live  in  Hades  after  the 
severance  of  death.  Jane  Harrison  has  conclusively  demonstrated 
that  while  the  religion  of  the  Olympic  gods  was  in  process  of  forma- 
tion, and  even  much  later,  the  Greeks  practised  rites  clearly  indic- 
ative of  the  belief  in  human  ghosts.3 

The  idea  of  manes,  essential  to  the  religion  of  the  old  Romans, 
is  a  "vague  conception  of  shades  of  the  dead  dwelling  bdow  the 
earth."4  If  one  is  to  believe  Lucretius,  and  there  seems  to  be  no 
reason  why  he  should  not  be  credited  in  this  particular,  the  Romans 
were  haunted  by  a  dread  of  the  judgment  to  come. 

If  the  presence  at  the  beginning  of  the  historical  period  of 
practices  indicative  of  a  belief  in  survival,  in  the  very  people  among 
whom  the  idea  of  immortality  is  said  to  have  appeared  late,  is  no 
longer  a  moot  point,  shall  we  hold  that  the  kind  of  continuation 

2  J.  B.  Carter,  The  Religious  Life  in  Ancient  Rome,  p.  72. 

3  Jane  Harrison,  Prolegomena  to  the  Study  of  Greek  Religion,  1st  ed.,  p.  11. 

4  W.  Ward  Fowler,  The  Religious  Experience  of  the  Roman  People,  p.  386 


6lO  THE  MONIST. 

after  death  which  our  historians  have  in  mind  when  they  deny  the 
existence  of  the  belief  in  immortality  at  the  beginning  of  the  histor- 
ical period  is  so  different  from  the  idea  entertained  by  the  savage 
that  they  do  not  take  that  belief  into  account?  The  present  paper 
will  show  that  the  early  conception  of  survival  after  death — let  it 
be  called  the  primitive  conception — is,  as  a  matter  of  fact,  radically 
different  from  the  modern  conception  in  point  of  origin,  of  nature, 
and  of  function. 

What  was  the  nature  of  the  primitive  belief  in  the  countries 
bordering  the  eastern  end  of  the  Mediterranean  Sea  at  the  period 
to  which  it  is  customary  to  trace  the  rise  of  the  belief  in  immortality  ? 
Let  us  begin  with  Egypt,  the  land  of  the  "religion  of  eternal  life." 
The  oldest  historical  documents  we  possess,  the  inscriptions  in  the 
passages  and  chambers  of  the  great  pyramids,  called  the  Pyramid 
texts,  belong  to  an  already  complex  civilization  although  they  date 
bask  to  about  3400  B.  C.  The  glimpses  of  earlier  belief  found  in 
these  texts  suffice,  however,  to  indicate  the  presence  of  a  religion 
of  the  underworld  according  to  which  the  dead  continued  in  unhappy 
existence  under  the  earth.  "The  prehistoric  Osiris  faith,"  writes 
Breasted,  "involved  a  forbidding  hereafter  which  was  dreaded." 
In  an  inscription  on  a  stela  addressed  by  a  dead  wife  to  her  husband 
we  read:  "Oh  my  comrade,  my  husband.  Cease  not  to  eat  and 
drink,  to  be  drunken,  to  enjoy  the  love  of  women,  to  hold  festivals. 
Follow  thy  longing  by  day  and  by  night.  Give  care  no  room  in  thy 
heart.  For  the  West  Land  (a  domain  of  the  dead)  is  a  land  of 
sleep  and  darkness,  a  dwelling-place  wherein  those  who  are  there 
remain."8 

The  Babylonian  dead  were  supposed  to  dwell  in  a  great  cave 
underneath  the  earth,  the  most  common  name  of  which  was  Arttla. 
It  "was  pictured  as  a  vast  place,  dark  and  gloomy ....  surrounded  by 
seven  walls  and  strongly  guarded,  it  was  a  place  to  which  no  living 
person  could  go  and  from  which  no  mortal  could  ever  depart  after 
once  entering  it."6  For  the  Babylonians  death  made  all  men  equal. 
There  were  no  distinctions  of  rank  in  the  underworld ;  kings,  priests, 
conjurers,  magicians,  and  common  people,  all  found  themselves  to- 
gether in  the  dry  and  dusty  kurnugea.  Everything  one  touched 
was  dusty.  Dust  and  earth  were  the  food,  the  muddy  water  the 

B  A.  Wiedemann,  The  Realms  of  the  Egyptian  Dead,  p.  28. 

6  Morris  Jastrow,  Aspects  of  Religious  Belief  and  Practice  in  Babylonia 
and  Assyria,  pp.  353,  356,  358. 


CRITICISMS  AND  DISCUSSIONS.  6l  I 

drink  of  those  living  the  shadowy  life  of  the  underworld.7  They 
lived  an  ineffective,  drowsy,  starved  existence. 

Sheol  of  the  Hebrews,  like  the  underworld  of  the  Babylonians, 
was  a  place  of  dread.  The  shades  were  forgotten  of  God.  Yahveh 
was  the  God  of  the  living,  not  of  the  dead.  "Go  thy  way,"  says 
Ecclesiastes,  "eat  thy  bread  with  joy,  and  drink  thy  wine  with  a 
merry  heart .'. .  . Let  thy  garments  be  always  white ;  and  let  not  thy 
head  lack  oil.  Live  joyfully  with  the  wife  thou  lovest  all  the  days 
of  thy  life  of  vanity.  ..  .for  there  is  no  work,  nor  device,  nor 
knowledge,  nor  wisdom,  in  Sheol  whither  thou  goest." 

In  Greece  also  the  souls  went  to  the  land  of  the  d  ad  bemoaning 
their  lot,  for  it  was  wretched.  From  that  dark  country  souls  never 
returned.  Homer  draws  a  repulsive  picture  of  the  dead  hovering 
in  the  dark  realm  of  Acheron,  hazily  conscious,  hollow  voiced,  weak, 
and  indifferent. 

Neither  the  Egyptians,  nor  the  Babylonians,  nor  the  Hebrews, 
nor  the  Greeks  could,  it  seems,  think  of  beings  deprived  of  a  vig- 
orous, effective  body  as  enjoying  a  happy  life.  The  few  fortunate 
individuals  who  were  translated  to  Elysium  or  elsewhere  without 
passing  through  death  and  lived  on  happily,  had  retained  their 
body.  The  knowledge  of  the  decomposition  of  the  body  after  death 
and  of  t-he  tenuous  unsubstantial  nature  of  ghostly  apparitions, 
account  naturally  enough  for  the  weakness  and  ineffectiveness  at- 
tributed to  ghosts. 

For  centuries  this  repulsive  and  hopeless  belief  oppressed  the 
millions  from  among  whom  was  to  rise  European  civilization.  A 
turning  point  had,  however,  been  reached  at  the  dawn  of  the  his- 
torical period.  The  primitive,  belief  was  apparently  doomed,  for 
the  leaders  in  those  nations  had  not  only  felt  the  social  danger  it 
threatened,  and  had  in  consequence  begun  to  deprecate  as  evil  the 
cult  addressed  to  ghosts,  but  they  had  also  become  clearly  conscious 
of  moral  cravings,  the  satisfaction  of  which  death  seemed  to  make 
impossible. 

Regarding  the  opposition  that  had  arisen  to  the  primitive  belief, 
we  may  recall  that  in  Israel,  the  religion  of  Yahveh  was  the  deter- 
mined enemy  of  the  cult  of  the  dead  in  all  its  forms.  And  of  the 
Greeks  we  are  told  by  Jane  Harrison  that  "that  which  was  in  the 

7  Friedrich  Delitzsch,  Das  Land  ohne  Heimkehr,  die  Gedanken  der  Baby- 
lonier-Assyrer  iiber  Tod  und  Jenseits,  p.  16.  He  thinks,  however,  that  as  early 
as  the  thirtieth  century  B.  C.  a  distinction  in  the  abode  of  the  shades  made  its 
appearance.  Some  of  them  lived  in  peace  and  comfort  in  a  country  provided 
with  water  (pp.  18-22). 


612  THE  MONIST. 

sixth  and  even  in  the  fifth  century  before  the  Christian  era  the 
real  religion  of  the  main  bulk  of  the  (Hellenic)  people,  a  religion 
not  of  cheerful  tendance  but  of  fear  and  deprecation,"  was  the 
same  that  Plutarch  centuries  later,  and  with  him  most  of  his  great 
contemporaries,  regarded  as  superstition.  Among  the  Romans, 
ghosts  had  so  far  lost  individuality  as  to  be  regarded  by  modern 
historians  as  impersonal  forces.  The  cult  had  become'to  an  amazing 
degree  a  matter  of  mere  conventional  behavior.8  Thus  a  period  of 
greatly  decreased  influence  among  the  people  of  the  primitive  belief 
in  immortality  and  of  definite  antagonism  to  it  by  the  leaders  had 
arrived. 

Simultaneously  with  this  opposition  to  the  old  belief,  the  con- 
sciousness of  the  insufficiency  of  this  life  to  satisfy  the  cravings 
of  the  heart  and  the  demands  of  conscience  manifested  itself  in 
various  and  increasingly  significant  ways.  One  notes  precursory 
signs :  for  instance,  the  averred  translation  of  Menelaus  to  Elysium ; 
of  Ganymede  to  Olympus ;  of  Parnapishtim  to  an  earthly  paradise 
somewhere  in  Mesopotamia ;  of  Enoch,  who  was  taken  up  unto 
his  Lord ;  and  of  Elijah,  who  was  carried  in  a  chariot  of  fire  by  a 
whirlwind  into  Heaven.  One  notes  also  the  appearance  among  the 
ancient  Hebrews  of  Messianic  hopes ;  in  particular,  of  the  belief  in 
the  day  of  Yahveh  when  the  righteous  who  had  descended  to  Sheol 
would  arise  and  participate  in  the  triumph  of  the  nation.  The  faith-- 
ful  were  to  be  resurrected,  not  in  order  to  live  a  blessed,  independent 
existence  elsewhere  than  on  this  earth,  but  in  order  to  be  reincorpo- 
rated  in  the  earthly  life  of  the  nation.  These  were  preliminary 
manifestations  of  needs  which  found  their  full  expression  in  the 
modern  conception  of  immortality. 

The  formation  of  that  conception,  as  it  took  place  among  the 
Hebrews,  is  exceedingly  interesting.  Lack  of  space  forbids  any- 
thing more  than  a  passing  reference  to  some  of  the  main  facts.  Job 
is  an  early  shining  instance  among  the  Hebrews  of  a  clear  con- 
sciousness of  the  moral  incompleteness  involved  in  the  limitation 
of  human  existence  to  earthly  life.  Yet  he  died  without  the  hope 
of  a  blessed  immortality.  His  nearest  approach  to  it  is  a  fleeting 
persuasion  or  hope  that  after  death  he  would  enjoy  for  a  moment 
a  vision  of  God,  who  would  then  vindicate  his  mysterious  ways. 

The  transformation  of  Yahveh,  the  God  of  the  nation,  into  a 
God  maintaining  individual  converse  with  the  members  thereof,  and 
holding  each  individual,  and  no  longer  the  nation  alone,  as  morally 
8  W.  Ward  Fowler,  loc.  at.,  pp.  386-388. 


CRITICISMS  AND  DISCUSSIONS.  613 

responsible  to  him,  is  intimately  connected  with  the  establishment 
among  the  Jews  of  the  modern  belief  in  immortality.  The  tragic 
inner  life  of  Jer.  miah  shows  us  how  circumstances  forced  him  into 
individual  relationship  with  Yahveh  (chapters  xv-xvii).  Ezekiel 
continued  the  development  of  Jeremiah's  thought.  From  the  exist- 
ence of  an  individual  relationship  with  a  just  God,  he  drew  the 
unavoidable  conclusion  that  each  individual  is  to  be  rewarded  or 
punished  according  to  his  desert.  This  new  doctrine  permeates  the 
Psalms  and  the  book  of  Proverbs.  But  when  limited  to  earthly 
existence,  the  doctrine  is  obviously  false.  Job  and  the  author  of 
Ecclesiastes  are  up  in  arms  against  this  truncated  truth :  "All  things 
come  alike  to  all,  there  is  one  event  to  the  righteous  and  to  the 
wicked :  to  the  good  and  to  the  clean,  and  to  the  unclean ;  to  him 
that  sacrificeth  and  to  him  that  sacrificeth  not ;  as  is  the  good,  so  is 
the  sinner;  and  he  that  sweareth,  as  he  that  feareth  an  oath." 
Ezekiel's  doctrine  could  be  made  true  only  by  positing  another  life 
after  death  in  which  the  injustice  of  this  life  would  be  repaired. 
This  has  remained  a  chief  argument  of  those  believers  in  immor- 
tality who  also  believe  in  a  benevolent  and  righteous  Creator. 

The  conception  of  and  the  belief  in  a  blessed  future  existence 
in  which  man's  deepest  and  noblest  yearnings  are  to  be  realized, 
followed  upon  the  appearance  of  a  deep  sense  of  the  worth  of  these 
cravings.  Whenever,  among  peoples  already  familiar  with  the  idea 
of  soul  or  ghost,  these  cravings  were  sufficiently  keenly  felt,  they 
seemed  to  have  given  rise  to  a  belief  similar  to  the  Christian  belief 
in  immortality. 

In  Egypt  in  the  religion  of  the  sun-god,  lort£  before  the  book 
of  Job  was  written,  a  glorious  existence  with  the  god  had  been  con- 
ceived. In  Greece,  Plato  taught  a  lofty  doctrine  of  successive 
earthly  incarnations  for  the  gradual  purification  of  souls  from  the 
pollution  which  comes  to  them  from  their  association  with  matter. 
Ultimately  souls  entered  the  glorious  world  of  pure  spirits.  But 
this  doctrine  did  not  originate  with  the  Greek  philosopher.  He 
tells  us  himself  that  he  got  it  from  the  Orphic  priests.  The  Orphic 
cult  was  addressed  to  Dionysos  by  a  sect  that  had  evolved  a  definite 
system  of  religio-philosophic  belief,  the  chief  article  of  which  was 
the  double  composition  of  man:  one  part  mortal,  coming  from  the 
Titans,  the  other  divine.  Man's  task  was  to  rid  himself  of  the 
Titanic  ebment,  which  corresponds  to  the  body,  in  order  to  return 
pure  to  God.  The  deliverance  of  the  soul  could  not  be  achieved 


614  THE  MONIST. 

suddenly  nor  without  the  helping  mediation  of  Orpheus,  who,  let 
it  be  noted,  demanded  a  pure  life  as  condition  of  salvation  from 
rebirth. 

The  nature  of  the  primary  conception  of  continuation  after 
death  gives  proof  that,  unlike  the  modern  conception,  it  was  not 
born  of  desires  for  the  fulfilment  in  another  existence  of  hopes 
frustrated  on  this  earth.  Had  it  had  that  origin,  it  would  neces- 
sarily have  been  conceived  of  in  a  form  designed  to  satisfy  these 
desires.  The  nature  of  the  belief  and  its  universality  among  sav- 
ages show  it  to  have  been  imposed,  regardless  of  man's  feeling 
toward  it,  as  irresistibly  as  the  belief  in  the  existence  of  any  object 
present  to  the  senses. 

Differences  in  origin  lead  to  differences  in  function.  In  the 
primary  belief,  the  ghosts,  even  those  of  friends,  are  on  the  whole 
sources  of  anxiety  and  fear,  and  the  relations  maintained  with  them 
aim  almost  exclusively  at  warding  off  their  interferences  in  human 
affairs.  No  one  loves  a  ghost  and,  speaking  generally,  no  one  de- 
sires to  become  one.  The  modern  belief  is,  on  the  contrary,  a  vivi- 
fying conviction  or  hope,  calling  forth  the  best  that  is  in  one's  per- 
sonality. 

To  consider  these  two  conceptions  as  bearing  to  each  other  the 
relation  of  the  seed  to  the  fruit,  is,  therefore,  to  disregard  their 
respective  nature  and  function  as  well  as  their  origin.  In  none  of 
these  respects  have  these  conceptions  anything  essential  in  common. 
That  is  why  the  primary  conception  had  to  be  discredited  and  dis- 
carded before  the  modern  one  of  a  glorious  life,  fulfilling  the  noblest 
human  demands,  could  be  formed  and  entertained. 

STATISTICS  OF  CONTEMPORARY  RELIGIOUS   BELIEFS. 

'  In  Part  II  of  my  book,  I  attempted  to  discover  what  propor- 
tions of  the  members  of  a  number  of  influential  classes  (physical 
scientists,  biological  scientists,  historians,  sociologists,  psychologists, 
and  college  students  of  non-technical  departments)  believe  in  per- 
sonal immortality  and  in  the  God  whose  existence  is  presupposed  by 
all  the  organized  religions,  i.  e.,  a  God  conceived  of  as  acting  upon  the 
physical  world  or  at  least  upon  man,  at  man's  request,  desire,  or  desert. 
It  appeared  to  me  of  great  interest  both  practically  and  scientifically 
to  find  out  definitely  the  percentages  of  believers,  disbelievers,  and 
doubters  among  these  classes,  and  to  correlate  eminence  in  them 
and  the  special  kinds  of  knowledge  possessed  by  their  members  with 
these  percentages. 


CRITICISMS  AND  DISCUSSIONS.  615 

I  was  aware  that  the  statistics  of  belief  so  far  gathered  have 
little  or  no  statistical  value.  When,  as  in  the  case  of  the  extensive 
inquiry  of  the  Society  for  Psychical  Research,  less  than  one-third 
of  those  who  were  solicited  answered,  no  particular  meaning  attaches 
to  the  discovery  that  two-thirds  of  that  one-third  believe  in  immor- 
tality. In  order  to  obtain  statistics  valid  for  the  whole  of  a  group, 
it  is  not  necessary,  it  is  true,  to  poll  every  member  of  the  group. 
It  is  sufficient  to  consider  a  part  of  that  group,  provided  that  every 
member  of  that  part  or  a  very  high  percentage,  answer  the  inquiry, 
and  that  the  selection  of  the  part  investigated  be  made  according  to 
chance.  The  statistics  of  that  part  may  then,  according  to  the  law 
of  probability,  be  held  valid  for  the  whole  group. 

The  statistical  defect  from  which  the  inquiry  of  the  Society 
for  Psychical  Research  suffers,  is  often  combined  with  an  insufficient 
definition  of  the  belief  under  investigation.  Not  long  ago  some  rash 
person  affirmed  in  the  English  press  that  "it  is  extremely  doubtful 
whether  any  scientist  or  philosopher  really  holds  the  doctrine  of  a 
personal  God."  Thereupon  a  Mr.  Tabrum  collected  from  among 
English  scientists  140  expressions  of  opinion  on  the  question,  "Is 
there  any  real  conflict  between  the  facts  of  science  and  the  funda- 
mentals of  Christianity?"  But  the  author  did  not  define  what  he 
meant  by  "fundamentals,"  neither  did  he  ask  his  correspondents  to 
state  the  meaning  they  attached  to  that  expression.  Strange  to  say, 
very  few  thought  it  necessary  to  be  explicit.  Lord  Rayleigh  wrote, 
for  instance,  "I  may  say  that  in  my  opinion  true  science  and  true 
religion  neither  are  nor  could  be  opposed."  This  has  the  appear- 
ance of  a  misplaced  pleasantry.  Any  one  may  make  that  statement ; 
its  significance  depends  altogether  upon  what  is  meant  by  "true 
religion."  You  may  have  in  mind  some  conception  of  religion 
which  would  tolerate  neither  the  Apostles'  nor  the  Nicean  creed, 
nor  even  a  personal  God! 

In  my  own  investigation  I  endeavored  to  avoid  the  two  major 
defects  illustrated  above,  and  succeeded,  I  think,  in  establishing 
statistics  of  belief  valid  for  the  entire  classes  named  above,  so  far 
as  the  United  States  is  concerned. 

The  student  of  human  development  will  be  interested  in  the 
possibility  now  opened  to  ascertain  the  statistical  history  of  re- 
ligious beliefs.  By  instituting  at  some  future  time  an  investigation 
similar  to  mine,  it  would  become  possible  to  express  with  a  high 
degree  of  exactness  the  changes  that  have  taken  place  in  the  spread 
of  the  beliefs  here  considered. 


6l6  THE   MONIST. 

If  I  cannot  enter  here  into  details  as  to  the  statistical  method 
I  have  followed,  the  results  secured,  and  their  interpretation,  I  may 
at  1  ast  add  in  conclusion  the  following  figures  and  some  brief  in- 
formation.9 • 

PHYSICAL        BIOLO-       HISTORI-     SOCIOLO-     PSVCHOL- 
BKLIBVKRS  IN  Goo  SCIKMISIS       GISTS  ANS  GISTS          OGISTS 

Lesser  Men   49.7      39.1       63.         29.2      32.1 

Greater  Men   .         .  34.8       16.9      32.9       19.4       13.2 


Lesser  Men  57.1      45.1      67.6      52.2      26.9 

Greater  Men   40.         25.4      35.3       27.1        8.8 

These  figures  show  that  in  every  class  of  persons  investigated 
the  number  of  believers  in  God  is  less,  and  in  most  classes  very 
much  less,  than  the  number  of  non-believers,  and  that  the  number 
of  believers  in  immortality  is  somewhat  larger  than  in  a  personal 
God ;  that  among  the  more  distinguished,  unbelief  is  very  much  more 
frequent  than  among  the  less  distinguished ;  and  finally  that  not  only 
the  degree  of  ability,  but  also  the  kind  of  knowledge  possessed  is 
significantly  relat  d  to  the  rejection  of  these  beliefs. 

"The  correlation  shown,  without  exception  in  every  one  of  our 
groups,  between  eminence  and  disbelief  appears  to  me  of  mom3ntous 
significance.  In  three  of  these  groups  (biologists,  historians  and 
psychologists)  the  number  of  belkvers  among  the  men  of  greater 
distinction  is  only  half,  or  less  than  half  the  number  of  believers 
among  the  less  distinguished  men.  I  do  not  see  any  way  of  avoiding 
the  conclusion  that  disbelief  in  a  personal  God  and  in  personal  im- 
mortality is  directly  proportional  to  abilities  making  for  success  in 
the  sciences  in  question.10 

"With  regard  to  the  kinds  of  knowledge  which  favor  disbelief, 
the  figures  show  that  the  historians  and  the  physical  scientists  pro- 
vide the  greater ;  and  the  psychologists,  the  sociologists  and  the 
biologists  the  smaller  number  of  believers.  The  explanation  is,  I 
think,  that  psychologists,  sociologists  and  biologists  in  very  large 
numbers  have  come  to  recognize  fixed  orderliness  in  organic  and 
psychic  life,  and  not  merely  in  inorganic  existence;  while  frequently 
physical  scientists  have  recognized  the  presence  of  invariable  law 
in  the  inorganic  world  only.  The  belief  in  a  personal  God  as  defined 
for  the  purpose  of  our  investigation  is,  therefore,  less  often  pos- 

9  These  figures  are  percentages  of  the  number  of  persons  who  answered  the 
questionnaire. 

10  Concerning  these  abilities  and  their  influence,  see  Chapter  X. 


CRITICISMS  AND  DISCUSSIONS.  617 

sible  to  students  of  psychic  and  of  organic  life  than  to  physical 
scientists. 

"The  place  occupi  d  by  the  historians  next  to  the  physical 
scientists  would  indicate  that  for  the  present  the  reign  of  law  is 
not  so  clearly  revealed  in  the  events  with  which  history  deals  as 
in  biology,  economics,  and  psychology.  A  large  number  of  his- 
torians continue  to  see  the  hand  of  God  in  human  affairs.  The  in- 
fluence, destructive  of  Christian  beliefs,  attributed  in  this  inter- 
pretation to  more  intimate  knowledge  of  organic  and  psychic  life, 
appears  incontrovertibly,  as  far  as  psychic  life  is  concerned,  in  the 
remarkable  fact  that  whereas  in  every  other  group  the  number  of 
believers  in  immortality  is  greater  than  that  in  God,  among  the 
psychologists  the  reverse  is  true;  the  numbT  of  believers  in  im- 
mortaHtv  among  the  greater  psychologists  sinks  to  8.8  percent. 

"One  may  affirm,  it  seems,  that  in  general  the  greater  the  ability 
of  the  psychologist,  the  more  difficult  it  becomes  for  him  to  believe 
in  thp  continuation  of  individual  life  aft~r  bodily  death. 

"The  students'  statistics  show  that  young  people  enter  college 
possessed  of  the  beliefs  still  aorpted,  more  or  less  perfunctorily, 
in  the  average  home  of  the  land,  and  that  as  their  mental  powers 
mature  and  their  horizon  widens  a  large  prec^ntage  of  them  aban- 
don the  cardinal  Christian  beliefs.  It  seems  probable  that  on  leaving 
coMege,  from  40  to  45  percent  of  the  students  with  whom  we  are 
concerned  deny  or  doubt  the  fundamental  dogmas  of  the  Christian 
religion.  The  marked  decrease  in  belief  that  takes  place  during 
the  later  adolescent  years  in  those  who  spend  those  years  in  study 
under  the  influence  of  persons  of  high  culture,  is  a  portentous  indi- 
cation of  the  fate  which,  according  to  our  statistics,  increased  knowl- 
edge and  the  possession  of  certain  capacities  leading  to  eminence 
reserve  to  the  beliefs  in  a  personal  God  and  in  personal  immor- 
tality."11 

To  the  statistical  data  are  added  a  large  number  of  letters  from 
my  correspondents  and  a  somewhat  full  study  of  the  religious  ideas 
of  students.  These  together  with  the  statistics  make  a  picture  of  the 
present  religious  situation  both  vivid  and  relatively  exact. 

J.  H.  LEUBA. 
BRYN  MAWR  COLLEGE. 

11  The  Belief  in  God  and  Immortality,  pp.  277-281. 


6l8  THE  MONIST. 


NOTES  ON  RECENT  WORK  IN  THE  PHILOSOPHY  OF 

SCIENCE. 

Federigo  Enriques  ("Sur  quelques  questions  soulevees  par  1'in- 
fini  mathematique,"  Rev.  de  metaphysique  et  de  morale,  March, 
1917,  Vol.  XXIV,  pp.  149-164)  points  out  that  experience,  when 
it  is  idealized  by  reason,  puts  before  us  two  kinds  of  infinity,  the 
actually  and  the  potentially  infinite ;  suppose  then  "that  we  are 
potentially  given  by  thought  an  infinity  of  objects,  the  question 
arises  as  to  whether  there  is  any  reason  to  consider  as  logically  de- 
fined a  new  object  of  thought  which  expresses  the  totality  or  the 
limit  of  these  objects  even  when  they  are  not  constructed  with 
respect  to  a  concept  of  the  kind  which  we  suppose  to  be  given 
a  priori."  The  answer  to  this  question  depends  on  a  fundamental 
tendency  of  the  mind ;  it  will  be  negative  or  in  some  degree  positive 
according  as  we  feel  ourselves  borne  toward  nominalism  or  toward 
realism.  Realist  doctrine — at  least  in  mathematics — in  its  first  his- 
torical form  rested  on  the  assumption  that  a  simple  passage  to  the 
limit  was  always  possible,  and  this  was  gradually  destroyed  by  the 
progress  of  the  infinitesimal  calculus.  The  realist  doctrine  in  its 
second  historical  form  rests  on  the  principle  that  (p.  159)  "every 
infinity  of  virtually  defined  objects  may  be  considered  as  a  totality 
forming  a  class  and  constituting  a  new  logical  object."  As  dis- 
tinguished from  realism  in  its  first  form,  in  this  new  realism  we 
conceive  that  the  properties  of  the  new  object  are  absolutely  new 
and  that  thus  we  cannot  state  them  a  priori  by  an  induction  extended 
from  the  finite  to  the  infinite.  This  new  form  of  realism  is  due,  in 
its  mathematical  form,  to  Georg  Cantor,  but  (p.  159)  "the  philos- 
opher B.  Russell  has  developed  in  the  widest  sense  the  philosophical 
consequences  of  the  realism  thus  introduced  into  mathematics." 
The  various  paradoxes  of  mathematical  logic  have  led  to  the  con- 
clusion that  there  are,,  in  certain  cases,  no  such  things  as  classes  of 
perfectly  definite  objects ;  and  this  realism,  in  its  second  form,  is 
partially  unsuccessful.  On  p.  163  we  read  that  "the  principle  of  an 
infinite  number  of  choices  is  adopted  by  Russell  and  by  Zermelo," 
and  so  we 'are  apparently  again  forced  to  the  conclusion  that  En- 
riques is  quite  unaware  of  the  tendency  shown  by  Russell's  work 
published  since  1905.  Since  the  question  is  rather  important,  per- 


CRITICISMS  AND  DISCUSSIONS.  619 

haps  the  present  reviewer  may  be  forgiven  for  dwelling  on  some 
work  on  the  paradoxes  in  question  since  1903. 

In  Russell's  Principles  of  Mathematics  (Cambridge,  1903) 
there  was  not  any  definite  suggestion  that  the  concept  of  class  should 
be  restricted,  though  there  was  certainly  a  more  or  less  vague  feeling 
that  some  classes  should  be  excluded  (see  Monist,  January,  1912. 
Vol.  XXII,  pp.  153,  157-158;  and  January,  1917,  Vol.  XXVII,  p. 
144).  The  merit  of  perceiving  that  a  restriction  was  necessary  and 
of  attempting  to  give  a  criterion  to  decide  which  classes  were  legit- 
imate seems  due  to  Jourdain,  in  a  paper  published  in  1904  (see 
Monist,  January,  1917,  Vol.  XXVII,  pp.  148-150).  The  question 
as  to  the  being  or  not-being  of  a  class  is  totally  different  from  the 
question  of  the  possibility  of  an  infinite  series  of  acts  of  selection, — 
which,  by  the  way,  neither  was  nor  is  assumed  by  Russell,  though 
it  is  believed  in  by  Zermelo  and  many  others.  The  merit  of  being 
the  first  to  publish  an  explicit  recognition  of  the  postulate  involved 
in  this  assumption  is  due  to  Zermelo  in  1904,  and  the  questions  re- 
lating to  Zermelo's  axiom  have  been  frequently  confused  with,  for 
example,  Jonrdain's  "proof"  by  even  eminent  people;  though  both 
Russell  and  Jourdain  pointed  out  repeatedly  that  the  questions 
involved  are  quite  different.  In  1905  and  later  Russell  published 
papers  gradually  showing  how  it  was  possible  to  work  through  a 
great  deal  of  Cantor's  theory  without  assuming  that  there  are  such 
things  as  classes  at  all,  and  a  thorough  exposition  of  this  theory  is 
one  of  the  most  important  parts  of  Whitehead  and  Russell's  Prin- 
cipla  Mathcmatica  (Vol.  I,  Cambridge,  1910).  Thus  it  is  obvious 
that  this  theory  of  Russell's  not  only  makes  the  considerations  of 
Jourdain  and  some  others  quite  superfluous,  but  also  makes  such 
criticisms  as  that  of  Enriques  entirely  off  the  point  (cf.  Monist, 
January,  1917,  Vol.  XXVII,  pp.  145-148). 

This  historical  sketch  is  also  relevant  to  our  consideration  of 
a  recent  paper  by  Dmitry  Mirimanoff  ("Les  antimonies  de  Russell 
et  de  Burali-Forti  et  le  probleme  fondamental  de  la  theorie  des 
ensembles,"  in  L'enseignemcnt  maihcmatique,  1917,  Vol.  XIX,  pp. 
37-52).  The  author  remarks  (p.  48)  that  we  can  find  in  the  works 
of  Bertrand  Russell,  Henri  Poincare,  and  Julius  Konig  (Neue 
Grundlagen  der  Logik.  Arithmctik  und  Mengcnlchre,  Leipsic,  1914) 
"a  profound  logical  and  psychological  analysis  of  the  Cantorian 
antinomies  and  of  the  notion  of  class,"  but  that  he  "will  not  have 
any  need  of  this  analysis  for  the  end  which"  he  lias  in  view.  His 


62O  THE  MONIST 

article  is  characterized  by  the  following  quotation  from  p.  33: 
"People  believed,  and  it  seemed  quite  evident,  that  the  existence  of 
individuals  necessarily  implies  that  of  the  class  of  them,  but  Burali- 
Forti  and  Russell  showed  by  different  examples  that  a  class  of 
individuals  may  be  non-existent,  although  the  individuals  exist.  As 
we  cannot  refuse  to  accept  this  new  fact,  we  are  obliged  to  conclude 
from  it  that  the  proposition  which  seemed  evident  and  which  was 
believed  always  to  be  true  is  only  true  under  certain  conditions. 
And  then  arises  the  problem  which  we  may  regard  as  the  funda- 
mental problem  of  the  theory  of  aggregates :  What  are  the  necessary 
and  sufficient  conditions  for  the  existence  of  a  class  of  individuals?" 
On  this  confusion  between  the  ideas  of  existence  and  entity,  see  the 
article  quoted  above  in,  The  Monist  for  January,  1917.  The  author 
gives  a  solution  of  this  problem  for  the  particular  case  of  classes 
that  he  calls  "ordinary"  classes,  and  his  deductions  rest  on  three 
postulates  (p.  49)  which  are  applied  by  some  in  the  study  of  prob- 
1  ms  of  the  theory  of  aggregates.  Further,  the  examples  of  Russell 
and  Burali-Forti  are  modified  (pp.  39-48)  in  a  way  that  seems 
advantageous  to  the  author,  and  the  author  announces  (pp.  39,  52) 
his  intention  to  give  in  a  future  article  the  reasons  which  determined 
him  not  to  adopt  in  this  paper  the  theory  of  Konig.  The  criterion 
which  the  author  arrives  at  (pp.  48-52)  for  deciding  whether  indi- 
viduals have  a  class  or  not  is  practically  that  suggested  by  Jourdain 
in  1904:  individuals  have  a  class  if  they  can  be  arranged  in  a  seg- 
ment of  the  series  W  of  all  ordinal  numbers  and  not  if  they  cannot 
be  so  arranged.  It  is  not  worth  while  to  enter  into  a  criticism  of 
this  suggested  criterion,  which  in  any  case  has  become  quite  super- 
fluous through  the  work  of  Russell  referred  to  above.  Mirimanoff 
(p.  52)  regrets  that  it  has  been  impossible  for  him  to  become  ac- 
quainted with  work  that  has  appeared  since  the  beginning  of  the 
war.  If  he  had  read — which  he  nowhere  gives  any  sign  of  having 
done — the  works  referred  to  above  of  1904  to  1910,  we  do  not  think 
that  it  would  have  been  necessary  to  write  this  paper. 


In  the  Revue  de  metaphysique  et  de  morale  for  January,  1917, 
there  is  an  address  given  by  the  late  Victor  Delbos  on  the  general 
characteristics  of  French  philosophy.  A  part  of  the  late  Louis 
Couturat's  Manuel  de  logislique,  which  he  wrote  about  1906  but 
which  is  not  yet  published,  is  printed.  These  extracts  form  the 
greater  part  of  the  second  chapter  of  this  book,  and  are  on  the 


CRITICISMS  AND  DISCUSSIONS,  621 

logical  relations  of  concepts  and  propositions.  An  interesting  fact 
about  them  is  that  the  work  of  Frege  seems  to  have  influenced 
Couturat  to  a  greater  extent  than  was  the  case  with  Couturat's 
Principes  of  1905.  This  contribution  is  fairly  elementary,  and  does 
not  deal  with  those  paradoxes  which  are,  perhaps,  of  the  greatest 
interest  to  logicians,  although  it  just  mentions  them.  F.  Colonna 
D'Istria  writes  on  the  logic  of  medicine  according  to  Cabanis's 
Rapports  dit  physique  et  du  moral  de  I'homme.  Arnold  Reymond 
makes  a  critical  study  of  the  new  and  recast  edition  of  Edouard 
Claparede's  Psychologic  dc  I  enfant  et  pedagoyie  experimentale 
(Geneva,  1916).  Thomas  Ruyssen  writes  on  "an  idea  in  peril: 
humanity,  humanitarianism,  humanism."'  Finally  there  are  obituary 
notices  of  Theodule  Ribot  (1839-1916),  the  eminent  psychologist, 
and  Henri  Dufumief. 

*        *       * 

In  the  number  of  the  Revue  de  metaphysique  et  de  morale  for 
May,  1917,  A.  Espinas  deals  with  the  initial  idea  of  the  philosophy 
of  Descartes,  and  the  late  Victor  Delbos's  lecture  on  method  in  the 
history  of  philosophy  forms  the  second  of  his  three  lectures  on  the 
history  of  philosophy.  A  manuscript  by  the  late  Louis  Couturat, 
which  was  certainly  written  before  1902  and  which  will  not  form 
part  of  the  projected  Manuel  dc  logistique,  is  printed  here  and  is  on 
the  algebra  of  logic  and  the  calculus  of  probabilities.  Finally  Ales- 
sandro  Padoa  has  a  paper  on  the  consequences  of  a  change  of 
primitive  ideas  in  any  deductive  theory  whatever. 

*       *       * 

In  a  paper  on  the  ''infinite  numbers"  which  Bernard  le  Bovier 
Fontenelle  tried  to  introduce  in  his  Elements  de  la  geometric  de 
I'infini  (Paris,  1727).  Branislav  Petronievics  ("Stir  les  nombres  in- 
finis  de  Fontenelle."  Rcndiconti  dclla  R.  Accademia  dei  Lincei 
[CJasse  di  science  fisiche.  iiiatcmatiche  e  naturali],  Vol.  XXVI, 
1917,  pp.  309-316)  tries  to  show  that  this  "first  attempt  at  a  rational 
theory  of  infinite  numbers,"  although  it  is  obviously  full  of  contra- 
dictions which  were  at  once  pointed  out  by  MacLaurin  and  others, 
possesses  a  historical  value  when  compared  with  the  theories  of 
Cantor  and  Veronese.  Cantor's  theory  has,  says  Petronievics,  an 
arithmetical  starting-point,  while  that  of  Veronese  has  a  geometrical 
one ;  and  Veronese  establishes  that  there  is  no  point  on  an  infinite 
straight  line  which  corresponds  to  the  first  transfinite  ordinal  num- 
ber of  Cantor,  so  that  "geometrical  application  of  the  transfinite 


622  THE  MONIST. 

numbers  of  Cantor  is  not  possible."  In  spite  of  the  fact  that  Fon- 
tenelle  introduced  his  "infinite  number  of  all  finite  numbers"  as 
"the  last  one"  of  this  series  itself,  Petronievics  maintains  that  the 
theory  of  Fontenelle  has  a  historical  value  in  that  it  "may  be  re- 
garded as  the  common  source  of  the  theories  of  Cantor  and  Vero- 
nese," and  that  "it  is  not  impossible  to  suppose,  in  view  of  the 
likenesses  between  the  theories,  that  Cantor  and  Veronese  both 
arrived  at  establishing  the  principles  of  their  theories  when  trying 
to  avoid  the  flagrant  contradictions  into  which  Fontenelle  fell." 

There  does  not  seem  to  the  reviewer  to  be  the  smallest  ground 
for  supposing  that  Cantor  was  led  to  his  theory  either  by  reading 
Fontenelle  or  by  setting  out  deliberately  to  generalize  arithmetic. 
Indeed,  one  of  the  points  of  the  long  introduction  to  the  trans- 
lation of  Cantor's  later  papers  published  under  the  title  of  Con- 
tributions to  the  Founding  of  the  Theory  of  Transfinite  Num- 
bers (Chicago  and  London,  1915)  is  to  show  that  Cantor  was  com- 
pelled to  generalize  the  idea  of  number  as  a  consequence  of  the 
natural  development  of  his  process  of  "derivation"  of  geometrical 
point-sets.  In  this  extension  it  appeared  clearly  that  the  transfinite 
numbers  began  beyond  the  whole  series  of  finite  numbers  in  oppo- 
sition to  Fontenelle's  notion  mentioned  above.  Fontenelle  says  on 
page  30  of  his  book :  "We  must  not  be  frightened  at  the  words  'last 
term'  in  this  connection.  It  is  a  last  finite  term  that  the  natural 
series  of  numbers  has  not,  but  not  to  have  a  last  finite  term  is  the 
same  thing  as  to  have  a  last  infinite  term."  This  is  a  charming  way 
of  turning  a  universal  negative  proposition  into  a  particular  affirma- 
tive one.  It  seems  that  the  first  time  that  Cantor  spoke  more  or  less 
publicly  of  Fontenelle's  theory  was  in  a  letter  of  1886  (cf.  Zur 
Lehre  vom  Transfiniten,  Halle,  1890,  p.  50),  and  therefore  long 
after  he  had  founded  his  absolutely  different  theory.  That  hearing 
of  one  theory  may  have  been  the  psychological  cause  of  Cantor's 
thinking  about  a  fundamentally  different  theory  is  of  course  both 
possibly  and  probably  irrelevant,  but  there  is  no  ground  for  sup- 
posing that  even  this  happened.  It  is  a  mistake  to  say  that  ordinal 
numbers  cannot  have  geometrical  applications :  an  illustration  of 
the  way  such  numbers  can  appear  is  given  by  this:  To  the  series 
on  the  .r-axis  formed  by  the  points  1,  1/2,  1/3,  .  . .  .,  1/n,. . .  .,  the 
point  0  bears  exactly  the  same  relation  as  the  first  transfinite  ordinal 
does  to  the  finite  ordinals  in  order  of  magnitude.  For  Cantor's 
remarks  on  Veronese  and  Veronese's  reply,  we  may  quote  the  above 


CRITICISMS  AND  DISCUSSIONS.  623 

Contributions  (pp.  117-118).  A  purely  analytical  exposition  of  the 
infinite  and  infinitesimal  numbers  of  Veronese  was  given  by  T. 
Levi-Civita  ("Infiniti  e  Infinitesimi  attnali,"  Atti  R.  Istituto  Veneto, 

1892). 

*       *       * 

Florian  Cajori  ("The  Zero  and  Principle  of  Local  Value  used 
by  the  Maya  of  Central  America,"  Science,  Vol.  XLIV,  1916,  pp. 
714-717)  draws  attention  to  the  fact,  hitherto  apparently  unnoticed 
by  mathematicians,  that  the  Maya  of  Central  America  and  southern 
Mexico  seem  to  have  used  a  symbol  for  zero  and  the  principle  of 
local  value  much  earlier  than  any  one  else.  The  material  for  Cajori's 
remarks  is  furnished  by  An  Introduction  to  the  Study  of  the  Maya 
Hieroglyphs,  by  Sylvanus  GriswoFd  Morley  (Bulletin  57  of  the 
Bureau  of  American  Ethnology,  Washington,  1915).  The  early 
Babylonians  possessed  the  principle  of  local  value,  but  so  far  as  we 
know  did  not  possess  a  zero.  About  200  B.  C.  they  did  have  a 
symbol  for  zero,  which,  as  Smith  and  Karpinski  (Hindu- Arabic 
Numerals,  Boston,  1911,  p.  51)  say,  was  "not  used  in  calculation, 
nor  does  it  always  occur  when  units  of  any  order  are  lacking." 
They  did  not  employ  it  systematically  in  writing  numbers  and  not 
at  all  in  performing  computations.  The  Hindus  certainly  did  not 
use  their  symbol  for  zero  systematically  before  probably  the  sixth 
century  A.  D.,  and  the  earliest  undoubted  occurrence  of  zero  in 
Indian  numerals  is  A.  D.  876  (cf.  also  G.  R.  Kaye,  Indian  Mathe- 
matics, Calcutta,  and  Simla,  1915,  p.  31,  for  the  date  of  the  appear- 
ance of  the  principle  of  local  value  in  India).  Now,  it  seems  that 
the  Maya  used  the  zero  and  the  principle  of  local  value  at  the  begin- 
ning of  the  Christian  era  if  not  much  earlier.  "As  far  as  is  known, 
the  Maya  used  their  numeral  systems  only  in  the  counting  of  time 
as  it  arose  in  their  calendar,  ritual,  and  astronomy."  Of  the  several 
Maya  numeral  notations  the  one  which  is  of  greatest  interest  as 
embodying  the  principle  of  local  value  and  the  symbol  for  zero  is 
found  in  Maya  codices  but  not  in  their  inscriptions.  The  number 
system  was  vigesimal,  with  the  solitary  break  that  18  (and  not  20) 
uinals  make  1  tun,  and  the  symbols  1  to  19,  both  inclusive,  are  ex- 
pressed by  bars  and  dots.  Each  bar  stands  for  five  units  and  each 
dot  for  one  unit,  and  the  dots  are  written  above  the  bars.  Thus 
19  is  written  as  three  bars  above  one  another  and  four  dots  on  the 
top.  "The  values  of  the  bars  and  dots  are  added  in  each  case.  The 
zero,  which  plays  a  leading  part  in  the  notations  found  on  inscrip- 


624  THE  MONIST. 

tions  as  well  as  those  on  codices,  is  represented  in  the  codices  by 
a  symbol  that  looks  roughly  like  a  half-closed  eye....  In  writing 
20.  ..  the  principle  of  local  value  enters  for  the  first  time.  It  is  ex- 
pressed by  a  dot  placed  over  the  symbol  for  zero.  The  numerals 
are  written,  not  horizontally,  but  vertically,  the  unit  of  the  lowest 
order  or  value  being  assigned  the  lowest  position.  Accordingly, 
37  was  expressed  by  the  symbols  for  17  (three  bars  and  two  dots) 
in  the  kin  [units]  place  and  one  dot,  representing  20,  placed  above 
the  17,  in  the  uinal  place.  The  number  300  is  expressed  by  three 
bars  drawn  above  the  symbol  for  zero  (3  x  5  x  20  =  300).  The  largest 
number  which  can  be  written  by  the  use  of  only  two  places  or  posi- 
tions is  17x20+19  =  359.  To  write  360,  the  Maya  drew  two  zeros, 
one  above  the  other,  with  one  dot  higher  up,  in  third  place.  Using 
three  places  to  represent  kins,  uinals,  and  tuns,  they  could  write  any 
number  not  larger  than  7199.  Proceeding  in  this  way  the  Maya 
wrote  numbers  in  very  compact  form.  The  highest  number  found 
in  codices  is  12,489,781.  . .  .  The  symbols  representing  this  number 
occupy  six  different  places,  one  above  the  other.  . . .  The  second 
numeral  notation  that  was  fully  developed  and  employed  by  the 
Maya  is  found  in  their  inscriptions.  It  employs  the  zero,  but  not 
the  principle  of  local  value.  Special  glyphs  are  employed  to  desig- 
nate the  different  units.  It  is  as  if  we  were  to  write  1203  as:  '1 
thousand,  2  hundreds,  0  tens,  3  ones.' " 

The  question  as  to  the  origin  of  the  arithmetical  notation 
that  we  call  "Hindu-Arabic"  has  received  a  new  and  unexpected 
contribution  from  Carra  de  Vaux  ("Sur  1'origine  des  chiffres," 
Scientia,  Vol.  XXI,  1917,  pp.  273-282.)  With  the  Arabs  these  figures  - 
are  called  hindi  and  the  usual  meaning  of  this  word  is  "Indian." 
Xow.  the  Arabian  historian  Masoudi,  writing  in  943  A.  D.,  said  that 
the  Hindu  numerals  were  discovered  by  a  congress  of  wise  men 
gathered  together  by  the  powerful  and  wise  king  Brahman  under 
whom  arts  and  sciences  flourished.  "People  who  are  even  slightly 
familiar  with  the  history  of  philosophy  will  recognize  this  at  once 
as  a  neo-Platonic  legend,"  and  a  mention  of  the  "Era  of  the 
Creation"  allowed  de  Vaux  to  conclude  that  this  legend  is  Persian, 
for  that  is  a  Persian  era.  Also  in  the  work  of  the  other  Arabic 
historian,  Albirouni,  we  have  a  remark  that  the  numerals  came  from 
India,  that  is  vague  and  contrasts  strongly  with  his  usual  exactness. 
This  seems  to  show  a  lack  of  definite  knowledge  on  Albirouni's  part. 

The  author  then  examines  the  word  hindi,  and  comes  to  the 


CRITICISMS  AND  DISCUSSIONS.  625 

conclusion  that  it  is  a  form  of  h'.ndasi  whose  root  is  the  Persian 
hid  and  which  means  metrical  or  arithmetical.  Thus  "signs  of 
h'nd"  means  "arithmitical  signs"  and  not  "signs,  of  India."  Con- 
sider this  example:  Apollonius  of  Perga,  who  was  not  an  Indian, 
was  said  to  be  el-hindi  in  some  Arabic  manuscripts,  thus  this  word 
must  evidently  be  translated  as  if  it  were  cl-h'ndasi,  the  geometer 
or  <:  ngineer.  It  is  to  be  noticed  that  in  Arabian  treatises  the  abacus 
is  called  taklit  which  is  a  Persian  name.  Thus  de  Vaux  concluded 
that  the  numerals  originated  in  the  Gre  k  world,  and  the  history 
of  their  slow  diffusion  is  easier  to  explain  if  we  admit  that  they 
are  a  neo-Platonic  "or  (so:t)  neo- Pythagorean"  invention,  for  the 
Pythagoreans  are  well  known  to  have  had  a  taste  for  secrecy.  From 
Greece  the  numerals  passed  to  Persia  and  the  Latin  world,  and  from 
Persia  to  India  and  afterward  to  Arabia. 

The  figures  themselves  were  not  formed  from  letters  of  the 
alphabet,  but  directly  by  means  of  very  simple  conventions.  These 
characters  are  due  to  the  neo-Platonists  and  were  known  in  the 
schools  of  Persia  before  they  were  in  Islam,  and  it  is  there  that  the 
Arabs  found  them.  From  Persia  also  they  passed  into  India. 


In  the  first  number  (January,  1917)  of  Vol.  XXIV  of  the  Amer- 
ican Mathematical  Monthly,  the  official  journal  of  the  Mathematical 
Association  of  America,  there  is  an  important  paper  by  Edward 
V.  Huntington  on  "The  Logical  Skeleton  of  Elementary  Dynam- 
ics" (pp.  1-16).  The  object  of  the  artic1e  is  to  outline  the  logical 
structure  of  elementary  dynamics.  Any  logically  d  veloped  science 
must  begin  with  undefined  concepts,  in  terms  of  which  all  the  other 
concepts  of  the  science  are  expressed ;  and  in  this  case  Huntington 
takes  them  to  be:  (1)  Space  and  time,  with  the  derived  concepts  of 
velocity  and  acceleration;  (2)  Forces,  "as  sugg  sted  by  the  tension 
and  compression  in  our  own  muscles"  (p.  1)  and  "as  measured 
by  a  spring  balance"  (p.  3)  ;  and  (3)  Inert  material  bodies,  on 
which  our  forces  act.  The  unproved  propositions,  from  which  all 
the  other  propositions  of  the  science  are  derivd,  are  only  four  in 
number.  The  first  is  (p.  4)  that  "a  free  particle,  when  acted  on 
by  a  force,  acquires  an  acceleration  in  the  direction  of  the  force; 
furthermore,  if  a  given  particle  is  acted  on  at  different  times  by  two 
forces  F  and  F',  and  if  a  and  a'  are  the  corresponding  accelerations, 
then  F/F'  =  a/a';  that  is,  the  accelerations  are  proportional  to  the 
forces."  This  principle  "is  best  regarded  as  a  scientific  hypothesis, 


626  THE  MONIST. 

the  truth  of  which  has  been  abundantly  verified  by  experiment." 
It  contains  the  answer  to  the  fundamental  question  of  dynamics: 
"If  a  force  gets  hold  of  a  free  particle,  and  proceeds  to  act  on  it, 
what  happens  to  the  particle?"  The  second  and  third  principles, 
which  cover  the  case  of  the  particle  being  acted  on  by  several  forces 
simultaneously,  are  the  principle  of  the  vector  addition  of  forces 
arid  the  principle  of  the  independence  of  two  perpendicular  forces 
(p.  5).  For  any  given  body  the  ratio  of  the  force  to  the  acceleration 
produced  is  constant,  and  the  value  of  this  ratio  "is  a  characteristic 
of  the  body,  which  may  be  called  its  inertia"  (p.  4).  "The  weight 
of  a  body  in  a  given  locality  with  respect  to  a  given  frame  of 
reference  is  best  defined  as  the  force  required  to  support  the  body 
at  rest  with  respect  to  that  frame  in  the  given  locality"  (p.  5).  We 
then  have  the  theorem  that  "if  W  is  the  weight  of  a  body  in  a  given 
locality  and  g  is  the  falling  acceleration  of  that  body  in  the  same 
locality,  then  the  ratio  W/g  is  independent  of  the  locality,  and  is  a 
correct  expression  for  the  inertia  of  the  body"  (p.  6).  The  proof 
for  the  case  of  fixed  axes  follows  immediately  from  the  first  two 
principles,  and  a  proof  for  the  case  of  moving  axes  "belongs  later 
in  the  course."  The  theorem  (p.  6)  that,  in  any  given  locality,  the 
falling  accelerations  of  all  bodies  are  equal,  "can  be  proved  from 
general  considerations ;  or,  if  preferred,  it  may  be  accepted  as  an 
empirical  fact."  The  words  "mass  of  three  pounds"  are  taken 
(p.  7)  as  meaning  the  same  thing  as  that  the  body  in  question  "has 
a  weight  of  3  Ibs."  in  the  standard  locality ;  the  weight  being  a 
multiple  of  the  unit  of  force  (Ib.  in  the  British  system).  The 
fourth  and  last  fundamental  principle  is  the  principle  of  action  and 
reaction  (p.  8)  :  "When  two  particles  are  in  contact  with  each  other, 
or  attract  or  repel  each  other  according  to  any  law  like  that  of 
gravitation  or  magnetism,  the  interaction  between  them  may  be 
represented  by  a  pair  of  twin  forces,  equal  in  magnitude  and  oppo- 
site in  direction — one  of  the  twins  acting  on  one  particle  and  one 
on  the  other,  along  their  joining  line."  The  definition  of  the 
"centroid  or  center  of  mass"  is  as  a  "weighted  average"  (p.  8), 
and  the  theorem  on  the  motion  of  the  center  of  mass  is  then  proved. 
It  is  emphasized  (p.  11)  that  the  difficulties  outside  the  four  funda- 
mental principles  are  of  a  mathematical  sort.  It  will  be  seen  that 
the  system  is  based  on  fundamental  units  of  force,  length,  and  time 
instead  of  on  units  of  mass,  length,  and  time,  and  the  author  shows 
by  tables  (pp.  15,  16)  the  higher  practical  value  of  the  system  of 


CRITICISMS  AND  DISCUSSIONS.  627 

derived  units  advocated  by  him.  This  is  "one  of  the  best  arguments 
in  favor  of  the  use  of  force  rather  than  mass  as  the  principal  unde- 
fined concept  of  dynamics.  The  only  reason  why  the  text-books 
so  insistently  base  their  derived  units  on  mass  instead  of  on  force 
is  apparently  that  a  standard  lump  of  metal  is  easier  to  preserve  in 
a  museum  than  a  standard  spring  balance.  But  this  is  no  argument 
for  the  logical  priority  of  mass  over  force.  As  a  matter  of  fact, 
the  fundamental  unit  of  force  is  as  easy  to  preserve  as  the  funda- 
mental unit  of  mass,  though  the  method  of  doing  so  does  not  consist 
in  simply  storing  away  a  spring  balance"  (p.  16).  The  name  of  the 
unit  of  force,  in  the  British  system,  is  "pound"  (lb.)  and  is  defined 
as  "the  force  required  to  support  a  carefully  preserved  lump  of 
metal,  called  the  'standard  pound  avoirdupois,'  in  vacuo,  in  the 
standard  locality"  (p.  14). 

The  reviewer  would  remark  that,  though  mass  under  the  name 
"inertia"  is  a  derived  unit  in  the  system  advocated  by  Huntington, 
we  have  to  use  the  unit  of  mass  as  a  practical  means  of  preserving 
the  unit  of  force.  It  is  quite  true  that  this  fact  is  no  argument  for 
the  logical  priority  of  mass:  it  is  merely  a  question  of  practical 
convenience.  But  in  either  of  the  two  systems  there  seem  to  be,  at 
first  sight,  three  fundamental  und  fined  units,  and  so,  from  a  logical 
point  of  view,  nothing  is  gained  by  replacing  mass  by  force  as  a 
fundamental  unit.  But  let  us  look  at  the  matter  more  closely.  As 
we  have  learned  from  the  work  of  Mach  (see,  e.  g.,  his  Mechanics, 
3d  edition,  Chicago  and  London,  1907,  p.  243),  "mass-ratio"  can 
be  defined  in  terms  of  the  mutual  accelerations  of  bodies,  and  so 
there  seems  to  be  a  logical  advantage  in  the  system  in  which  force 
is  not  regarded  as  fundamental,  but  is  defined.  Further,  even  in 
Huntington's  system,  "force"  can  be  defined  by  the  property  that  F/a 
is  constant,  and  then  his  system  and  Mach's  seem  to  be  identical. 
By  the  way,  the  forces  we  use  in  dynamics  are  not  all  "suggested 
by  the  tension  and  compression  in  our  own  muscles" :  the  attraction 
of  the  sun  is  not :  and  it  is  both  logically  objectionable  and  rather 
confusing  to  a  student  to  have  various  concepts  with  a  single  name. 

It  must  be  added  that  stress  is  (pp.  7-8)  rightly  laid  on  the 
difficulty  which  beginners  have  in  realizing  that,  when  a  particle 
describes  a  curve,  there  is  actually  an  acceleration  along  the  normal. 

There  is  an  interesting  review  of  Florian  Cajori's  William 
Oughtred  on  pp.  29-30  written  by  Louis  C.  Karpinski,  where  it  is 
stated  that  Cajori's  remark  that  Napier  was  the  first  to  use  a  decimal 


628  THE  MONIST. 

point   (1616  and  1617)   is  an  error:  it  was  first  used  by  Pitiscus 
in  1612. 

In  the  February  number  is  a  short  account  (pp.  54-55)  of 
Cajori's  presidential  address  to  the  annual  meeting  of  the  Mathemat- 
ical Association  of  America  in  New  York  City  at  the  end  of  1916 
entitled  "Discussions  of  Fluxions  from  Berkeley  to  Woodhouse." 
This  address  was  a  shortened  account  of  a  book  by  Cajori  which 
will  appear  before  very  long  in  the  "Open  Court  Classics  of  Science 
and  Philosophy."  At  a  meeting  of  the  Council  it  was  decided, 
among  other  things,  to  consider  the  question  of  possible  assistance 
for  the  Revue  semestriclle  and  the  Jahrbuch  iiber  die  Fortschritte 
der  Mathematik  in  view  of  the  difficulties  that  must  attend  publi- 
cation owing  to  the  war  (p.  64).  Very  much  the  same  discussion 
was  held  by  the  Chicago  Section  of  the  American  Mathematical 
Society  (p.  97).  David  Eugene  Smith,  in  "On  the  Origin  of  Certain 
Typical  Problems"  (pp.  65-71),  has  a  very  learned  article  on  the 
history  of  the  problems  of  (1)  filling  a  cistern  of  water,  (2)  the 
Josephspiel,  or  the  problem  of  the  Turks  and  Christians,  (3)  the 
testament  of  a  man  about  to  die,  dividing  his  estate,  (4)  the  problem 

of  pursuit. 

*  *       * 

Frank  Egleston  Robbins  (Amer.  Math.  Monthly,  March,  1917, 
Vol.  XXIV,  pp.  121-123)  gives  an  interesting  and  critical  review 
of  George  Johnson's  partial  translation  of  and  commentary  on  the 
Introduction  to  Arithmetic  of  Nicomachus,  in  his  dissertation  on 
The  Arithmetical  Philosophy  of  Nicomachus  of  Gerasa  (Lancaster, 
Pa.,  1916).  The  essay  of  Nicomachus  is  of  course  the  earliest  ex- 
tant attempt  at  a  systematization  of  the  Greek  science  of  theoret- 
ical, as  distinguished  from  practical,  arithmetic. 

*  *       * 

In  the  American  Mathematical  Monthly  for  May,  1917,  W.  H. 
Bussey  ("The  Origin  of  Mathematical  Induction,"  Vol.  XXIV,  pp. 
199-207)  points  out  that  Moritz  Cantor  is  mistaken  about  the  use 
of  complete  induction  both  in  his  Geschichte  der  Mathematik  (Vol. 
II,  2d  ed.,  1900,  p.  749)  and  his  note,  correcting  this  mistake, 
on  Maurolycns  in  the  Zeitschrift  filr  mathematischen  und  natur- 
wisscnschaftlichen  Unterricht  (Vol.  XXXIII,  1902,  p.  536).  In 
the  note  Cantor  said  that  he  had  found  that  Maurolycus  described 
and  used  the  method  in  his  Arithmetic  or  um  libri  duo  (Venice,  1575), 
and  that  Pascal  had  expressly  borrowed  the  method  from  Mauroly- 


CRITICISMS  AND  DISCUSSIONS.  629 

cus;  Bussey  shows  that  there  is  an  error,  in  a  minor  respect,  in 
Cantor's  references.  Bussey  quotes  a  number  of  Maurolycus's 
theorems.  In  his  sixth  proposition,  that  any  integer  (n)  added  to 
the  preceding  one  is  equal  to  the  "collateral  odd  number"  (the  nth 
odd  number,  2n-  1),  Maurolycus's  proof,  freely  translated  is:  "The 
integer  2  added  to  unity  makes  the  integer  3,  but  when  added  to  3 
it  makes  an  amount  greater  by  2  and  this ....  is  the  next  odd  integer, 
namely  5.  Again,  since  the  integer  3  added  to  2  makes  5,  which  is 
the  collateral  [third]  odd  integer,  when  it  is  added  to  4  the  result 
will  be  greater  by  2,  that  is.... it  will  be  the  next  odd  integer, 
which  is  7.  And  in  like  manner  to  infinity  as  the  proposition  states." 
On  this  proof  Bussey  remarks  (p.  201) :  "This  is  not  a  very  clear 
statement  of  a  proof  by  mathematical  induction  but  the  idea  is 
there."  The  eleventh  proposition,  that  every  triangular  number  added 
to  the  preceding  triangular  number  is  equal  to  the  collateral  square 
number,  or,  in  modern  notation,  n(n+  l)/2+  (n-  l)n/2  =  n2,  is  the 
one  which  Cantor  said,  in  the  above  note,  Pascal  got  from  Mauro- 
lycus  and  which  Matirolycus  proved  by  complete  induction.  But 
Cantor  is  mistaken  in  saying  that  this  theorem  is  proved  by  com- 
plete induction:  the  first  undoubted  case  of  a  proof  by  complete 
induction  is  the  fifteen  proposition,  that  the  sum  of  the  first  n  odd 
integers  is  equal  to  the  nth  square  number.  Maurolycus's  proof  is 
(p.  203)  :  "By  a  previous  proposition  the  first  square  number 
(unity)  added  to  the  following  odd  number  (3)  makes  the  following 
square  number  (4)  ;  and  this  second  square  number  (4)  added  to 
the  third  odd  number  (5)  makes  the  third  square  number  (9)  ; 
and  likewise  the  third  square  number  (9)  added  to  the  fourth  odd 
number  (7)  makes  the  fourth  square  number  (16)  ;  and  so  succes- 
sively to  infinity.  ..."  Pascal  mentioned  in  a  letter  to  Carcavi  the 
fact  that  he  borrowed  from  Maurolycus,  and  he  repeatedly  used  the 
method  of  complete  induction  in  connection  with  his  arithmetical 
triangle  and  its  applications.  Bussey  then  gives  two  interesting 
examples  of  Pascal's  use  of  the  method  of  complete  induction,  and 
finally  gives  some  other  and  more  recent  uses  of  it. 

*       *       * 

In  the  same  number  of  the  Monthly,  David  Eugene  Smith 
("Mathematical  Problems  in  Relation  to  the  History  of  Economics 
and  Commerce,"  pp.  221-223)  maintains  that  "a  very  good  history 
of  civilization  could  be  written  from  the  wide  range  of  problems 
of  mathematics."  In  the  subject  of  commercial  and  economic  his- 


630  THE  MONIST. 

tory,  for  example,  he  mentions  that  the  problems  in  the  manuscripts 
and  early  printed  books  on  arithmetic  in  the  fifteenth  century  tell 
us  that  Venice  was  then  the  center  of  the  silk  trade,  although  Bo- 
logna, Genoa,  and  Florence  were  then  prominent ;  the  problems  also 
tell  us  the  cost  of  the  luxuries  and  necessities  of  life;  the  rent  of 
houses ;  the  changes  in  commercial  customs  and  the  rise  in  standards 
of  business  integrity.  "Not  only  to  the  economist  and  the  student 
of  commerce  is  the  field  a  rich  one,  but  it  is  well  worth  the  study 
of  any  one  who  may  be  possessed  of  doubt  as  to  the  relation  of 
mathematics  to  the  daily  life  of  the  race.  Not  only  can  the  history 
of  the  problem  easily  be  made  the  history  of  commerce  and  econom- 
ics, but  the  history  of  mathematics  can  easily  be  made  the  history 
of  civilization." 


In  the  number  of  the  Bulletin  of  the  American  Mathematical 
Society  for  March,  1917  (Vol.  XXIII),  there  are  two  interesting 
papers  by  Edward  V.  Huntington  on  the  logical  postulates  for 
order.  In  "Complete  Existential  Theory  of  the  Postulates  for  Serial 
Order"  (pp.  276-280)  Huntington  establishes  the  "complete  inde- 
pendence"— in  the  sense  defined  by  E.  H.  Moore  of  Chicago  in  his 
Introduction  to  a  Form  of  General  Analysis  of  1910 — of  each  of 
three  different  sets  of  postulates  for  serial  order.  The  first  set  is 
new  and  very  convenient  for  many  purposes ;  the  second  set  dates 
back  to  Vailati  (1892)  ;  the  third  set  is  a  modification  of  the  second 
set  and  was  introduced  in  Huntington's  well-known  paper  on  "The 
Continuum  as  a  Type  of  Order"  in  the  Annals  of  Mathematics  for 
1905.  In  "Complete  Existential  Theory  of  the  Postulates  for  Well 
Ordered  Sets"  (pp.  280-282)  Huntington  gives  three  sets  of  in- 
dependent postulates  for  well-ordered  systems,  each  of  these  three 
sets,  being  "completely  independent"  in  the  above  sense.  R.  L. 
Borger  ("A  Theorem  in  the  Analysis  of  Real  Variables,"  pp.  287- 
290)  gives  a  theorem  on  two  real  functions  of  two  real  variables 
which  is  derived  from  a  theorem  in  Kowalewski's  Die  komplexen 
V  eranderlichen  and  ihre  Funktionen,  and  deduces  from  it  the  ex- 
ceedingly fundamental  and  important  theorem  that  if  any  function 
of  a  complex  variable  possesses  a  finite  derivative  at  each  point  of  a 
simply  connected  closed  region,  then  this  derivative  is  continuous, 
all  the  derivatives  of  the  function  exist,  and  the  function  may  be 
represented  by  a  power-series.  Mathematicians  who  are  acquainted 
with  the  nature  of  the  progress  brought  about  by  Goursat's  proof 


CRITICISMS  AND  DISCUSSIONS.  63! 

of  Cauchy's  theorem  will  at  once  see  how  important  this  note  is. 
J.  R.  Kline  ("Concerning  the  Complement  of  a  Countable  Infinity 
of  Point  Sets  of  a  Certain  Type,"  pp.  290-292)  proves  a  theorem 
which  is  a  general  case  of  the  theorem  proved  by  Hausdorff  in  his 
Grundziige  der  Mengenlchre  of  1914  that,  if  E  denotes  a  Euclidean 
space  of  two  or  more  dimensions  while  R  is  an  enumerable  set  of 
points  belonging  to  E,  then  E  -  R  is  a  connected  set.  Kline's  theorem 
was  proved  by  Robert  L.  Moore  (Trans.  Amer.  Math.  Soc.  for 
1916)  on  the  basis  of  a  system  of  axioms  proposed  by  him. 


The  number  of  the  Bulletin  of  the  American  Mathematical 
Society  for  May,  1917  (Vol.  XXIII,  No.  8),  contains  several  articles 
of  interest  to  those  who  cultivate  the  philosophical  and  historical 
aspects  of  mathematics.  Samuel  Beatty  ("The  Inversion  of  an 
Analytic  Function,"  pp.  347-353)  proves  the  existence  of  the  inverse 
of  an  analytic  function  when  the  conception  of  an  analytic  function 
which  is  due  to  Goursat  is  the  starting-point.  In  the  theory  of 
Weierstrass  this  proof  is  made  to  depend  on  the  representation  by  a 
series  of  powers  and  in  Cauchy's  theory  on  the  Jacobian  of  the  real 
and  imaginary  parts  of  the  function  with  reference  to  the  real  and 
imaginary  parts  of  the  variable.  It  is  well  known  that  Goursat 
showed  in  1900  how  the  fundamental  proposition  on  complex  inte- 
gration in  Cauchy's  theory  could  be  proved  merely  from  the  assump- 
tion that  the  function  in  question  has  a  finite  derivative  at  each  point 
of  a  simply  connected  domain,  without  any  assumption  of  the  con- 
tinuity of  this  derivative.  This  continuity  was  then  proved  as  a 
consequence  of  the  Cauchy-Goursat  theorem.  The  method  of 
Beatty 's  proof  makes  use  of  the  theory  of  sets  of  points.  Thomas 
S.  Fiske  ("Emory  McClintock,"  pp.  353-357)  gives  a  biography  of 
Emory  McClintock  (1840-1916).  McClintock 's  first  paper  on  pure 
mathematics  entitled  "An  Essay  on  the  Calculus  of  Enlargement," 
in  the  American  Journal  of  Mathematics  for  1879  "was  an  effort 
to  present  the  theory  of  finite  differences  and  the  differential  cal- 
culus from  a  unified  point  of  view.  The  paper  may  be  regarded  as 
a  precursor  of  recent  attempts  to  consider  difference  equations  as 
differential  equations  of  infinite  order.  His  other  more  important 
papers  were  a  series  of  researches  on  solvable  quintic  equations 
published  in  the  American  Journal  of  Mathematics  [for  1884,  1885 
and  1898]  and  a  paper  on  the  theory  of  numbers  ['On  the  Nature 
and  Use  of  the  Functions  Employed  in  the  Recognition  of  Quad- 


632  THE   MONIST. 

ratic  Residues']  published  in  the  third  volume  [1902]  of"  the  Trans- 
actions of  the  American  Mathematical  Society  (p.  355).  "When 
one  considers  that  McClintock  made  no  use  of  the  powerful  labor- 
saving  machinery  which  has  revolutionized  modern  analysis,  the 
results  obtained  by  him  in  his  researches  on  quintic  equations,  as 
well  as  some  of  his  other  achievements,  appear  to  indicate  a  truly 
wonderful  power  of  manipulation  and  clearness  of  vision"  (p.  356). 
A  list  of  McClintock's  mathematical  publications  is  given. 

J.  H.  Weaver  ("On  Foci  of  Conies,"  pp.  357-365)  gives  (1) 
a  short  historical  sketch  of  the  development  of  the  properties  of 
conies  connected  with  the  foci,  and  (2)  some  of  the  theorems  from 
Pappus  which  have  a  bearing  on  foci  and  tangents.  According  to 
Zeuthen  (Geschichte  der  Mathematlk  im  Alterthum  und  Mittelalter, 
Copenhagen,  1896,  p.  211)  it  seems  that  the  focus  for  the  parabola 
may  have  b:en  known  to  Euclid.  However,  we  have  no  mention 
of  such  points  or  of  any  of  their  properties  until  the  work  of  Apol- 
lonius  on  conies  (Book  III,  Probs.  45-52),  but  Apollonius  did  not 
use  or  mention  in  any  way  a  focus  for  the  parabola.  Pappus  gave 
the  first  recorded  use  and  proofs  of  the  focus-directrix  definition 
of  conies.  Johann  Kepler  named  the  points  in  question  in  a  work 
of  1604,  and  part  of  the  short  account  of  the  conic  sections  that  he 
gave  (Opera  Omnla,  ed.  Frisch,  Frankfort,  1859,  Vol.  II,  p.  185) 
is  freely  translated  by  Weaver  (p.  359)  as  follows:  "There  are 
among  these  curves  certain  points  of  especial  consideration,  which 
have  a  certain  definition  but  no  name,  unless  they  usurp  for  name 
the  definition  of  some  property.  For  if  from  these  points  lines  are 
drawn  to  the  points  of  contact  of  tangents  to  the  section,  th°s?  lines 
make  equal  angles  with  the  tangents. . .  .We,  because  of  the  prop- 
erties of  light  and  the  eye,  from  the  viewpoint  of  mechanics  shall 
call  these  points  foci.  We  might  have  called  them  centers,  because 
they  are  on  the  axis  of  the  s'ction,  if  authors,  in  the  hyperbola  and 
ellipse,  were  not  accustomed  to  calling  another  point  the  center.  In 
the  circle  there  is  one  focus,  the  center.  In  the  ellipse  there  are 
two  foci  equally  distant  from  the  center,  and  more  removed  in  the 
more  acute.  In  the  parabola,  one  focus  is  within  the  section  and 
the  other  may  be  considered  either  within  or  without  the  section 
and  removed  to  an  infinite  distance  from  the  first  focus,  so  that  if 
a  line  drawn  from  this  caecus  [blind]  focus  to  a  point  of  the  section 
will  be  parallel  to  the  axis.  In  the  hyperbola,  the  external  focus 
becomes  nearer  the  internal  focus  as  the  hyperbola  becomes  more 


CRITICISMS  AND  DISCUSSIONS.  633 

obtuse."  The  method  of  the  work  of  Kepler  was  developed  and 
added  to  by  Desargues,  and  important  work  on  foci  was  done  by 
Maclaurin,  Poncelet,  Pliicker,  and  many  others. 

Finally,  there  is  a  short  and  extremely  interesting  paper  by 
Jekuthial  Ginsburg  ("New  Light  on  Our  Numerals,"  with  an  in- 
troductory note  by  David  Eugene  Smith,  pp.  366-369).  That  our 
common  numerals  are  of  Hindu  origin  seems  to  the  author  to  be  a 
well-established  fact,  and  that  Europe  received  them  from  the  Arabs 
seems  equally  certain,  but  how  and  when  these  numerals  reached  the 
Arabs  is  a  question  that  has  never  been  satisfactorily  answered. 
The  article  calls  attention  to  a  paper  by  the  French  orientalist  F. 
Nau  in  the  Journal  asiatique  for  1910  (Series  X,  Vol.  XVI)  showing 
that  the  Hindu  numerals  were  known  to  and  appreciated  by  the 
Syrian  writer  Severus  Sebokht  who  lived  in  the  second  half  of  the 
seventh  century ;  that  is,  about  one  hundred  years  before  the  date 
of  the  first  definite  trace  that  we  have  hitherto  had  of  the  introduc- 
tion of  the  system  into  Bagdad.  Sebokht  says,  after  asserting  that 
the  Greeks,  in  astronomy,  were  merely  the  pupils  of  the  Babylonians : 
"I  will  omit  all  discussion  of  the  science  of  the  Hindus,  a  people 
not  the  same  as  the  Syrians ;  their  subtle  discoveries  in  this  science 
of  astronomy,  discoveries  that  are  more  ingenious  than  those  of 
the  Greeks  and  the  Babylonians ;  their  valuable  methods  of  calcula- 
tion ;  and  their  computing  that  surpasses  description.  I  wish  only  to 
say  that  this  computation  is  done  by  means  of  nine  signs.  If  those 
who  belr.ve,  because  they  speak  Greek,  that  they  have  reached  the 
limits  of  science  should  know  these  things  they  would  be  convinced 
that  there  are  also  others  who  know  something"  (p.  363).  On  this 
fragment  Ginsburg  remarks  (pp.  363-369)  that  it  "clearly  shows 
that  not  only  did  Sebokht  know  something  of  the  numerals,  but 
that  he  understood  their  full  significance,  and  may  even  have  known 
the  zero  as  Rabbi  ben  Esra  did,  in  spite  of  the  fact  that  he,  too, 
speaks  of  nine  numerals."  However,  Smith  (pp.  366-367)  remarks 
that  the  artich  "shows  that  the  zero  was  probably  not  in  the  system 
as  then  mentioned,  showing  at  least  that  its  value  was  not  generally 
comprehended  in  the  seventh  century  and  possibly  confirming  the 
impression  that  the  symbol  had  not  yet  been  invented."  With  re- 
gard to  the  question  as  to  how  Sebokht  could  have  obtained  in- 
formation about  the  Hindu  numerals,  Ginsburg  remarks  (p.  369) 
that  the  city  where  Severus  lived,  in  the  northeast  part  of  Meso- 
potamia, "was  situated  in  a  rich  and  fruitful  country,  was  long  the 


634  THE   MONIST. 

center  of  a  very  extensive  trade,  and  was  the  great  northern  em- 
porium for  the  merchandise  of  the  east  and  west ;"  and  "the  ex- 
change of  goods  is  always  accompanied  by  the  exchange  of  ideas." 
Further,  the  weight  of  the  evidence  is  (p.  369)  in  favor  of  Sebokht's 
work  being  at  least  one  of  the  agencies  by  means  of  which  the 
knowledge  of  the  numerals  was  transmitted  to  the  Arabs. 


Raphael  Demos  ("A  Discussion  of  a  Certain  Type  of  Negative 
Proposition,"  Mind,  Vol.  XXVI,  1917,  pp.  188-196)  applies  to  par- 
ticular negative  propositions  the  treatment  which  Bertrand  Russell 
(cf.  Russell  and  Whitehead's  Principia  Mathematica,  Vol.  I,  Cam- 
bridge, 1910)  has  applied  to  "descriptive  phrases"  or  "incomplete 
symbols."  Russell  "found  himself  confronted  with  the  fact  that  to 
accept  descriptive  phrases  as  significant  in  their  given  form  would 
be  to  people  the  world  of  things  with  the  apparent  objects  of  such 
self -contradictory  and  fantastic  descriptions  as  'round-square,'  'cen- 
taur/ etc." ;  and  Demos  "was  faced  with  the  fact  that  to  accept 
negative  propositions  at  their  face  value  would  be  to  people  the 
world  of  objects  with  negative  facts,  a  type  of  objects  which  ex- 
perience fails  to  disclose."  Demos,  somewhat  like  Russell,  by  view- 
ing the  negative  proposition  as  an  incomplete  symbol,  was  led  to 
declare  it  meaningless  in  its  apparent  form,  and  its  apparent  object 
— the  negative  fact — to  be  nothing.  In  this  article  he  stated,  first, 
that  a  particular  simple  negative  proposition  is  an  objective  entity 
whose  peculiarity  as  negative  is  not  dependent  upon  the  mind's 
attitude  toward  it.  He  then  argued  that  the  negative  proposition 
cannot  be  construed  in  the  form  which  it  apparently  possesses,  inas- 
much as  such  construction  would  make  it  formally  different  from 
positive  propositions  and  would  endow  it  with  purely  negative  ob- 
jects, which  are,  it  seems,  nowhere  to  be  found  in  experience.  He 
concluded  that  some  special  interpretation  must  be  given  to  the 
negative  proposition,  and  showed  that  its  negative  element  is  a 
modification,  not  of  any  distinct  constituent  (such  as  the  predicate) 
in  the  proposition,  but  of  the  whole  content  of  it.  Thus  any  nega- 
tive proposition  is  a  modification,  in  terms  of  "not,"  of  the  rest 
of  its  content,  and — since  the  latter  is  positive — a  modification  of 
some  particular  positive  proposition.  He  stated  the  meaning  of 
"not"  to  be  "opposite" — a  relational  qualification  in  terms  of  the 
familiar  relation  of  opposition  or  contrariety  among  positive  propo- 
sitions— and  hence  the  meaning  of  the  whole  proposition  "not-/>"  to 


CRITICISMS  AND  DISCUSSIONS.  635 

* 

be  "opposite  of  p."  He  argued  that,  so  stated,  a  negative  proposi- 
tion is  an  ambiguous  description  of  some  positive  proposition,  and 
that,  completely  stated,  it  is  of  the  form  "an  opposite  of  p  is  true," 
or  "some  q  is  true  which  is  an  opposite  of  />."  Thus  he  defined  a 
particular  simple  negative  proposition  as  an  ambiguous  description 
of  some  true  positive  proposition  in  terms  of  the  latter's  relation 
of  opposition  to  a  certain  other  positive  proposition,  such  that,  in 
terms  of  the  former,  reference  is  achieved  to  the  latter.  Lastly,  he 
explained  that  negative  knowledge  is  knowledge  of  a  true  positive 
proposition  by  description  in  terms  of  its  opposition  to  some  other 
proposition,  and  hence  must  be  characterized  as  positive  in  reference 
but  not  in  content,  inasmuch  as  the  proposition  referred  to  is  not  a 
constituent  of  the  complex  of  assertion  or  knowledge.  "Substan- 
tially the  above  definition  of  simple  negative  propositions  applies 
to  double  and  'w-ple'  negatives  as  well ;  the  latter,  too,  are  descrip- 
tions of  positive  propositions  which  are  true  in  terms  of  what  they 
oppose.  There  is  this  difference,  however,  that  whereas  simple  nega- 
tives are  functions  of  a  positive  content,  double  and  other  negatives 
are  functions  of  a  negative  content,  such  that  any  negative  propo- 
sition in  the  nth  power  is  a  function  of  a  content  which  is  negative 
in  the  («— l)th  power."  4> 


C.  E.  Hooper  publishes  "The  Meaning  of  the  Universe"  (Mind, 
April,  1917),  the  first  instalment  of  an  article  of  massive  appearance. 
The  definition  is  as  follows :  the  Universe  means  the  totality  of  real 
thought-objects  (or  object-matters)  considered  under  four  related 
aspects:  (1)  space,  (2)  time,  (3)  the  variety  in  unity  of  natural 
characters,  i.  e.,  real  thought-objects  as  particulars  having  natures 
of  their  own,  but  natures  agreeing  in  various  specific  and  generic 
respects  with  the  natures  of  other  particulars,  (4)  unity  in  variety 
of  natural  causation.  Time  and  space  are  both  objective.  Mr. 
Hooper  goes  on  to  define  thought-object,  reality  and  aspect.  A 
thought-object  is  apparently  an  intended  object,  whether  or  not  a 
reality  corresponds  to  the  intention  (e.  g..  Kant's  noumenoti).  Real- 
ity is  contrasted  not  with  appearance  but  with  "mental  figment," 
and  includes  subsistent  as  well  as  existent  objects.  It  is  difficult  to 
tell  how  far  the  term  "thought-object"  has  an  idealistic  bias  in  Mr. 
Hooper's  mind,  but  reality,  at  all  events,  seems  to  be  merely  a  sum 
or  system  of  objects  which  are  severally  real.  The  universe  is  thus 
a  real  thought  which  contains  all  other  real  thought-objects  in  their 


636  THE  MO1STIST. 

manifold  relations.  Symbolic  entities  (ideas,  signs)  are  compre- 
hended, but  whether  per  se  or  only  as  reflected  upon  (made  objects 
of  thought)  is  not  stated.  It  would  seem  that  imaginary  or  incon- 
ceivable thought-objects,  such  as  Meinong's  pets,  the  golden  moun- 
tain and  the  round  square,  are  to  have  no  place  in  the  universe,  but 
are  discarded  as  "figments."  While  the  universe  contains  finite 
thought-objects  and  symbols,  it  does  so  only  in  fact,  not  in  nature. 
Of  the  four  modes  or  aspects,  space  and  time  may  be  classed  to- 
gether as  coincidentals,  while  the  systems  of  natural  characters  and 
natural  causation  may  be  termed  co-essentials.  On  the  other  hand 
space  and  nature  may  be  classed  together  as  static,  time  and  causa- 
tion as  dynamic.  We  find  some  difficulty  in  understanding  how 
Mr.  Hooper  accounts  for  the  universe's  being  known  at  all.  "It 
cannot,  like  a  finite  object,  be  actually  related  to  some  fellow  object. 
It  is  as  related  to  the  mind  or  system  of  subjective  ideas  that  we 
know  all  that  is  possible  to  know  about  it."  But  the  "mind"  if 
genuinely  symbolic  is  a  thought-object;  and  if  the  universe  cannot 
be  related  to  a  finite  object  which  is  part  of  itself  we  do  not  know 
how  it  can  be  related  to  the  mind.  But  criticism  of  so  substantial 
an  article  should  be  deferred  until  its  completion  in  succeeding 
issues.  17 

*       *       * 

Agnes  Cuming  ("Lotze,  Bradley,  and  Bosanquet,"  Mind,  April, 
1917)  declares  Lotze's  logic  to  be  a  partial  revolt  against  the  in- 
tellectualism  of  Hegel.  Our  intelligent  experience,  according  to 
Lotze,  is  only  a  small  part  of  the  real  world  and  thought  is  only  a 
small  part  of  our  intelligent  experience.  Thought  is  a  tool,  a  sub- 
stitute for  adequate  perceptive  intuition.  Bradley 's  and  Bosanquet's 
logic  are  similar  in  so  far  as  each  is  influenced  by  Lotze.  They  hold 
an  almost  identical  definition  of  "idea,"  and  agree  in  their  theories 
of  judgment.  Bradley  however  arrives  at  reality  ontologically  and 
Bosanquet  epistemologically.  "Knowledge  for  Bosanquet  is  the 
system  of  reality  progressively  demonstrated  before  our  eyes.... 
In  this  emphasis  on  system  as  the  postulate  of  knowledge.  . .  .Bo- 
sanquet is  in  advance  of  Bradley."  Lotze  insists  on  feeling  as  a 
criterion,  and  is  thus  very  far  from  Bosanquet  with  his  conception 
of  system,  but  he  admits  the  essential  of  Bosanquet's  position,  which 
is  the  inadequacy  of  feeling.  Lotze  is  a  dualist :  he  divides  sharply 
the  feeling  which  supplies  the  material  from  thought,  exercising  a 
formal  activity  upon  it.  In  Bradley  the  dualism  becomes  a  gloomy 


CRITICISMS  AND  DISCUSSIONS.  637 

scepticism  ;  thought  and  its  object  are  forever  sundered.  But  Bo- 
sanquet  bridges  the  gulf.  In  both  Bosanquet  and  Bradley  the  sepa- 
ration of  thought  and  reality  is  inherited  from  Lotze  with  his  idea 
of  the  "scaffold"  of  thought.  The  only  possible  criterion  of  knowl- 
edge is  immanent  —  a  criticism  of  a  lower  from  a  higher  point  of 
view.  Miss  Cuming  considers  that  Bosanquet  has  improved  upon 
both  Lotze  and  Bradley  ;  the  direction  which  she  believes  to  be 
progress  seems  to  be  almost  a  return  to  more  orthodox  H.gelianism. 


In  the  same  number  B.  M.  Laing  ("Schopenhauer  and  Individ- 
uality") considers  that  Schopenhauer  fails  to  appreciate  the  meta- 
physical claims  of  individuality.  He  dethrones  reason,  making  it  a 
mere  temporary  organ  of  the  will.  He  interprets  Kant  in  such  a 
way  as  to  make  Kant  assert  that  the  mind  creates  the  world  of 
things,  instead  of  merely  conditioning  it.  This  perversion  of  the 
Kantian  doctrine  leads  Schopenhauer  to  hold  (in  contrast  to  Kant) 
that  the  world  of  space  and  time  is  an  illusion.  Hence  he  is  unable 
to  conserve  individuality,  and  tends  to  confuse  individuality  with 
(temporal  and  spatial)  individnation.  Schopenhauer's  monism  is  a 
mere  prejudice  against  multiplicity,  and  his  will  a  purs  abstraction. 
Furthermore,  he  confuses  the  will  with  bodily  wants  and  cravings. 
Schopenhauer  exposes  himself  on  every  side  to  such  destructive 
criticism:  but  while  Mr.  Laing  seizes  upon  some  of  his  weakest 
points  in  his  interpretation  of  Kant,  the  view  of  individuality  which 
Schopenhauer  represents,  and  which  is  more  abiding  then  Schopen- 
hauer, cannot  be  said  to  be  demolished.  ij 

*       *       * 

Scientia  for  February,  1917,  opens  with  an  article  by  Gino 
Loria  on  the  history  of  imaginary  numbers.  He  takes  as  his  text 
Kronecker's  aphorism  "Die  ganzen  Zahlen  hat  der  Hebe  Gott  ge- 
macht,  alles  andere  ist  Menschenwerk."  There  is  no  square  root  of 
a  negative  quantity,  said  the  Viga  Ganita,  for  it  is  not  a  square,  and 
the  mysterious  quantities  remained  an  enigma  and  defied  concrete 
interpretation  until  a  memoir  appeared  about  the  end  of  the  eight- 
eenth century,  written  by  an  unknown  Danish  land-surveyor,  Caspar 
Wessel  by  name.  It  seems  to  have  suffered  a  fate  like  that  of  Swin- 
burne's Queen  Rosamund,  of  which  not  a  copy  was  asked  for  or 
sold.  Mendel's  discovery  remained  unheeded  for  forty  years,  but 
Wessel's  was  not  unearthed  until  a  century  after  his  death.  But  in 


638  THE  MONIST. 

this  summary  is  no  more  room  for  comment  on  Loria's  charming 
paper.  P.  Zeeman  writes  upon  the  hypothesis  of  the  immovable 
ether,  describing  the  experiments  of  Fizeau,  Michelsen  and  Morley, 
Eichenwald,  and  himself,  and  concluding  that  it  seems  impossible 
by  any  imaginable  means  to  measure  absolute  velocities.  He  points 
the  lesson  that  the  most  important  scientific  principles  are  the  results 
of  boldly  generalized  experiments.  J.  R.  Carracido  discusses  the 
foundations  of  biochemistry,  and  the  lines  research  must  follow  in 
the  pursuit  of  organic  synthesis.  He  does  not  see  why  success 
should  not  be  reaped  before  the  end  of  the  century,  and  success  it 
will  be,  even  if  limited  to  the  synthesis  of  the  most  rudimentary 
form  of  living  matter. 

The  number  for  March,  1917,  contains  an  admirable  article  by 
Gaston  Milhaud,  in  which  he  attacks  the  very  difficult  problem  as 
to  the  extent  to  which  Descartes  was  influenced  by  Bacon.  M. 
Cantone  discusses  the  present  trend  of  physical  research  in  a  paper 
surveying  the  work  of  those  whose  discoveries  have  in  thirty  or 
forty  years  revolutionized  our  outlook  on  the  world  of  matter. 
Etienne  Rabaud  writes  on  the  life  and  death  of  species.  An  analysis 
of  the  current  doctrine  of  "means  of  defense"  prepares  for  the 
question  as  to  how  it  is  that  species  persist  in  spite  of  the  daily 
hecatombs  of  individuals.  y 


In  Scicntia  for  April,  1917,  we  have,  from  the  pen  of  Francisco 
Iniguez,  of  the  Observatory  of  Madrid,  a  slight  but  interesting 
sketch  of  what  we  know  about  stellar  spectra,  their  classification, 
and  the  light  they  throw  on  the  subject  of  the  evolution  of  the  stars. 
We  are  warned  of  the  limits  we  must  set  to  our  inferences  in  con- 
sidering the  nebulae,  for  what  we  know  as  yet  of  these  celestial 
bodies  does  not  justify  our  indulging  in  theories  on  the  subject.  The 
author  then  indicates  how  we  may  infer  the  existence  of  dark  stars, 
how  their  evolution  still  continues,  and  points  out  their  connection 
with  meteorites  and  cosmic  dust.  Etienne  Rabaud  brings  to  a  close 
his  paper  on  the  life  and  death  of  species,  of  which  this  second 
instalment  deals  with  the  conditions  of  the  persistence  and  of  the 
disappearance  of  species.  He  finds  the  affinity  of  organisms  to  be 
the  crux  of  the  problem.  This  leads  to  the  consideration  of  the 
conditions  of  attraction  and  repulsion,  and  to  a  short  discussion  of 
parasitism  and  symbiosis,  with  the  cosmic  influences  which,  often 
of  great  complexity,  determine  the  life  of  a  species.  The  whole 


BOOK  REVIEWS  AND  NOTES.  639 

forms  a  graphic  picture  of  the  variations  of  the  relative  proportion 
of  individuals  and  species.  The  slightest  change  in  the  conditions 
of  normal  life  may  lead  to  the  disappearance  of  the  last  member  of 
a  species,  or  on  the  contrary  the  species  may  thrive  and  continue  to 
thrive.  Species  persist,  they  increase  immeasurably  in  numbers,  or 
their  numbers  fall  off,  and  they  disappear,  subject  but  to  the  inter- 
vention of  two  sets  of  influences,  affinity  and  the  circumstances  that 
determine  their  displacements  in  space.  "These  modifications,  no 
doubt,  sometimes  involve  other  important  modifications  in  the  con- 
ditions of  life ;  variations  may  ensue  which  find  their  repercussion 
in  the  aggregate  of  the  interaction.  Thus,  linked  to  one  another 
and  to  the  world  from  which  they  come,  the  life,  the  transforma- 
tions, and  the  death  of  organisms  are  functions  of  their  interde- 
pendence." y 


BOOK  REVIEWS  AND  NOTES. 

DIDEROT'S  EARLY  PHILOSOPHICAL  WORKS.  Translated  and  edited  by  Margaret 
Jourdain.  "The  Open  Court  Classics  of  Science  and  Philosophy,"  No.  4. 
Chicago  and  London :  The  Open  Court  Publishing  Co.,  1916.  Pp.  vi,  246. 
Price  $1.25  or  4s.  6d.  net. 

Among  the  documents  of  the  renaissance  of  the  eighteenth  century,  none 
are  of  more  interest  than  the  early  informal  contributions  to  ethics  and  philos- 
ophy of  Diderot,  written  with  much  of  the  incoherence  of  the  epistolary  form. 
They  are,  as  he  claims  again  and  again,  Letters;  and  they  are  letters  written  in 
a  hurry.  The  Philosophic  Thoughts,  which  is  the  only  one  of  Diderot's  works 
in  this  selection  not  in  the  epistolary  form,  is  said  to  have  been  thrown  to- 
gether between  Good  Friday  and  Easter  Monday  of  1746.  Yet  they  are  not 
philosophic  journalism,  no  mechanical  transmission  of  the  current  philosophical 
coin  of  the  day.  It  is  for  their  originality  of  outlook  that  they  have  been 
closely  studied  in  Germany,  while  in  England  there  is  Lord  Morley's  study  of 
Diderot  in  relation  to  the  movement  centered  in  the  Encyclopedic,  Diderot  and 
the  Encyclopedists. 

This  selection  includes  the  Philosophic  Thoughts,  a  breviary  of  eighteenth- 
century  scepticism,  a  copy  of  which  was  found  in  the  possession  of  the  un- 
fortunate La  Barre,  and  in  which  Diderot  appears  as  a  Deist,  to  whom  the 
argument  from  design  (Thought  XX,  pp.  56-58)  is  still  of  weight:  "I  am 
greatly  deceived  (he  writes)  if  this  proof  is  not  well  worth  the  best  that  has 
ever  issued  from  the  schools."  That  very  argument  is  very  differently  treated 
in  the  Letter  on  the  Blind  (p.  109)  by  Diderot's  mouthpiece,  the  blind  mathe- 
matician, Nicholas  Sattnderson,  who  conjectures  a  world  in  its  early  stages 
"in  a  state  of  ferment,"  without  any  vestiges  of  that  "intelligent  Being  whose 
wisdom  fills  you  with  such  wonder  and  admiration  here....  What  is  our 
world,  but  a  complex,  subject  to  cycles  of  change,  all  of  which  show  a  continual 


640  THE   MONISt. 

tendency  to  destruction ;  a  rapid  succession  of  beings  that  appear  one  by  one, 
flourish  and  disappear;  a  merely  transitory  symmetry  and  momentary  appear- 
ance of  order?" 

In  the  brilliant  passages  in  which  Diderot  sketches  the  probability  of  evo- 
lution he  appears  as  a  forerunner  of  thinkers  such  as  Erasmus  Darwin  in 
England  and  Lamarck  in  France.  Transformism  only  needed  the  partial 
scientific  confirmation  it  received  from  Lamarck  and  Geoffrey  St.  Hilaire  in 
the  early  decades  of  the  nineteenth  century,  "to  pass  from  the  realm  of  sys- 
tematic philosophy  into  that  of  scientific  controversy." 

The  Letter  on  the  Deaf  and  Dumb,  a  criticism  addressed  to  the  Abbe 
Batteaux,  author  of  the  Fine  Arts  Reduced  to  a  Single  Principle,  has  its  in- 
terest as  a  forerunner  of  Lessing's  Laokoon,  in  esthetics.  It  also  contains  the 
idea  of  a  muet  de  convention  (theoretical  mute),  which  is  closely  paralleled  by 
Condillac's  Statue  in  the  Treatise  on  the  Sensations,  published  three  years  after 
Diderot's  Letter.  Condillac's  treatment  of  the  idea,  however,  was  far  more 
systematic  and  detailed  than  Diderot's,  and  he  did  not  by  his  own  account  owe 
the  suggestion  of  his  statue  to  Diderot. 

Diderot,  the  most  German  of  French  authors,  as  far  as  his  style  is  con- 
cerned, bears  translation  well.  He  has  been  neglected  by  translators,  however, 
until  this  edition,  which  includes  all  that  is  of  permanent  value  in  his  early 
works  of  1751,  the  date  of  the  Letter  on  the  Deaf  and  Dumb,  excluding  the 
relatively  uninteresting  Sceptic's  Walk.  ft 


THE  NEW  PHILOSOPHY  OF  HENRI  BERGSON.  By  Edouard  Le  Roy.  Translated 
from  the  French  by  Vincent  Benson,  M.A.  New  York:  Holt.  Pp.  235. 
Price  $1.25  net;  by  mail  $1.35. 

This  interpreter  of  Bergson's  philosophy  is  also  the  author  of  the  article 
"What  is  a  Dogma?"  in  the  body  of  this  issue  of  The  Monist.  He  is  particu- 
larly fitted  for  the  present  task  because  though  not  a  pupil  of  Bergson's  he 
had  followed  much  the  same  trains  of  thought  quite  independently  so  that 
when  he  became  acquainted  with  Bergson  he  recognized  in  his  work,  as  he 
himself  says,  "the  striking  realization  of  a  presentiment  and  a  desire."  That 
M.  Le  Roy  has  comprehensively  grasped  Bergson's  spirit  and  conclusions  so 
that  the  present  volume  furnishes  a  valuable  prolegomenon  to  the  study  of  the 
famous  Frenchman^  thought  is  attested  by  the  following  lines  in  the  Preface 
in  which  Bergson  himself  has  set  the  seal  of  his  approval  on  the  task.  M. 
Bergson  wrote  to  M.  Le  Roy:  "Underneath  and  beyond  the  method  you  have 
caught  the  intention  and  the  spirit....  Your  study  could  not  be  more  con- 
scientious or  true  to  the  original.  As  it  advances,  condensation  increases  in 
a  marked  degree :  the  reader  becomes  aware  that  the  explanation  is  undergoing 
a  progressive  involution  similar  to  the  involution  by  which  we  determine  the 
reality  of  Time.  To  produce  this  feeling,  much  more  has  been  necessary  than 
a  close  study  of  my  works :  it  has  required  deep  sympathy  of  thought,  the 
power,  in  fact,  of  rethinking  the  subject  in  a  personal  and  original  manner. 
Nowhere  is  this  sympathy  more  in  evidence  than  in  your  concluding  pages, 
where  in  a  few  words  you  point  out  the  possibilities  of  further  developments 
of  the  doctrine.  In  this  direction  I  should  myself  say  exactly  what  you  have 
said."  f 


B 

1 

M7 

v.27 


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